Floating cage aquaculture production in Indonesia: Assessment of opportunities and challenges in Lake Maninjau

Junaidi; Hafrijal Syandri; Azrita; Abdullah Munzir; Junaidi; Hafrijal Syandri; Azrita; Abdullah Munzir

doi:10.3934/environsci.2022001

AIMS Environmental Science

2022, Volume 9, Issue 1: 1-15. doi: 10.3934/environsci.2022001

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Research article Special Issues

Floating cage aquaculture production in Indonesia: Assessment of opportunities and challenges in Lake Maninjau

1.
Department of Fisheries Resources Utilization, Faculty of Fisheries and Marine Sciences, Universitas Bung Hatta Padang, West Sumatera, 25133
2.
Department of Aquaculture, Faculty of Fisheries and Marine Science, Bung Hatta University, Padang, Indonesia
3.
Department of Biology Education, Faculty of Education, Bung Hatta University, Padang, Indonesia

Received: 08 October 2021 Revised: 18 December 2021 Accepted: 22 December 2021 Published: 18 January 2022

Aquaculture in floating cages in Lake Maninjau has recorded significant growth, even as the largest contributor to total annual aquacultural production in West Sumatra Province. In this study, we assessed the output of floating net cages in Lake Maninjau, Indonesia. We analyzed the characteristics of fish farming, fish fry, feed supply, and production, and the challenges and opportunities for increasing aquacultural production in the future. We used purposive sampling in this study with an interview questionnaire to obtain information from 80 fish-cultivating households in Lake Maninjau. We then used descriptive statistical methods of data analysis. The results showed that in 2018, there were 17596 floating net cages. The majority (n = 33, 41.25%) of fish farmers have 20 to 40 floating net cages per household, and 67.5% (n = 54) are used for tilapia cultivation. We recorded that 77.5% (n = 62) of fingerlings were sourced from private hatcheries. Six companies supply commercial feed pellets in an amount of 2000 tons per month for aquaculture activities. Japfa Comfeed Indonesia Ltd. provides 35% of the feed. The fish species cultivated were Nile tilapia, common carp, giant gourami, Clarias catfish, and pangasius catfish, with gross yields (kg/m³/cycle) of 12, 11.5, 10.4, 7.88, and 8.89, respectively. Fish farmers face challenging conditions: poor water quality, mass mortality of tilapia, high fish feed prices and low fish sale prices, and noncash payments. We recommend ensuring the development of floating net cages in Lake Maninjau for a more sustainable future. Therefore, it is necessary to operate as many as 6000 nets to meet guidelines for carrying capacity and cultivation based on the Regional Regulation of Agam Regency Number 5 of 2014 concerning the management of Maninjau Lake, which is accessible proportionally by eight villages. Giant gourami is prioritized for cultivation because it is resistant to poor water quality and high market prices.

Keywords:

Citation: Junaidi, Hafrijal Syandri, Azrita, Abdullah Munzir. Floating cage aquaculture production in Indonesia: Assessment of opportunities and challenges in Lake Maninjau[J]. AIMS Environmental Science, 2022, 9(1): 1-15. doi: 10.3934/environsci.2022001

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Abstract

1. Introduction and preliminaries

The Gröbner basis theory for commutative algebras was introduced by Buchberger ^[2] which provided a solution to the reduction problem for commutative algebras. In ^[3], Bergman generalized this theory to the associative algebras by proving the diamond lemma. On the other hand, the parallel theory of the Gröbner basis was developed for Lie algebras by Shirshov in ^[4]. In ^[5], Bokut noticed that Shirshov's method works for also associative algebras. Hence Shirshov's theory for Lie and their universal enveloping algebras is called the Gröbner-Shirshov basis theory. We may refer the papers ^{[6,7,8,9,10,11,12,13]} for some recent studies over Gröbner-Shirshov bases in terms of algebraic ways, the papers ^[14,15] related to Hilbert series and the paper ^[16] in terms of graph theoretic way. Furthermore citation ^[17] can be used to understand normal forms for the monoid of positive braids by using Gröbner-Shirshov basis.

The word, conjugacy and isomorphism problems (shortly decision problems) have played an important role in group theory since the work of M. Dehn in early 1900's. Among them, especially the word problem has been studied widely in groups (see ^[18]). It is well known that the word problem for finitely presented groups is not solvable in general; that is, given any two words obtained by generators of the group, there may not be an algorithm to decide whether these words represent the same element in this group.

The method Gröbner-Shirshov basis theory gives a new algorithm to obtain normal forms of elements of groups, monoids and semigroups, and hence a new algorithm to solve the word problem in these algebraic structures (see also ^[19], for relationship with word problem for semigroups and ideal membership problem for non-commutative polynomail rings). By considering this fact, our aim in this paper is to find a Gröbner-Shirshov basis of the symmetric inverse monoid in terms of the dex-leg ordering on the related words of symmetric inverse monoids.

Symmetric inverse monoids are partial bijections and they are very well known in combinatorics. Easdown et al. ^[20] studied a presentation for the symmetric inverse monoid $I_n$ . By adding relations $\sigma _1^2$ $= \sigma _2^2$ $= \cdots$ $= \sigma _{n-1}^2 = 1$ into the presentation of the braid group described in terms of Artin's Theorem, it is obtained the well-known Moore presentation for the symmetric group as defined in ^[21]. Using this in the Popova's description ^[22] for the presentation of the symmetric inverse monoid $I_n$ yields the following presentation:

$\begin{eqnarray} I_n = \left\langle {\varepsilon}, \sigma _1, \sigma _2, \cdots, \sigma _{n-1} \right. &;& \left. {\sigma _i}{\sigma _j} = {\sigma _j}{\sigma _i}\: \: \left( {\left| {i - j} \right| \gt 1} \right), \right. \\ && \left. {\sigma _i}{\sigma _{i + 1}}\sigma _{i} = {\sigma _{i + 1}}{\sigma _i}{\sigma _{i+1}}\: \: (1 \leq i \leq n - 2), \right. \\ && \left. \sigma _1^2 = \sigma _2^2 = \cdots = \sigma _{n-1}^2 = 1, \: {\varepsilon ^2} = \varepsilon, \right. \\ && \left. \varepsilon {\sigma _i} = {\sigma _i}\varepsilon \: \: (1 \leq i \leq n - 2), \right. \\ && \left. \varepsilon {\sigma _{n-1}}\varepsilon = {\sigma _{n-1}}\varepsilon {\sigma _{n-1}}\varepsilon = \varepsilon {\sigma _{n-1}}\varepsilon {\sigma _{n-1}} \right\rangle . \end{eqnarray}$

(1.1)

In ^[23], the author has also studied presentations of symmetric inverse and singular part of the symmetric inverse monoids.

2. Gröbner-Shirshov bases and the composition-diamond lemma

Let $k$ be a field and $k\langle X\rangle$ be the free associative algebra over $k$ generated by $X$ . Denote by $X^{\ast}$ the free monoid generated by $X$ , where the empty word is the identity which is denoted by $1$ . For a word $w\in X^{\ast}$ , let us denote the length of $w$ by $|w|$ . Also assume that $X^{\ast}$ is a well ordered monoid. A well-ordering $\leq$ on $X^{\ast }$ is called a monomial ordering if for $u, v\in X^{\ast }$ , we have $u\leq v\Rightarrow w_{1}uw_{2} \leq w_{1}vw_{2}$ , for all $w_{1}, w_{2}\in X^*$ . A standard example of monomial ordering on $X^*$ is the deg-lex ordering, in which two words are compared first by the degree and then lexicographically, where $X$ is a well-ordered set.

Every nonzero polynomial $f\in k\left\langle X\right\rangle$ has the leading word $\overline{f}$ . If the coefficient of $\overline{f}$ in $f$ is equal to $1$ , then $f$ is called monic. The following fundamental materials can be found in ^{[3,5,6,7,8,10,11,12,24]}.

Let $f$ and $g$ be two monic polynomials in $k\langle X\rangle$ . Therefore we have two compositions between $f$ and $g$ as follows:

$1.$ If $w$ is a word such that $w = \overline{f}b = a\overline{g}$ for some $a, b\in X^{\ast }$ with $|\overline{f}|+|\overline{g}| > |w|$ , then the polynomial $(f, g)_{w} = fb-ag$ is called the intersection composition of $f$ and $g$ with respect to $w$ (and denoted by $f\wedge g$ ). In here, the word $w$ is called an ambiguity of the intersection.

$2.$ If $w = \overline{f} = a\overline{g}b$ for some $a, b\in X^{\ast }$ , then the polynomial $(f, g)_{w} = f-agb$ is called the inclusion composition of $f$ and $g$ with respect to $w$ (and denoted by $f\vee g$ ). In this case, the word $w$ is called an ambiguity of the inclusion.

If $g$ is a monic polynomial, $\overline{f} = a\overline{g}b$ and $\alpha$ is the coefficient of the leading term $\overline{f}$ , then the transformation $f\mapsto f-\alpha agb$ is called an elimination of the leading word (ELW) of $g$ in $f$ .

Let $S\subseteq k\left\langle X\right\rangle$ with each $s\in S$ monic. Then the composition $(f, g)_{w}$ is called trivial modulo $(S, w)$ if $(f, g)_{w} = \sum\alpha_{i}a_{i}s_{i}b_{i}$ , where each $\alpha _{i}\in k, \:a_{i}, b_{i}\in X^{\ast }, \:s_{i}\in S$ and $a_{i}\overline{s_{i}}b_{i} < w$ . If this is the case, then we write $(f, g)_{w}\equiv 0\ mod(S, w)$ . In general, for $p, q\in k\langle X\rangle$ , we write $p\equiv q\ mod(S, w)$ which means that $p-q = \sum\alpha_i a_{i}s_{i}b_{i}$ , where each $\alpha _{i}\in k, \:a_{i}, b_{i}\in X^{\ast }, \:s_{i}\in S$ and $a_{i}\overline{s_{i}}b_{i} < w$ .

A set $S$ with the well ordering $\leq$ is called a Gröbner-Shirshov basis for $k\left\langle X\mid S\right\rangle$ if every composition $(f, g)_{w}$ of polynomials in $S$ is trivial modulo $S$ and the corresponding $w$ .

The following lemma was proved by Shirshov ^[4] for free Lie algebras (with deg-lex ordering) in 1962 (^[24]). In 1976, Bokut ^[5] specialized the Shirshov's approach to associative algebras (see also ^[3]). On the other hand, for commutative polynomials, this lemma is known as the Buchberger's Theorem (cf. ^[2,25]).

Lemma 1 (Composition-Diamond Lemma). Let $k$ be a field,

$A = k\left\langle X\mid S\right\rangle = \ k\langle X\rangle / Id(S)$

and $\leq$ a monomial order on $X^{\ast }$ , where $Id(S)$ is the ideal of $k\langle X\rangle$ generated by $S$ . Then the following statements are equivalent:

1. $S$ is a Gröbner-Shirshov basis.

2. $f\in Id(S)\Rightarrow \overline{f} = a\overline{s}b$ for some $s\in S$ and $a, b\in X^{\ast }$ .

3. $Irr(S) = \{u\in X^{\ast }\mid u\neq a\overline{s}b, s\in S, a, b\in X^{\ast }\}$ is a basis of the algebra $A = k\left\langle X\mid S\right\rangle$ .

If a subset $S$ of $k\langle X\rangle$ is not a Gröbner-Shirshov basis, then we can add to $S$ all nontrivial compositions of polynomials of $S$ , and by continuing this process (maybe infinitely) many times, we eventually obtain a Gröbner-Shirshov basis $S^{comp}$ . Such a process is called the Shirshov algorithm.

If $S$ is a set of "semigroup relations" (that is, the polynomials in $S$ are of the form $u-v$ , where $u, v\in X^*$ ), then a nontrivial composition will have the same form. As a result, the set $S^{comp}$ also consists of semigroup relations.

Let $M = sgp\left\langle X\mid S\right\rangle$ be a semigroup presentation. Then $S$ is a subset of $k\langle X\rangle$ and hence one can find a Gröbner-Shirshov basis $S^{comp}$ . The last set does not depend on $k$ , and as mentioned before, it consists of semigroup relations. We will call $S^{comp}$ a Gröbner-Shirshov basis of $M$ . This is the same as a Gröbner-Shirshov basis of the semigroup algebra $kM = k\left\langle X\mid S\right\rangle$ . If $S$ is a Gröbner-Shirshov basis of the semigroup $M = sgp\left\langle X\mid S\right\rangle$ , then $Irr\left(S\right)$ is a normal form for $M$ ^[9,26].

3. A Gröbner-Shirshov basis for the monoid $I_n$

The target of this section is to obtain a Gröbner-Shirshov basis for the symmetric inverse monoid $I_n$ by taking into account the presentation given in (1.1). After that we will indicate the solvability of the word problem over $I_n$ .

By ordering the generators as $\varepsilon > {\sigma _{n - 1}} > {\sigma _{n - 2}} > {\sigma _{n - 3}} > \cdots > {\sigma _2} > {\sigma _1}$ in (1.1), we have the following main result of this paper.

Theorem 2. A Gröbner-Shirshov basis for the symmetric inverse monoid consists of the following relations:

$\begin{eqnarray*} &(1)&\:\sigma _i^2 = 1\:(1 \leq i \leq n - 1), \qquad\qquad\qquad (2)\:{\sigma _i}{\sigma _j} = {\sigma _j}{\sigma _i}\:\left( {\left| {i - j} \right| \gt 1} \right), \\ &(3)&\:{\varepsilon ^2} = \varepsilon, \qquad\qquad\qquad\qquad\qquad\:\qquad (4)\:\varepsilon {\sigma _i} = {\sigma _i}\varepsilon \:(1 \leq i \leq n - 2), \\ &(5)&\:{\sigma _{i + 1}}{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots\sigma _1^{{M_1}}{\sigma _{i + 1}} = {\sigma _i}{\sigma _{i + 1}}{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots\, \sigma _1^{{M_1}}\\ &&(1 \leq i \leq n - 2, \:{M_k} = \left\{ {0, 1} \right\}\, \left( {1 \leq k \leq i - 1\, } \right)), \\ &(6)&\:{\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _1^{{P_1}}\varepsilon = \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _1^{{P_1}}\varepsilon \, \, \, ({P_k} = \left\{ {0, 1} \right\}\, \, \left( {1 \leq k \leq n - 2} \right)), \\ &(7)&\:{\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}^{{Q_{n - 3}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon = \varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}^{{Q_{n - 3}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \: (j \gt i, \, 1 \leq i \leq n - 3, \, \, \, 2 \leq j \leq n - 2, \, {Q_{{k_1}}}, {\varphi _{{k_2}}} = \left\{ {0, 1} \right\}\\ &&\left( {i \leq {k_1} \leq n - 3, \, j \leq {k_2} \leq n - 2} \right)), \\ &(7')&\:{\sigma _{n - p}}\varepsilon^r {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}\sigma _{n-4}^{{Q_{n-4}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n-3}^{{\varphi _{n-3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \cdots {\sigma _{n - 1}}\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}\varepsilon = \\ && \varepsilon^r {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}\sigma _{n-4}^{{Q_{n-4}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n-3}^{{\varphi _{n-3}}}\cdots \sigma _j^{{\varphi _j}}\varepsilon \cdots {\sigma _{n - 1}}\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}\varepsilon \\ &&(2 \lt p \lt n, \:r = \left\{ {0, 1} \right\}, \:s\geq \cdots \geq j \geq i, \, 1 \leq i \leq n - 3, \, \, \, 2 \leq j, s \leq n - 2, \\ &&{Q_{{k_1}}}, {\varphi _{{k_2}}}, {\lambda _{{k_3}}} = \left\{ {0, 1} \right\}\: \left( {i \leq {k_1} \leq n - 4, \, j \leq {k_2} \leq n - 3, \, s \leq {k_3} \leq n - 2} \right)), \\ &(8)&\:\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}} = \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon \sigma _{n - 1}^{{L_{n - 1}}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\\ &&\sigma _{n - t + 1}^{{L_{n - t + 1}}}\:(2 \leq t \leq n - 1, \:{L_k} = \left\{ {0, 1} \right\}\, \, \left( {1 \leq k \leq n - 1} \right)), \\ &(9)&\:{\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\cdots\sigma _1^{{U_1}})(\sigma _{(n - k) + 2}^{{V_{(n - k) + 2}}}\sigma _{(n - k) + 1}^{{V_{(n - k) + 1}}}\cdots\sigma _1^{{V_1}})\cdots(\sigma _{n - 1}^{{S_{n - 1}}}\sigma _{n - 2}^{{S_{n - 2}}}\cdots\sigma _1^{{S_1}})\\ &&(\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\cdots\sigma _1^{{T_1}})\varepsilon = ({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) + 1}^{{U_{(n - k) + 1}}}\cdots\sigma _1^{{U_1}})(\sigma _{(n - k) + 2}^{{V_{(n - k) +2}}}\sigma _{(n - k) + 1}^{{V_{(n - k) + 1}}}\cdots\sigma _1^{{V_1}})\cdots\\ &&(\sigma _{n - 1}^{{S_{n - 1}}}\sigma _{n - 2}^{{S_{n - 2}}}\cdots\sigma _1^{{S_1}})(\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\cdots\sigma _1^{{T_1}})\varepsilon\\ &&(2 \leq k \leq n - 1, \:{U_{{k_1}}}, {V_{{k_2}}}, {S_{{k_3}}}, {T_{{k_4}}} = \left\{ {0, 1} \right\}\\ &&\left( {1 \leq {k_1} \leq (n - k) - 1, \:1 \leq {k_2} \leq (n - k) + 2}, \: {1 \leq {k_3} \leq n - 1}, \:{1 \leq {k_4} \leq n - 2})\right), \\ &(10)&\:{\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\cdots\sigma _1^{{X_1}})(\sigma _{(n - k) + 2}^{{Y_{(n - k) + 2}}}\sigma _{(n - k) + 1}^{{Y_{(n - k) + 1}}}\cdots\sigma _1^{{Y_1}})\cdots(\sigma _{n - 2}^{{Z_{n - 2}}}\sigma _{n - 3}^{{Z_{n - 3}}}\cdots\sigma _1^{{Z_1}})\\ &&(\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}) (\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{R_{n - 2}}}\cdots\sigma _j^{{R_j}})\varepsilon = ({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\cdots\sigma _1^{{X_1}})(\sigma _{(n - k) + 2}^{{Y_{(n - k) + 2}}}\\ &&\sigma _{(n - k) + 1}^{{Y_{(n - k) + 1}}}\cdots\sigma _1^{{Y_1}})\cdots(\sigma _{n - 2}^{{Z_{n - 2}}}\sigma _{n - 3}^{{Z_{n - 3}}}\cdots\sigma _1^{{Z_1}}) (\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}})(\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{R_{n - 2}}}\cdots\sigma _j^{{R_j}})\varepsilon \\ &&(j \gt i, \:1 \leq i \leq n - 2, \:2 \leq j \leq n - 2, \:3 \leq k \leq n -1, \:{X_{{k_1}}}, {Y_{{k_2}}}, {Z_{{k_3}}}, {W_{{k_4}}}, {R_{{k_5}}} = \left\{ {0, 1} \right\}\\ &&({1 \leq {k_1} \leq (n - k) - 1, \:1 \leq {k_2} \leq (n - k) + 2, \:1 \leq {k_3} \leq n - 2, \:1 \leq {k_4} \leq n - 3, \:1 \leq {k_5} \leq n - 2}). \end{eqnarray*}$

We also have the following additional conditions.

● For the relation $(5)$ : For $1\leq k < i-1$ , to take $M_k = 1$ it is necessary $M_{k+1} = 1$ .

● For the relation $(6)$ : For $1\leq k < n-2$ , to take $P_k = 1$ it is necessary $P_{k+1} = 1$ .

● For the relation $(7)$ : For $i\leq k < n-3$ , to take $Q_k = 1$ it is necessary $Q_{k+1} = 1$ .

For $j\leq k < n-2$ , to take $\varphi_k = 1$ it is necessary $\varphi_{k+1} = 1$ .

● For the relation $(7')$ : For $i\leq k < n-4$ , to take $Q_k = 1$ it is necessary $Q_{k+1} = 1$ .

For $j\leq k < n-3$ , to take $\varphi_k = 1$ it is necessary $\varphi_{k+1} = 1$ .

For $s\leq k < n-2$ , to take $\lambda_k = 1$ it is necessary $\lambda_{k+1} = 1$ .

● For the relation $(8)$ : For $n-t\leq k < n-2$ , to take $L_k = 1$ it is necessary $L_{k+1} = 1$ .

● For the relation $(9)$ : For $1\leq t < (n-k)-1$ , to take $U_t = 1$ it is necessary $U_{t+1} = 1$ .

For $1\leq t < (n-k)+2$ , to take $V_t = 1$ it is necessary $V_{t+1} = 1$ .

For $1\leq t < n-1$ , to take $S_t = 1$ it is necessary $S_{t+1} = 1$ .

For $1\leq t < n-2$ , to take $T_t = 1$ it is necessary $T_{t+1} = 1$ .

● For the relation $(10)$ : For $1\leq t < (n-k)-1$ , to take $X_t = 1$ it is necessary $X_{t+1} = 1$ .

For $1\leq t < (n-k)+2$ , to take $Y_t = 1$ it is necessary $Y_{t+1} = 1$ .

For $1\leq t < n-2$ , to take $Z_t = 1$ it is necessary $Z_{t+1} = 1$ .

For $i\leq t < n-3$ , to take $W_t = 1$ it is necessary $W_{t+1} = 1$ .

For $j\leq t < n-2$ , to take $R_t = 1$ it is necessary $R_{t+1} = 1$ .

Proof. Relations given for $I_n$ in (1.1) provide relations among (1)–(10). Now we need to prove that all compositions among relations (1)–(10) are trivial. To do that, firstly, we consider intersection compositions of these relations. Hence we have the following ambiguties $w$ :

$\begin{eqnarray*} &(1) \wedge (1)&:\:w = \sigma _i^3\:\:(1 \leq i \leq n - 1), \qquad\quad(1) \wedge (2)\::w = \sigma _i^2{\sigma _j}\:\:(\left| {i - j} \right| \gt 1), \\ &(1) \wedge (5)&:w = \sigma _{i + 1}^2{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}\, {\sigma _{i + 1}}\:\:(1 \leq i \leq n - 2), \\ &(1) \wedge (6)&:w = \sigma _{n - 1}^2\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\sigma _{n - 3}^{{P_{n - 3}}}\cdots\sigma _1^{{P_1}}\varepsilon, \\ &(1) \wedge (7)&:w = \sigma _{n - 2}^2\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{Q_{n - 2}}}\sigma _{n - 3}^{{Q_{n - 3}}}\cdots\sigma _i^{{Q_i}}\, \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\sigma _{n - 3}^{{\varphi _{n - 3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon, \\ &(1) \wedge (7')&:w = \sigma _{n - p}^2\varepsilon^r {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}\sigma _{n - 4}^{{Q_{n - 4}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{\varphi _{n - 3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \cdots {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}\varepsilon, \\ &(1) \wedge (9)&:w = \sigma _{n - k}^2({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}\cdots\sigma _1^{{U_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\sigma _{n - 3}^{{T_{n - 3}}}\cdots\sigma _1^{{T_1}}\varepsilon, \\ &(1) \wedge (10)&:w = \sigma _{n - k}^2({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{X_{(n - k) - 2}}}\cdots\sigma _1^{{X_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\\ &&\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{R_{n - 2}}}\sigma _{n - 3}^{{R_{n - 3}}}\cdots\sigma _j^{{R_j}}\varepsilon\: \:(j \gt i), \end{eqnarray*}$

$\begin{eqnarray*} &(2) \wedge (1)&:w = {\sigma _i}\, \sigma _j^2\:\:(\left| {i - j} \right| \gt 1), \\ &(2) \wedge (2)&:w = {\sigma _i}\, {\sigma _j}\, {\sigma _k}\:\:(\left| {i - j} \right| \gt 1, \left| {j - k} \right| \gt 1), \\ &(2) \wedge (5)&:w = {\sigma _i}{\sigma _{j + 1}}{\sigma _j}\sigma _{j - 1}^{{M_{j - 1}}}\cdots\sigma _1^{{M_1}}\, {\sigma _{j + 1}}\:\:\, (\, \left| {i - (j + 1) \gt 1} \right|), \\ &(2) \wedge (7')&:w = \sigma_i\sigma _{n - p}\varepsilon^r {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}\sigma _{n - 4}^{{Q_{n - 4}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{\varphi _{n - 3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \cdots {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}\varepsilon\quad (\, \left| {i - (n-p) \gt 1} \right|), \\ &(2) \wedge (9)&:w = {\sigma _i}{\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}..\, \, \sigma _1^{{U_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\sigma _{n - 3}^{{T_{n - 3}}}\cdots\sigma _1^{{T_1}}\varepsilon\\ &&(\left| {i - (n - k) \gt 1} \right|), \\ &(2) \wedge (10)&:w = {\sigma _i}{\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{X_{(n - k) - 2}}}\cdots\sigma _1^{{X_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\\ &&\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{R_{n - 2}}}\sigma _{n - 3}^{{R_{n - 3}}}\cdots\sigma _j^{{R_j}}\varepsilon \:\:(j \gt i, \left| {i - (n - k) \gt 1} \right|), \end{eqnarray*}$

$\begin{eqnarray*} &(3) \wedge (3)&:\, w = {\varepsilon ^3}, \qquad (3) \wedge (4):\, w = {\varepsilon ^2}{\sigma _i}\:\:(1 \leq i \leq n - 2), \\ &(3) \wedge (8)&:\, w = {\varepsilon ^2}{\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\, \sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\, \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\, \sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}, \end{eqnarray*}$

$\begin{eqnarray*} &(4) \wedge (1)&:\, w = \varepsilon \sigma _i^2\:\:\:(1 \leq i \leq n - 2), \qquad\:(4) \wedge (2):\, w = \varepsilon {\sigma _i}{\sigma _j}\, \, \, (\left| {i - j} \right| \gt 1), \\ &(4) \wedge (5)&:\, w = \varepsilon {\sigma _{i + 1}}{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\sigma _{i - 2}^{{M_{i - 2}}}\cdots\sigma _1^{{M_1}}{\sigma _{i + 1}}\:\:(1 \leq i \leq n - 3), \\ &(4) \wedge (7)&:w = \varepsilon {\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{Q_{n - 2}}}\sigma _{n - 3}^{{Q_{n - 3}}}\cdots\sigma _i^{{Q_i}}\, \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\sigma _{n - 3}^{{\varphi _{n - 3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon, \\ &(4) \wedge (7')&:w = \varepsilon\sigma _{n - p}\varepsilon^r {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}\sigma _{n - 4}^{{Q_{n - 4}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{\varphi _{n - 3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \cdots {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}\varepsilon, \\ &(4) \wedge (9)&:w = \varepsilon {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}\cdots\sigma _1^{{U_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\sigma _{n - 3}^{{T_{n - 3}}}\cdots\sigma _1^{{T_1}}\varepsilon, \\ &(4) \wedge (10)&:w = \varepsilon {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{X_{(n - k) - 2}}}\cdots\sigma _1^{{X_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\varepsilon\\ &&{\sigma _{n - 1}}\sigma _{n - 2}^{{R_{n - 2}}}\sigma _{n - 3}^{{R_{n - 3}}}\cdots\sigma _j^{{R_j}}\varepsilon, \end{eqnarray*}$

$\begin{eqnarray*} &(5) \wedge (1)&:w = {\sigma _{i + 1}}{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots\sigma _1^{{M_1}}\sigma _{i + 1}^2\:\:(1 \leq i \leq n - 2), \\ &(5) \wedge (2)&:w = {\sigma _{i + 1}}{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots\sigma _1^{{M_1}}{\sigma _{i + 1}}{\sigma _j}\:\:(\left| {i + 1 - j} \right| \gt 1), \\ &(5) \wedge (5)&:w = {\sigma _{i + 1}}{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots\sigma _1^{{M_1}}\, {\sigma _{i + 1}}{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots\sigma _1^{{M_1}}\, {\sigma _{i + 1}}\:\:(1 \leq i \leq n - 2), \\ &(5) \wedge (6)&:w = {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}^{{M_{n - 3}}}\sigma _{n - 4}^{{M_{n - 4}}}\cdots\sigma _1^{{M_1}}\, {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _1^{{P_1}}\, \varepsilon, \\ &(5) \wedge (7)&:w = {\sigma _{n - 2}}{\sigma _{n - 3}}\sigma _{n - 4}^{{M_{n - 4}}}\sigma _{n - 5}^{{M_{n - 5}}}\cdots\sigma _1^{{M_1}}\, {\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{Q_{n - 2}}}\cdots\sigma _i^{{Q_i}}\, \varepsilon {\sigma _{n - 1}} \sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \\ &&(j \gt i), \\ &(5) \wedge (7')&:w = {\sigma _{n - p}}{\sigma _{n-p-1}}\sigma _{n-p-2}^{{M_{n - p -2}}}\cdots\sigma _1^{{M_1}}\, \sigma _{n - p}\varepsilon^r {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}\sigma _{n - 4}^{{Q_{n - 4}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\\ &&\sigma _{n - 2}\sigma _{n - 3}^{{\varphi _{n - 3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon\cdots {\sigma _{n - 1}}\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}\varepsilon, \\ &(5) \wedge (9)&:w = {\sigma _{n - k}}{\sigma _{(n - k) - 1}}\sigma _{(n - k) - 2}^{{M_{(n - k) - 2}}}\sigma _{(n - k) - 3}^{{M_{(n - k) - 3}}}\cdots\sigma _1^{{M_1}}({\sigma _{n - k}}{\sigma _{(n - k) - 1}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}\cdots\sigma _1^{{U_1}})\cdots\varepsilon \\ &&{\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\sigma _{n - 3}^{{T_{n - 3}}}\cdots\sigma _1^{{T_1}}\varepsilon, \\ &(5) \wedge (9)&:w = {\sigma _{n - k}}{\sigma _{(n - k) - 1}}\sigma _{(n - k) - 2}^{{M_{(n - k) - 2}}}\sigma _{(n - k) - 3}^{{M_{(n - k) - 3}}}\cdots\sigma _1^{{M_1}}{\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\\ &&\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\cdots\sigma _1^{{U_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\sigma _{n - 3}^{{T_{n - 3}}}\cdots\sigma _1^{{T_1}}\varepsilon, \\ &(5) \wedge (10)&:w = {\sigma _{n - k}}{\sigma _{(n - k) - 1}}\sigma _{(n - k) - 2}^{{M_{(n - k) - 2}}}\cdots\sigma _1^{{M_1}}{\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{X_{(n - k) - 2}}}\cdots\sigma _1^{{X_1}})\\ &&\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{R_{n - 2}}}\sigma _{n - 3}^{{R_{n - 3}}}\cdots\sigma _j^{{R_j}}\varepsilon \:\:(j \gt i), \\ &(5) \wedge (10)&:w = {\sigma _{n - k}}{\sigma _{(n - k) - 1}}\sigma _{(n - k) - 2}^{{M_{(n - k) - 2}}}\cdots\sigma _1^{{M_1}}{\sigma _{n - k}}{\sigma _{(n - k) - 1}}\sigma _{(n - k) - 2}^{{X_{(n - k) - 2}}}\cdots\sigma _1^{{X_1}}\\ &&\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{R_{n - 2}}}\sigma _{n - 3}^{{R_{n - 3}}}\cdots\sigma _j^{{R_j}}\varepsilon \:\:(j \gt i), \end{eqnarray*}$

$\begin{eqnarray*} &(6) \wedge (3)&:w = {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\sigma _{n - 3}^{{P_{n - 3}}}\cdots\sigma _1^{{P_1}}\, {\varepsilon ^2}, \\ &(6) \wedge (4)&:w = {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\sigma _{n - 3}^{{P_{n - 3}}}\cdots\sigma _1^{{P_1}}\varepsilon {\sigma _i}\:\:(1 \leq i \leq n - 2), \\ &(6) \wedge (6)&:\, w = {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\, \sigma _{n - 3}^{{P_{n - 3}}}\cdots\sigma _1^{{P_1}}\varepsilon, \\ &(6) \wedge (7)&:\, w = {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}^{{Q_{n - 3}}}\sigma _{n - 4}^{{Q_{n - 4}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \:\:(j \gt i), \\ &(6) \wedge (8)&:\, w = {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\, \sigma _{n - 3}^{{P_{n - 3}}}\cdots\sigma _1^{{P_1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\, \sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}, \\ &(6) \wedge (8)&:\, w = {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{n-t}^{{P_{n-t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{n - t}^{{P_{n - t}}}, \end{eqnarray*}$

$\begin{eqnarray*} &(7) \wedge (3)&:w = {\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{Q_{n - 2}}}\cdots\sigma _i^{{Q_i}}\, \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}{\varepsilon ^2}\:\:(j \gt i), \\ &(7) \wedge (4)&:w = {\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{Q_{n - 2}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon {\sigma _t}, \\ &&(j \gt i, \:1 \leq i \leq n - 2, \:2 \leq j \leq n - 2, \:1 \leq t \leq n - 2), \\ &(7) \wedge (6)&:w = {\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}^{{Q_{n - 3}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _1^{{P_1}}\varepsilon\: \:(1 \leq i \leq n - 2), \\ &(7) \wedge (7)&:w = {\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}^{{Q_{n - 3}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\varepsilon{\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}^{{Q_{n - 3}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon\:\:(j \gt i, \:1 \leq i \leq n - 3, \:2 \leq j \leq n - 2), \\ &(7) \wedge (8)&:w = {\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{Q_{n - 2}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{L_{n - 2}}}\sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\:\: (j \gt i, \:1 \leq i \leq n - 2, \:2 \leq j \leq n - 2), \\ &(7) \wedge (8)&:w = {\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{Q_{n - 2}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}, \\ &(7') \wedge (7')&:w = {\sigma _{n - p}}\varepsilon^r {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}\sigma _{n-4}^{{Q_{n-4}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n-3}^{{\varphi _{n-3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \cdots {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}\varepsilon{\sigma _{n - 1}}\sigma _{n - 2}^{{\phi _{n - 2}}}\cdots\sigma _p^{{\phi _p}}\varepsilon, \\ &(7')\wedge (8)&:\, w = {\sigma _{n - p}}\varepsilon^r {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}\sigma _{n-4}^{{Q_{n-4}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n-3}^{{\varphi _{n-3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \cdots {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}\varepsilon{\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}, \\ &(7')\wedge (8)&:\, w = {\sigma _{n - p}}\varepsilon^r {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}\sigma _{n-4}^{{Q_{n-4}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n-3}^{{\varphi _{n-3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \cdots {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}\varepsilon{\sigma _{n - 1}}\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}, \end{eqnarray*}$

$\begin{eqnarray*} &(8) \wedge (1)&:\, w = \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\sigma _{n - 3}^{{L_{n - 3}}}\cdots{(\sigma _{n - t}^{{L_{n - t}}})^2}, \\ &(8) \wedge (2)&:\, w = \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}{\sigma _j}\:\:(\left| {n - t - j} \right| \gt 1), \\ &(8) \wedge (5)&:\, w = \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}{\sigma _{(n - t) - 1}}\sigma _{(n - t) - 2}^{{M_{(n - t) - 2}}}\cdots\sigma _1^{{M_1}}\\ &&{\sigma _{n - t}}\:\:(1 \leq t \leq n - 2), \\ &(8) \wedge (6)&:\, w = \varepsilon {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\sigma _{n - 3}^{{P_{n - 3}}}\cdots\sigma _1^{{P_1}}\varepsilon, \\ &(8) \wedge (6)&:\, w = \varepsilon {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\sigma _{n - 3}^{{P_{n - 3}}}\cdots\sigma _1^{{P_1}}\varepsilon, \\ &(8) \wedge (7)&:w = \varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{Q_{n - 3}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \:\:(j \gt i), \\ &(8) \wedge (7)&:w = \varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}^{{Q_{n - 3}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \:\:(j \gt i), \\ &(8) \wedge (7')&:w = \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon^r {\sigma _{n - 1}}\\ &&{\sigma _{n - 2}}\sigma _{n - 3}\sigma _{n-4}^{{Q_{n-4}}}\cdots\sigma _i^{{Q_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n-3}^{{\varphi _{n-3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \cdots {\sigma _{n - 1}}\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}\varepsilon, \\ &(8) \wedge (7')&:w = \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}\sigma _{n - 4}^{{L_{n - 4}}}\cdots\sigma _{n - t}^{{L_{n - t}}} \varepsilon{\sigma _{n - 1}}\\ && {\sigma _{n - 2}}\sigma _{n-3}^{{\varphi _{n-3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \cdots {\sigma _{n - 1}}\sigma _{n - 2}^{{\lambda _{n - 2}}}\cdots\sigma _s^{{\lambda _s}}\varepsilon, \\ &(8) \wedge (8)&:w = \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}, \\ &(8) \wedge (9)&:w = \varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\cdots{\sigma _{n - k}}\varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\cdots{\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}\\ &&\cdots\sigma _1^{{U_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\cdots\sigma _1^{{T_1}}\varepsilon, \\ &(8) \wedge (10)&:w = \varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\cdots{\sigma _{n - k}}\varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\cdots{\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\cdots\sigma _1^{{X_1}})\\ &&\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{R_{n - 2}}}\cdots\sigma _j^{{R_j}}\varepsilon \:\:(j \gt i), \end{eqnarray*}$

$\begin{eqnarray*} &(9) \wedge (3)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}\cdots\sigma _1^{{U_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\sigma _{n - 3}^{{T_{n - 3}}}\cdots\sigma _1^{{T_1}}{\varepsilon ^2}, \\ &(9) \wedge (4)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}\cdots\sigma _1^{{U_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\sigma _{n - 3}^{{T_{n - 3}}}\cdots\sigma _1^{{T_1}}\varepsilon {\sigma _i}\\ &&(1 \leq i \leq n - 2), \\ &(9) \wedge (6)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}\cdots\sigma _1^{{U_1}})\cdots(\sigma _{n - 1}^{S_{n - 1}}\sigma _{n - 2}^{{S_{n - 2}}}\sigma _{n - 3}^{{S_{n - 3}}}\cdots\sigma _1^{{S_1}})\varepsilon \\ &&{\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\sigma _{n - 3}^{{P_{n - 3}}}\cdots\sigma _1^{{P_1}}\varepsilon, \\ &(9) \wedge (7)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}\cdots\sigma _1^{{U_1}})\cdots(\sigma _{n - 1}^{{S_{n - 1}}}\sigma _{n - 2}^{{S_{n - 2}}}\sigma _{n - 3}^{{S_{n - 3}}}\cdots\sigma _1^{{S_1}})\varepsilon\\ && {\sigma _{n - 1}}{\sigma _{n - 2}}\varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}^{{Q_{n - 3}}}\sigma _{n - 4}^{{Q_{n - 4}}}\cdots\sigma _i^{{Q_i}}\varepsilon{\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\sigma _{n - 3}^{{\varphi _{n - 3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon\: \:(j \gt i), \\ &(9) \wedge (7)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}\cdots\sigma _1^{{U_1}})\cdots\sigma _{n - 1}^{{S_{n - 1}}}\sigma _{n - 2}^{{S_{n - 2}}}\varepsilon\\ && {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}^{{Q_{n - 3}}}\cdots\sigma _i^{{Q_i}}\varepsilon{\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\sigma _{n - 3}^{{\varphi _{n - 3}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon\: \:(j \gt i), \\ &(9) \wedge (8)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}\cdots\sigma _1^{{U_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\sigma _{n - 3}^{{T_{n - 3}}}\cdots\sigma _1^{{T_1}}\varepsilon\\ &&{\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\sigma _{n - 3}^{{L_{n - 3}}}\cdots\sigma _{n - t}^{{L_{n - t}}}, \\ &(9) \wedge (8)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{U_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{U_{(n - k) - 2}}}\cdots\sigma _1^{{U_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\sigma _{n - 3}^{{T_{n - 3}}}\cdots\sigma _1^{{T_1}}\varepsilon\\ &&{\sigma _{n - 1}}\sigma _{n - 2}^{{T_{n - 2}}}\sigma _{n - 3}^{{T_{n - 3}}}\cdots\sigma _1^{{T_1}}, \end{eqnarray*}$

$\begin{eqnarray*} &(10) \wedge (3)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{X_{(n - k) - 2}}}\cdots\sigma _1^{{X_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\varepsilon{\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{R_{n - 2}}}\cdots\sigma _j^{{R_j}}{\varepsilon ^2}\:\:(j \gt i), \\ &(10) \wedge (4)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{X_{(n - k) - 2}}}\cdots\sigma _1^{{X_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\varepsilon {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{R_{n - 2}}}\cdots\sigma _j^{{R_j}}\varepsilon {\sigma _t}\:\:(1 \leq t \leq n - 2), \\ &(10) \wedge (6)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{X_{(n - k) - 2}}}\cdots\sigma _1^{{X_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\varepsilon{\sigma _{n - 1}}\\ &&\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _1^{{P_1}}\varepsilon, \\ &(10) \wedge (7)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{X_{(n - k) - 2}}}\cdots\sigma _1^{{X_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\varepsilon {\sigma _{n - 1}}\\ &&{\sigma _{n - 2}} \varepsilon {\sigma _{n - 1}}{\sigma _{n - 2}}\sigma _{n - 3}^{{Q_{n - 3}}}\cdots\sigma _t^{{Q_t}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{\varphi _{n - 2}}}\cdots\sigma _j^{{\varphi _j}}\varepsilon \:\:(j \gt t), \\ &(10) \wedge (7')&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\cdots\sigma _1^{{X_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\varepsilon {\sigma _{n - 1}}\\ &&{\sigma _{n - 2}}^{{R_{n - 2}}}\cdots\sigma _j^{{R_j}}\varepsilon{\sigma _{n - 1}}\sigma _{n - 2}^{{S_{n - 2}}}\cdots\sigma _l^{{S_l}}\varepsilon \:\:(l \gt j \gt i), \\ &(10) \wedge (8)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{X_{(n - k) - 2}}}\cdots\sigma _1^{{X_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\varepsilon {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{R_{n - 2}}}\cdots\sigma _j^{{R_j}}\varepsilon {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{L_{n - 2}}}\cdots\sigma _{n - t}^{{L_{n - t}}}\:\:(j \gt i), \\ &(10) \wedge (8)&:w = {\sigma _{n - k}}({\sigma _{(n - k) + 1}}{\sigma _{n - k}}\sigma _{(n - k) - 1}^{{X_{(n - k) - 1}}}\sigma _{(n - k) - 2}^{{X_{(n - k) - 2}}}\cdots\sigma _1^{{X_1}})\cdots\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}\sigma _{n - 3}^{{W_{n - 3}}}\cdots\sigma _i^{{W_i}}\varepsilon {\sigma _{n - 1}}\\ &&\sigma _{n - 2}^{{R_{n - 2}}}\cdots\sigma _j^{{R_j}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{R_{n - 2}}}\cdots\sigma _j^{{R_j}}\:\:(j \gt i), \end{eqnarray*}$

All these ambiguities are trivial. Let us show some of them as in the following.

$\begin{eqnarray*} (1)\wedge(5):w& = &\sigma _{i + 1}^2{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}{\sigma _{i + 1}}, \:(1 \leq i \leq n - 2), \\ (f, g)_{w}& = &(\sigma _{i+1}^2-1){\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}{\sigma _{i + 1}} \\ & & -{\sigma _{i + 1}}({\sigma _{i + 1}}{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots\sigma _1^{{M_1}}{\sigma _{i + 1}}-{\sigma _i}{\sigma _{i + 1}}{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots\, \sigma _1^{{M_1}})\\ & = &\sigma _{i + 1}^2{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}{\sigma _{i + 1}}-{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}{\sigma _{i + 1}}\\ && - \sigma _{i + 1}^2{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}{\sigma _{i + 1}}+{\sigma _{i + 1}}{\sigma _i}{\sigma _{i + 1}}{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots\, \sigma _1^{{M_1}} \end{eqnarray*}$

$\begin{eqnarray*} & = &{\sigma _{i + 1}}{\sigma _i}{\sigma _{i + 1}}{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots\, \sigma _1^{{M_1}}-{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}{\sigma _{i + 1}}\\ &\equiv&{\sigma _i}{\sigma _{i+1}}{\sigma _i}^{2}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}-{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}{\sigma _{i + 1}}\\ &\equiv&{\sigma _i}{\sigma _{i+1}}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}-{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}{\sigma _{i + 1}}\\ &\equiv& {\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}{\sigma _{i + 1}}-{\sigma _i}\sigma _{i - 1}^{{M_{i - 1}}}\cdots \sigma _1^{{M_1}}{\sigma _{i + 1}}\\ &\equiv& 0\:\: mod(S, w). \end{eqnarray*}$

$\begin{eqnarray*} (2)\wedge(2):w& = &{\sigma _i}{\sigma _j}{\sigma _k}\:(\left| {i - j} \right| \gt 1, \left| {j - k} \right| \gt 1), \\ (f, g)_{w}& = &({\sigma _i}{\sigma _j}-{\sigma _j}{\sigma _i}){\sigma _k}-{\sigma _i}({\sigma _j}{\sigma _k}-{\sigma _k}{\sigma _j})\\ & = &{\sigma _i}{\sigma _j}{\sigma _k}-{\sigma _j}{\sigma _i}{\sigma _k}-{\sigma _i}{\sigma _j}{\sigma _k}+{\sigma _i}{\sigma _k}{\sigma _j}\\ & = &{\sigma _i}{\sigma _k}{\sigma _j}-{\sigma _j}{\sigma _i}{\sigma _k}\\ &\equiv& {\sigma _k}{\sigma _i}{\sigma _j}-{\sigma _j}{\sigma _k}{\sigma _i}\\ &\equiv&{\sigma _k}{\sigma _j}{\sigma _i}-{\sigma _k}{\sigma _j}{\sigma _i}\\ &\equiv& 0\:\:mod(S, w). \end{eqnarray*}$

$\begin{eqnarray*} (6)\wedge(4):w& = &{\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i}^{{P_{i}}}\cdots\sigma _1^{{P_1}}\varepsilon {\sigma _i}\:(1 \leq i \leq n - 2), \\ (f, g)_{w}& = &({\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i}^{{P_{i}}}\cdots\sigma _1^{{P_1}}\varepsilon - \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i}^{{P_{i}}}\cdots\sigma _1^{{P_1}}\varepsilon){\sigma _i}\\ && -{\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i}^{{P_{i}}}\cdots\sigma _1^{{P_1}}(\varepsilon{\sigma _i}-{\sigma _i}\varepsilon) \\ & = &{\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i}^{{P_{i}}}\cdots\sigma _1^{{P_1}}\varepsilon {\sigma _i}-\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i}^{{P_{i}}}\cdots\sigma _1^{{P_1}}\varepsilon{\sigma _i}\\ && -{\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i}^{{P_{i}}}\cdots\sigma _1^{{P_1}}\varepsilon {\sigma _i}+{\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i}^{{P_{i}}}\cdots\sigma _1^{{P_1}}{\sigma _i}\varepsilon \\ & = & {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i}^{{P_{i}}}\cdots\sigma _1^{{P_1}}{\sigma _i}\varepsilon-\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i}^{{P_{i}}}\cdots\sigma _1^{{P_1}}\varepsilon{\sigma _i} \\ &\equiv& {\sigma _{n - 1}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots \sigma _{i-1}^{{P_{i-1}}}\sigma _i^{{P_i}}\sigma _{i-1}^{{P_{i-1}}}\cdots\sigma _1^{{P_1}}\varepsilon- \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i}^{{P_{i}}}\cdots\sigma _1^{{P_1}}{\sigma _i}\varepsilon \\ &\equiv& \sigma _{i-1}^{{P_{i-1}}} {\sigma _{n - 1}}\varepsilon{\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _1^{{P_1}}\varepsilon - \varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _{i-1}^{{P_{i-1}}}\sigma _{i}^{{P_{i}}}\sigma _{i-1}^{{P_{i-1}}}\cdots\sigma _1^{{P_1}}\varepsilon \\ &\equiv& \sigma _{i-1}^{{P_{i-1}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _1^{{P_1}}\varepsilon - \sigma _{i-1}^{{P_{i-1}}}\varepsilon {\sigma _{n - 1}}\sigma _{n - 2}^{{P_{n - 2}}}\cdots\sigma _1^{{P_1}}\varepsilon\\ &\equiv& 0\:\: mod(S, w). \end{eqnarray*}$

It is seen that there are no any inclusion compositions among relations (1)–(10). This ends up the proof.

As a consequence of Lemma 1 and Theorem 2, we have the following result.

Corollary 3. Let $C(u)$ be a normal form of a word $u\in I_{n}$ . Then $C(u)$ is of the form

$\begin{eqnarray*} {W_1}{\varepsilon ^{{k_1}}}{W_2}{\varepsilon ^{{k_2}}}\cdots{W_n}{\varepsilon ^{{k_n}}}\, , \end{eqnarray*}$

where ${{k_p} = \left\{ {0, 1} \right\}}$ $(1 \leq p \leq n)$ . In this above expression,

● if ${k_p} = 1\, \, \left({1 \leq p \leq n - 1} \right)$ then the word ${W_{p\, + 1\, }}$ which begins with ${\sigma _{n - 1}}$ and generated by $\sigma _i$ $(1 \leq i \leq n - 1)$ is actually a reduced word. Moreover the word ${W_{1}}$ generated by $\sigma _i$ $(1 \leq i \leq n - 1)$ is an arbitrary reduced word.

● if ${k_p} = 0\, \, \left({1 \leq p \leq n - 1} \right)$ then the word $W_pW_{p+1}$ is also reduced.

In addition, subwords of the forms $W_i\varepsilon^{k_i}W_{i+1}\varepsilon^{k_{i+1}}$ $\left({1 \leq i \leq n - 1} \right)$ , $W_j\varepsilon^{k_j}W_{j+1}\varepsilon^{k_{j+1}}W_{j+2}\varepsilon^{k_{j+2}}$ $\left({1 \leq j \leq n - 2} \right)$ , $W_r\varepsilon^{k_r}W_{r+1}\varepsilon^{k_{r+1}}W_{r+2}\varepsilon^{k_{r+2}}W_{r+3}\varepsilon^{k_{r+3}}$ $\left({1 \leq r \leq n - 3} \right)$ and $\varepsilon^{k_s}W_{s+1}\varepsilon^{k_{s+1}}W_{s+2}$ $\left({1 \leq s \leq n - 2} \right)$ must be reduced.

By Corollary 3, we can say that the word problem is solvable for symmetric inverse monoid $I_{n}$ .

Remark 4. We note that if we change the orderings on words we find another Gröbner-Shirshov bases related to chosen orederings. Thus we get normal form for given algebraic structure depending on ordering. To get this normal form it is used third item of Composition-Diamond Lemma. It is known that to get normal form structure implies solvability of the word problem. If one can not obtain a Gröbner-Shirshov basis according to chosen ordering on words, this does not mean that the word problem is not solvable.

4. An application

As an application of Theorem 2, we will give the following Example 5 which describes a Gröbner-Shirshov basis for the symmetric inverse monoid $I_4$ . The accuracy and efficiency of this example can be seen by " $GBNP$ package in GAP ^[1] which computes Gröbner bases of non-commutative polynomials as follows.

$\mathtt{gap \gt \ LoadPackage("GBNP",\ "0",\ {\bf{false}});}\\ \mathtt{{\bf{true}}}\\ \mathtt{gap \gt \ SetInfoLevel(InfoGBNP,\ 1);}\\ \mathtt{gap \gt \ SetInfoLevel(InfoGBNPTime,\ 1);}\\ \mathtt{gap \gt \ A: =\ FreeAssociativeAlgebraWithOne(Rationals,\ "s1",\ "s2",\ "s3",\ "e");}\\ \mathtt{ \lt algebra-with-one\ over\ Rationals,\ with\ 4\ generators \gt }\\ \mathtt{gap \gt \ g: =\ GeneratorsOfAlgebra(A);}\\ \mathtt{[\ (1)^{*} \lt identity\ ... \gt ,\ (1)^{*}s1,\ (1)^{*}s2,\ (1)^{*}s3,\ (1)^{*}e\ ]}\\ \mathtt{gap \gt \ s1: =\ g[2];;s2: =\ g[3];;s3: =\ g[4];;e: =\ g[5];;o: =\ g[1];}\\ \mathtt{(1)^{*} \lt identity\ ... \gt }\\ \mathtt{gap \gt \ GBNP.ConfigPrint(A);}\\ \mathtt{gap \gt \ twosidrels: =\ [s1} \wedge \mathtt{2-o,\ s2} \wedge \mathtt{2-o,\ s3} \wedge \mathtt{2-o,\ (s1^{*}s2)} \wedge \mathtt{3-o,\ (s2^{*}s3)} \wedge \mathtt{3-o,\ (s1^{*}s3)} \wedge \mathtt{2-o,}\\ \mathtt{e} \wedge \mathtt{2-e,\ s1^{*}e-e^{*}s1,\ s2^{*}e-e^{*}s2,\ e^{*}s3^{*}e-(e^{*}s3)} \wedge \mathtt{2,\ e^{*}s3^{*}e-(s3^{*}e)} \wedge \mathtt{2];}\\ \mathtt{[\ (-1)^{*} \lt identity\ ... \gt +(1)^{*}s1} \wedge \mathtt{2,\ (-1)^{*} \lt identity\ ... \gt +(1)^{*}s2} \wedge \mathtt{2,}\\ \mathtt{(-1)^{*} \lt identity\ ... \gt +(1)^{*}s3} \wedge \mathtt{2,\ (-1)^{*} \lt identity\ ... \gt +(1)^{*}(s1^{*}s2)} \wedge \mathtt{3,}\\ \mathtt{(-1)^{*} \lt identity\ ... \gt +(1)^{*}(s2^{*}s3)} \wedge \mathtt{3,\ (-1)^{*} \lt identity\ ... \gt +(1)^{*}(s1^{*}s3) \wedge 2,}\\ \mathtt{(-1)^{*}e+(1)^{*}e} \wedge \mathtt{2,\ (1)^{*}s1^{*}e+(-1)^{*}e^{*}s1,\ (1)^{*}s2^{*}e+(-1)^{*}e^{*}s2,}\\ \mathtt{(1)^{*}e^{*}s3^{*}e+(-1)^{*}(e^{*}s3)} \wedge \mathtt{2,\ (1)^{*}e^{*}s3^{*}e+(-1)^{*}(s3^{*}e)} \wedge \mathtt{2\ ]}\\ \mathtt{gap \gt \ prefixrels: =\ [e,\ s1-o,\ s2-o,\ s3^{*}e-s3];}\\ \mathtt{[\ (1)^{*}e,\ (-1)^{*} \lt identity\ ... \gt +(1)^{*}s1,\ (-1)^{*} \lt identity\ ... \gt +(1)^{*}s2,\ (-1)^{*}s3+(1)^{*}s3^{*}e\ ]}\\ \mathtt{gap \gt \ PrintNPList(GBR.ts);}\\ \mathtt{s1} \wedge \mathtt{2\ -\ 1}\\ \mathtt{s2}\mathtt{\wedge\ 2\ -\ 1}\\ \mathtt{s3s1\ -\ s1s3}\\ \mathtt{s3}\mathtt{\wedge\ 2\ -\ 1}\\ \mathtt{es1\ -\ s1e}\\ \mathtt{es2\ -\ s2e}\\ \mathtt{e} \wedge \mathtt{2\ -\ e}\\ \mathtt{s2s1s2\ -\ s1s2s1}\\ \mathtt{s3s2s3\ -\ s2s3s2}\\ \mathtt{s3s2s1s3\ -\ s2s3s2s1}\\ \mathtt{s3es3e\ -\ es3e}\\ \mathtt{es3es3\ -\ es3e}\\ \mathtt{s3es3s2e\ -\ es3s2e}\\ \mathtt{s2s3s2es3e\ -\ s3s2es3e}\\ \mathtt{s3es3s2s1e\ -\ es3s2s1e}\\ \mathtt{es3s2es3s2\ -\ es3s2es3}\\ \mathtt{s2s3s2s1es3e\ -\ s3s2s1es3e}\\ \mathtt{s2s3s2es3s2e\ -\ s3s2es3s2e}\\ \mathtt{s2es3s2es3e\ -\ es3s2es3e}\\ \mathtt{s1s2s1s3s2es3e\ -\ s2s1s3s2es3e}\\ \mathtt{s2s3s2s1es3s2e\ -\ s3s2s1es3s2e}\\ \mathtt{s2s3s2es3s2s1e\ -\ s3s2es3s2s1e}\\ \mathtt{s2es3s2s1es3e\ -\ es3s2s1es3e}\\ \mathtt{es3s2s1es3s2s1\ -\ es3s2s1es3s2}\\ \mathtt{s1s2s1s3s2s1es3e\ -\ s2s1s3s2s1es3e}\\ \mathtt{s1s2s1s3s2es3s2e\ -\ s2s1s3s2es3s2e}\\ \mathtt{s1s2s1es3s2es3e\ -\ s2s1es3s2es3e}\\ \mathtt{s2s3s2s1es3s2s1e\ -\ s3s2s1es3s2s1e}\\ \mathtt{s2es3s2s1es3s2e\ -\ es3s2s1es3s2e}\\ \mathtt{s1s2s1s3s2s1es3s2e\ -\ s2s1s3s2s1es3s2e}\\ \mathtt{s1s2s1s3s2es3s2s1e\ -\ s2s1s3s2es3s2s1e}\\ \mathtt{s1s2s1es3s2s1es3e\ -\ s2s1es3s2s1es3e}\\ \mathtt{s1s3s2s1es3s2es3e\ -\ s3s2s1es3s2es3e}\\ \mathtt{s1s2s1s3s2s1es3s2s1e\ -\ s2s1s3s2s1es3s2s1e}\\ \mathtt{s1s2s1es3s2s1es3s2e\ -\ s2s1es3s2s1es3s2e}\\ \mathtt{s1s3s2s1es3s2s1es3e\ -\ s3s2s1es3s2s1es3e}\\ \mathtt{s1es3s2s1es3s2es3e\ -\ es3s2s1es3s2es3e}\\ \mathtt{s1s3s2s1es3s2s1es3s2e\ -\ s3s2s1es3s2s1es3s2e}$

We note that by $GBNP$ package program one can compute Gröbner-Shirshov basis of symmetric inverse monoids for small sizes, for example $I_4$ and $I_5$ . But there are no any other computer programs that compute a Gröbner-Shirshov basis for general size of symmetric inverse monoids. For this reason, it is worth to study and obtain a Gröbner-Shirshov basis for this important structure.

Example 5. The presentation of $I_4$ is as follows.

$\begin{eqnarray*} \left\langle {\varepsilon}, \sigma _1, \sigma _2, \sigma _{3} \right. &;& \left. \sigma _1^2 = \sigma _2^2 = \sigma _{3}^2 = 1, \: {\sigma _3}{\sigma _1} = {\sigma _1}{\sigma _3}, \:{\varepsilon ^2} = \varepsilon, \:\varepsilon {\sigma _1} = {\sigma _1}\varepsilon, \:\varepsilon {\sigma _2} = {\sigma _2}\varepsilon, \right.\\ && \left.{\sigma _3}{\sigma _{2}}\sigma _{3} = {\sigma _{2}}{\sigma _3}{\sigma _{2}}, {\sigma _2}{\sigma _{1}}\sigma _{2} = {\sigma _{1}}{\sigma _2}{\sigma _{1}}, \:{\sigma _{3}}\varepsilon {\sigma _{3}}\varepsilon = \varepsilon {\sigma _{3}}\varepsilon, \:\varepsilon {\sigma _{3}}\varepsilon{\sigma _{3}} = \varepsilon {\sigma _{3}}\varepsilon \right\rangle . \end{eqnarray*}$

We use deg-lex order induced by $\sigma _1 < \sigma _2 < \sigma _{3} < \varepsilon$ . By this ordering, a Gröbner-Shirshov basis for symmetric inverse monoid $I_4$ consists of the following $38$ relations.

$\begin{eqnarray*} &(1)&\:\sigma _1^2 = 1, \:\:\: \sigma _2^2 = 1, \:\:\: \sigma _{3}^2 = 1, \\ &(2)&\:{\sigma _3}{\sigma _1} = {\sigma _1}{\sigma _3}, \\ &(3)&\:{\varepsilon ^2} = \varepsilon, \\ &(4)&\:\varepsilon {\sigma _1} = {\sigma _1}\varepsilon, \:\:\:\varepsilon {\sigma _2} = {\sigma _2}\varepsilon, \\ &(5)&\: {\sigma _3}{\sigma _{2}}\sigma _{3} = {\sigma _{2}}{\sigma _3}{\sigma _{2}}, \:\:\: {\sigma _2}{\sigma _{1}}\sigma _{2} = {\sigma _{1}}{\sigma _2}{\sigma _{1}}, \:{\sigma _3}{\sigma _{2}}\sigma_1\sigma _{3} = {\sigma _{2}}{\sigma _3}{\sigma _{2}}\sigma_1, \\ &(6)&\:{\sigma _{3}}\varepsilon {\sigma _{3}}\varepsilon = \varepsilon {\sigma _{3}}\varepsilon, \:\:{\sigma _{3}}\varepsilon {\sigma _{3}}\sigma_2\varepsilon = \varepsilon {\sigma _{3}}\sigma_2\varepsilon, \:\: {\sigma _{3}}\varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon = \varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon, \\ &(7)&\:{\sigma _{2}}\varepsilon {\sigma _{3}}\sigma_2\varepsilon\sigma_3\varepsilon = \varepsilon {\sigma _{3}}\sigma_2\varepsilon\sigma_3\varepsilon, \:\: {\sigma _{2}}\varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\varepsilon = \varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\varepsilon, \\ &&{\sigma _{2}}\varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\sigma_2\varepsilon = \varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\sigma_2\varepsilon, \\ &(7')&\:\sigma_1{\sigma _{3}}\sigma_2\sigma_1\varepsilon {\sigma _{3}}\sigma_2\varepsilon\sigma_3\varepsilon = {\sigma _{3}}\sigma_2\sigma_1\varepsilon {\sigma _{3}}\sigma_2\varepsilon\sigma_3\varepsilon, \\ &&\sigma_1{\sigma _{3}}\sigma_2\sigma_1\varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\varepsilon = {\sigma _{3}}\sigma_2\sigma_1\varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\varepsilon, \\ &&\sigma_1{\sigma _{3}}\sigma_2\sigma_1\varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\sigma_2\varepsilon = {\sigma _{3}}\sigma_2\sigma_1\varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\sigma_2\varepsilon, \\ &&\sigma_1\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon {\sigma _{3}}\sigma_2\varepsilon\sigma_3\varepsilon = \varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon {\sigma _{3}}\sigma_2\varepsilon\sigma_3\varepsilon, \\ &(8)&\:\varepsilon {\sigma _{3}}\varepsilon{\sigma _{3}} = \varepsilon {\sigma _{3}}\varepsilon, \:\: \varepsilon {\sigma _{3}}\sigma_2\varepsilon{\sigma _{3}}\sigma_2 = \varepsilon {\sigma _{3}}\sigma_2\varepsilon{\sigma _{3}}, \\ &&\varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\sigma_1 = \varepsilon {\sigma _{3}}\sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2, \\ &(9)&\:{\sigma _{2}}\sigma_3\sigma_2\varepsilon{\sigma _{3}}\varepsilon = {\sigma _{3}}\sigma_2\varepsilon{\sigma _{3}}\varepsilon, \:\: {\sigma _{2}}\sigma_3\sigma_2\sigma_1\varepsilon{\sigma _{3}}\varepsilon = {\sigma _{3}}\sigma_2\sigma_1\varepsilon{\sigma _{3}}\varepsilon, \\ &&{\sigma _{2}}\sigma_3\sigma_2\varepsilon{\sigma _{3}}\sigma_2\varepsilon = {\sigma _{3}}\sigma_2\varepsilon{\sigma _{3}}\sigma_2\varepsilon, \:\: \sigma_1{\sigma _{2}}\sigma_1\sigma_3\sigma_2\varepsilon{\sigma _{3}}\varepsilon = {\sigma _{2}}\sigma_1\sigma_3\sigma_2\varepsilon{\sigma _{3}}\varepsilon, \\ &&\sigma_2{\sigma _{3}}\sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\varepsilon = {\sigma _{3}}\sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\varepsilon, \:\: \sigma_2{\sigma _{3}}\sigma_2\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon = {\sigma _{3}}\sigma_2\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon, \\ &&\sigma_1\sigma_2\sigma_1{\sigma _{3}}\sigma_2\sigma_1\varepsilon{\sigma _{3}}\varepsilon = {\sigma _{2}}\sigma_1\sigma_3{\sigma _{2}}\sigma_1\varepsilon{\sigma _{3}}\varepsilon, \: \sigma_1\sigma_2\sigma_1{\sigma _{3}}\sigma_2\varepsilon{\sigma _{3}}\sigma_2\varepsilon = {\sigma _{2}}\sigma_1\sigma_3{\sigma _{2}}\varepsilon{\sigma _{3}}\sigma_2\varepsilon, \\ &&\sigma_2\sigma_3\sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon = \sigma_3\sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon, \: \sigma_1\sigma_2\sigma_1{\sigma _{3}}\sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\varepsilon = {\sigma _{2}}\sigma_1\sigma_3{\sigma _{2}}\sigma_1\varepsilon{\sigma _{3}}\sigma_2\varepsilon, \\ &&\sigma_1\sigma_2\sigma_1{\sigma _{3}}\sigma_2\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon = {\sigma _{2}}\sigma_1\sigma_3{\sigma _{2}}\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon, \\ &&\sigma_1\sigma_2\sigma_1{\sigma _{3}}\sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon = {\sigma _{2}}\sigma_1\sigma_3{\sigma _{2}}\sigma_1\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon, \\ &(10)&\: \sigma_1\sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\varepsilon\sigma_3\varepsilon = \sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\varepsilon\sigma_3\varepsilon, \\ &&\sigma_1\sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\varepsilon = \sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\varepsilon, \\ &&\sigma_1\sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\sigma_2\varepsilon = \sigma_2\sigma_1\varepsilon{\sigma _{3}}\sigma_2\sigma_1\varepsilon\sigma_3\sigma_2\varepsilon. \end{eqnarray*}$

5. Conclusions

The idea of Gröbner-Shirshov basis theory plays a significant role in several fields of mathematics (algebra, graph theory, knot theory), computer sciences (computational algebra) and information sciences. From algebraic way the method Gröbner-Shirshov basis theory gives a new algorithm to obtain normal forms of elements of groups, monoids, semigroups and various type of algebras, and hence a new algorithm to solve the word problem in these algebraic structures.

In this study, we obtained a Gröbner-Shirshov basis for a special type of braid monoids, namely the symmetric inverse monoid $I_n$ , in terms of the dex-leg ordering on the related elements of monoid. As known symmetric inverse monoids are partial bijections and they are very well known and important in combinatorics. By taking into account the Gröbner-Shirshov basis, we achieved the normal form structure of this important monoid. This normal form gave us the solution of the word problem. At the final part of this study, we presented an application of our main result which find out a Gröbner-Shirshov basis for the symmetric inverse monoid $I_4$ by using a package program, $GBNP$ , in GAP. Since $GBNP$ is a restricted package program in point of size of symmetric inverse monoids it is worth to study and obtain a Gröbner-Shirshov basis for general size of this important structure.

In the future, the result of this work can be expanded to some other algebraic, computational structures and associated to graph theory, growth, Hilbert series and knot theory.

Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No (130-211-D1439). The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors would like to thank to the referees for their suggestions and valuable comments.

Conflict of interest

The authors declare that they have no conflict of interest.

References

[1]	FAO (2018) The state of world fisheries and aquaculture 2018: contributing to food security and nutrition for all, Rome.
[2]	Tran N, Rodriguez UP, Chan CY, et al. (2017) Indonesian aquaculture futures: An analysis of fish supply and demand in Indonesia to 2030 and role of aquaculture using the Asia Fish model. Mar Policy 79: 25-32. https://doi.org/10.1016/j.marpol.2017.02.002 doi: 10.1016/j.marpol.2017.02.002
[3]	Senff P, Partelow S, Indriana LF, et al. (2018) Improving pond aquaculture production on Lombok, Indonesia. Aquaculture 497: 64-73. https://doi.org/10.1016/j.aquaculture.2018.07.027 doi: 10.1016/j.aquaculture.2018.07.027
[4]	Pouil S, Samsudin R, Slembrouck J, et al. (2019) Nutrient budgets in a small-scale freshwater fish pond system in Indonesia. Aquaculture 504: 267-274. https://doi.org/10.1016/j.aquaculture.2019.01.067 doi: 10.1016/j.aquaculture.2019.01.067
[5]	CDSI, Central Data Statistic Indonesia (2016) Ministry of Marine and Fisheries Republic of Indonesia. CSI. Center for data, statistics and information (CSI). Marine and Fisheries in Figures. Ministry of Marine and Fisheries Republic of Indonesia. 2018. https://kkp.go.id/wp-content/uploads/2018/01/KKP-Dirjen-PDSPKP-FMB-Kominfo-19-Januari-2018.pdf
[6]	Syandri H, Azrita, Junaidi, et al. (2015) Social Status of the fish-farmers of floating-net-cages in Lake Maninjau, Indonesia. J Aquac Res Dev 7: 1.
[7]	Rimmer MA, Sugama, Rakhmawati K, et al. (2013) A review and SWOT analysis of aquaculture development in Indonesia. Rev Aquac 5: 255-279. https://doi.org/10.1111/raq.12017 doi: 10.1111/raq.12017
[8]	Henriksson PJG, Tran N, Mohan CV, et al. (2017) Indonesian aquaculture futures evaluating environmental and socio-economic potentials and limitations. J Clean Prod 162: 1482-1490. https://doi.org/10.1016/j.jclepro.2017.06.133 doi: 10.1016/j.jclepro.2017.06.133
[9]	Syandri H, Azrita, Mardiah A, et al. (2018) Nitrogen and phosphorus waste production from different fish species cultured at floating net cages in Lake Maninjau, Indonesia. Asian J Sci Res 11: 287-294. https://doi.org/10.3923/ajsr.2018.287.294 doi: 10.3923/ajsr.2018.287.294
[10]	Syandri H, Azrita, Sumiarsih E, et al. (2021) Nutrient loading and farm characteristics of giant gourami fish aquaculture systems in Lake Maninjau, Indonesia: Basic knowledge of production performance. F1000Research, 10: 378. https://doi.org/10.12688/f1000research.52613.1.
[11]	Syandri H, Azrita, Niagara, et al. (2016) Trophic Status and load capacity of water pollution waste fish culture with floating net cages in Maninjau Lake, Indonesia. Eco Env Cons 22: 469-476. http://www.envirobiotechjournals.com/article_abstract.php?aid = 6717&iid = 207&jid = 3
[12]	Ahmed N, Thompson S (2019) The blue dimensions of aquaculture: A global synthesis. Sci. Total Environ 652: 851-861. https://doi.org/10.1016/j.scitotenv.2018.10.163. doi: 10.1016/j.scitotenv.2018.10.163
[13]	Aura CM, Musa S, Yongo E, et al. (2017) Integration of mapping and socio-economic status of cage culture: Toward balancing lake-use and culture fisheries in Lake Victoria Kenya. Aquac Res 49: 532-545. https://doi.org/10.1111/are.13484 doi: 10.1111/are.13484
[14]	CSA, Central Statistics Agency (2018) Agam District, West Sumatera Province, Indonesia. https://agamkab.bps.go.id/
[15]	Syandri H, Junaidi, Azrita, et al. 2014. State of aquatic resources Maninjau Lake West Sumatra Province, Indonesia. J Ecol Environ Sci 1: 109-114.
[16]	Central Bureau of Statistics (CBS) of Agam Regency. (2017). Agam Regency in Figures 2017.
[17]	Opiyo MA, Marijani E, Muendo P, et al. (2018) A review of aquaculture production and health management practices of farmed fish in Kenya. Int J Vet Sci Med 6: 141-148. https://doi.org/10.1016/j.ijvsm.2018.07.001 doi: 10.1016/j.ijvsm.2018.07.001
[18]	Mbowa S, Odokonyero T, Munyaho AT, et al. (2017) Harnessing ﬂoating cage technology to increase fish production in Uganda, Research Series No. 138.
[19]	Hasimuna OJ, Maulu S, Monde C, et al. (2019) Cage aquaculture production in Zambia: Assessment of opportunities and challenges on Lake Kariba, Siavonga district. Egypt. J Aquat Res 45: 281-285. https://doi.org/10.1016/j.ejar.2019.06.007
[20]	Thongprajukaew K, Kovitvadhi S, Kovitvadhi U, et al. (2017) Effects of feeding frequency on growth performance and digestive enzyme activity of sex-reversed Nile tilapia, Oreochromis niloticus (Linnaeus, 1758). Agric Nat Resources 51: 292-298. https://doi.org/10.1016/j.anres.2017.04.005 doi: 10.1016/j.anres.2017.04.005
[21]	Prem R, Tewari VK (2020) Development of human-powered fish feeding machine for freshwater aquaculture farms of developing countries. Aquac Eng 88: 102028. https://doi.org/10.1016/j.aquaeng.2019.102028 doi: 10.1016/j.aquaeng.2019.102028
[22]	Syandri H, Azrita, Mardiah A, et al. (2020) Water Quality Status and Pollution Waste Load from Floating Net Cages at Maninjau Lake, West Sumatera Indonesia. IOP Conf Ser Earth Environ Sci 430: 01203. https://doi.org/10.1088/1755-1315/430/1/012031 doi: 10.1088/1755-1315/430/1/012031
[23]	Ji B, Qin H, Guo S, et al. (2018) Bacterial communities of four adjacent fresh lakes at different trophic status. Ecotoxicol Environ Saf 157: 388-394. https://doi.org/10.1016/j.ecoenv.2018.03.086 doi: 10.1016/j.ecoenv.2018.03.086
[24]	Burgos MJG, Romero JL, Pulido RP, et al. (2018) Analysis of potential risks from the bacterial communities associated with air-contact surfaces from tilapia (Oreochromis niloticus) fish farming. Environ Res 160: 385-390. https://doi.org/10.1016/j.envres.2017.10.021 doi: 10.1016/j.envres.2017.10.021
[25]	Zhao M, Xie S, Zhu X, et al. (2006) Effect of dietary cyanobacteria on growth and accumulation of microcystins in Nile tilapia (Oreochromis niloticus). Aquaculture 261: 960-966. https://doi.org/10.1016/j.aquaculture.2006.08.019 doi: 10.1016/j.aquaculture.2006.08.019
[26]	de Oliveira TF, Queiroz GA, Teixeira JP, et al. (2018) Recurrent Streptococcus agalactiae infection in Nile tilapia (Oreochromis niloticus) treated with florfenicol. Aquaculture 493: 51-60. https://doi.org/10.1016/j.aquaculture.2018.04.037 doi: 10.1016/j.aquaculture.2018.04.037
[27]	Nicholson P, Mon-on N, Jaemwimol P, et al. (2019) Co-infection of tilapia lake virus and Aeromonas hydrophila synergistically increased mortality and worsened the disease severity in tilapia (Oreochromis spp.). Aquaculture 520: 734746. https://doi.org/10.1016/j.aquaculture.2019.734746
[28]	Hounmanou YMG, Mdegela RH, Dougnon TV, et al. (2018) Tilapia lake virus threatens tilapiines farming and food security: Socio-economic challenges and preventive measures in Sub- Saharan Africa. Aquaculture 493: 123-129. https://doi.org/10.1016/j.aquaculture.2018.05.001 doi: 10.1016/j.aquaculture.2018.05.001
[29]	Ferguson HW, Kabuusu R, Beltran S, et al. (2014) Syncytial hepatitis of farmed tilapia, Oreochromis niloticus (L.): a case report. J Fish Dis 37: 583-589. https://doi.org/10.1111/jfd.12142 doi: 10.1111/jfd.12142
[30]	Tsofack JEK, Zamostiano R, Watted S, et al. (2017) Detection of Tilapia Lake Virus in Clinical Samples by Culturing and Nested Reverse Transcription-PCR. J Clin Microbiol 55: 759-767. https://doi.org/10.1128/JCM.01808-16 doi: 10.1128/JCM.01808-16
[31]	Amal MNA, Koh CB, Nurliyana M, et al. (2018) A case of natural co-infection of Tilapia Lake Virus and Aeromonas veronii in a Malaysian red hybrid tilapia (Oreochromis niloticus×O. mossambicus) farm experiencing high mortality. Aquaculture 485: 12-16. https://doi.org/10.1016/j.aquaculture.2017.11.019 doi: 10.1016/j.aquaculture.2017.11.019
[32]	Mugimba KK, Chengula, AA Wamala S, et al. (2018) Detection of tilapia lake virus (TiLV) infection by PCR in farmed and wild Nile tilapia (Oreochromis niloticus) from Lake Victoria. J Fish Dis https://doi.org/10.1111/jfd.12790
[33]	Ali H, Rahman MM, Murshed-e-Jaha K, et al. (2018) Production economics of striped catfish (Pangasianodon hypophthalmus, Sauvage, 1878) farming under polyculture system in Bangladesh. Aquaculture 491: 281-390. https://doi.org/10.1016/j.aquaculture.2017.12.004 doi: 10.1016/j.aquaculture.2017.12.004
[34]	Verdegem MCJ, Bosma RH (2009) Water withdrawal for brackish and inland aquaculture and options to produce more fish in ponds with present water use. Water Policy 11: 52-68 Supplement 1. https://doi.org/10.2166/wp.2009.003 doi: 10.2166/wp.2009.003
[35]	David GS, Carvalho ED, Lemos D, et al. (2015) Ecological carrying capacity for intensive Tilapia (Oreochromis niloticus) cage aquaculture in a large hydroelectrical reservoir in Southeastern Brazil. Aquac Eng 66: 30-40. https://doi.org/10.1016/j.aquaeng.2015.02.003 doi: 10.1016/j.aquaeng.2015.02.003
[36]	Musinguzi L, Lugya J, Rwezawula P, et al. (2019) The extent of cage aquaculture, adherence to best practices and reﬂections for sustainable aquaculture on African inland waters. J Great Lakes Res 45: 1340-1347. https://doi.org/10.1016/j.jglr.2019.09.011 doi: 10.1016/j.jglr.2019.09.011
[37]	Jamu D, Banda M, Njaya F, et al. (2011) Challenges to sustainable management of the lakes of Malawi. J Great Lakes Res 37: 3-14. https://doi.org/10.1016/j.jglr.2010.11.017 doi: 10.1016/j.jglr.2010.11.017
[38]	Jia P, Zhang W, Liu Q, et al. (2013) Lake fisheries in China: Challenges and opportunities. Fish Res 140: 66-72. https://doi.org/10.1016/j.fishres.2012.12.007 doi: 10.1016/j.fishres.2012.12.007
[39]	Asche F, Roll KH, Tveteras R, et al. (2009) Economic inefficiency and environmental impact: An application to aquaculture production. J Environ Econ Manage 58: 93-105. https://doi.org/10.1016/j.jeem.2008.10.003 doi: 10.1016/j.jeem.2008.10.003
[40]	Lindim C, Becker A, Grüneberg B, et al. (2015) Modelling the effects of nutrient loads reduction and testing the N and P control paradigm in a German shallow lake. Aquac Eng 82: 418-457. https://doi.org/10.1016/j.ecoleng.2015.05.009 doi: 10.1016/j.ecoleng.2015.05.009
[41]	Syandri H, Azrita, Junaidi, et al. (2017) Levels of available nitrogen-phosphorus before and after fish mass mortality in Maninjau Lake of Indonesia. J Fish Aquat Sci 12: 191-196. https://doi.org/10.3923/jfas.2017.191.196 doi: 10.3923/jfas.2017.191.196
[42]	Du H, Chen Z, Mao G, et al. (2019) Evaluation of eutrophication in freshwater lakes: A new non-equilibrium statistical approach. Ecol Indic 102: 686-692. https://doi.org/10.1016/j.ecolind.2019.03.032 doi: 10.1016/j.ecolind.2019.03.032
[43]	Ni Z, Wu X, Li L, et al. (2017) Pollution control and in situ bioremediation for lake aquaculture using an ecological dam. J Clean Prod 172: 2256-2265. https://doi.org/10.1016/j.jclepro.2017.11.185 doi: 10.1016/j.jclepro.2017.11.185
[44]	Young N, Brattland C, Digiovanni, et al. (2019) Limitations to growth: Social-ecological challenges to aquaculture development in five wealthy nations. Mar Policy 104: 216-224. https://doi.org/10.1016/j.marpol.2019.02.022 doi: 10.1016/j.marpol.2019.02.022
[45]	Holden JS, Collicutt B, Covernton G, et al. (2019) Synergies on the coast: Challenges facing shellfish aquaculture development on the central and north coast of British Columbia. Mar Policy 101: 108-117. https://doi.org/10.1016/j.marpol.2019.01.001. doi: 10.1016/j.marpol.2019.01.001
[46]	Weitzman J (2019) Applying the ecosystem services concept to aquaculture: A review of approaches, definitions, and uses. Ecosyst Serv 35: 194-206. https://doi.org/10.1016/j.ecoser.2018.12.009. doi: 10.1016/j.ecoser.2018.12.009

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