On the Caputo-Hadamard fractional IVP with variable order using the upper-lower solutions technique

1.   Introduction

A semiring is an algebra with two associative binary operations $+, \cdot$, in which $+$ is commutative and $\cdot$ distributive over $+$ from the left and right. Such an algebra is a common generalization of both rings and distributive lattices. It has broad applications in information science and theoretical computer science (see ^[5,6]). In this paper, we shall investigate some small-order semirings which will play a crucial role in subsequent follows.

The semiring A with addition and multiplication table (see ^[12])

The semiring B with addition and multiplication table (see ^[4])

Eight 2-element semirings with addition and multiplication table (see ^[2])

For any semiring $S$, we denote by $S^{0}$ the semiring obtained from $S$ by adding an extra element $0$ and where $a = 0+a = a+0, 0 = 0a = a0$ for every $a\in S$. For any semiring $S$, $S^{\ast}$ will denote the (multiplicative) left-right dual of $S$. In 2005, Pastijn et al. ^[4,9,10] studied the semiring variety generated by $B^{0}$ and $(B^{0})^{\ast}$ (Denoted by ${\bf{Sr}}(2, 1)$). They showed that the lattice of subvarieties of this variety is distributive and contains 78 varieties precisely. Moreover, each of these is finitely based. In 2016, Ren et al. ^[12,13] studied the variety generated by $B^{0}, (B^{0})^{\ast}$ and $A^{0}$ (Denoted by ${\bf{Sr}}(3, 1)$). They showed that the lattice of subvarieties of this variety is distributive and contains 179 varieties precisely. Moreover, each of these is finitely based. From ^[4,10], we have ${\bf{HSP}}(L_{2}, R_{2}, M_{2}, D_{2})\subsetneqq{\bf{HSP}}(B^{0}, (B^{0})^{\ast}).$ So

In 2016, Shao and Ren ^[15] studied the variety ${\bf{HSP}}(L_{2}, R_{2}, M_{2}, D_{2}, Z_2, W_2)$ (Denoted by ${\bf{S}}_6$). They showed that the lattice of subvarieties of this variety is distributive and contains 64 varieties precisely. Moreover, each of these is finitely based. Recently, Ren and Zeng ^[14] studied the variety generated by $B^{0}, (B^{0})^{\ast}, N_{2}, T_{2}$. They proved that the lattice of subvarieties of this variety is a distributive lattice of order 312 and that each of its subvarieties is finitely based. In ^[16], Wang, Wang and Li studied the variety generated by $B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}$. They proved that the lattice of subvarieties of this variety is a distributive lattice of order 716 and that each of its subvarieties is finitely based. It is easy to check

So semiring variety ${\bf{HSP}}{(B^{0}, (B^{0})^{\ast}, A^0, N_{2}, T_{2})}$ is a proper subvariety of the semiring variety ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$. The main purpose of this paper is to study the variety ${\bf{HSP}}(B^{0}, (B^{0})^{\ast},$ $A^{0}, N_{2}, T_{2}, Z_2, W_2)$. We show that the lattice of subvarieties of this variety is a distributive lattice of order 2327. Moreover, we show this variety is hereditarily finitely based.

2.   Preliminaries

By a variety we mean a class of algebras of the same type that is closed under subalgebras, homomorphic images and direct products (see ^[11]). Let ${\bf{W}}$ be a variety, let $\cal{L}({\bf{W}})$ denote the lattice of subvarieties of ${\bf{W}}$ and let ${\rm{Id}}_{\bf W}(X)$ denote the set of all identities defining ${\bf{W}}$. If ${\bf{W}}$ can be defined by finitely many identities, then we say that $\bf W$ is finitely based (see ^[14]). In other words, $\bf W$ is said to be finitely based if there exists a finite subset $\Sigma$ of ${\rm{Id}}_{\bf W}(X)$ such that for any $p\approx q\in {\rm{Id}}_{\bf W}(X)$, $p\approx q$ can be derived from $\Sigma$, i.e., $\Sigma\vdash p\approx q$. Otherwise, we say that ${\bf W}$ is nonfinitely based. Recall that ${\bf{W}}$ is said to be hereditarily finitely based if all members of ${\cal L}({\bf{W}})$ are finitely based. If a variety ${\bf{W}}$ is finitely based and ${\cal L}({\bf{W}})$ is a finite lattice, then ${\bf{W}}$ is hereditarily finitely based (see ^[14]).

A semiring is called an additively idempotent semiring (ai-semiring for short) if its additive reduct is a semilattice, i.e., a commutative idempotent semigroup. It is also called a semilattice-ordered semigroup (see ^[3,8,12]). The variety of all semirings (resp. all ai-semirings) is denoted by ${\bf{SR}}$ (resp. AI). Let $X$ denote a fixed countably infinite set of variables and $X^{+}$ the free semigroup on $X$ (see ^[8]). A semiring identity (SR-identity for short) is an expression of the form $u\approx v$, where $u$ and $v$ are terms with $u = u_1+\cdots +u_k$, $v = v_1+\cdots+v_\ell$, where $u_i, v_j\in X^+$. Let $\underline{k}$ denote the set $\{1, 2, \ldots, k\}$ for a positive integer $k$, $\Sigma$ be a set of identities which include the identities determining AI (Each identity in $\Sigma$ is called an AI-identity) and $u\approx v$ be an AI-identity. It is easy to check that the ai-semiring variety defined by $u\approx v$ coincides with the ai-semiring variety defined by the identities $u\approx u+v_{j}, v\approx v+u_i, i\in \underline{k}, j\in \underline{\ell}$. Thus, in order to show that $u\approx v$ is derivable from $\Sigma$, we only need to show that $u\approx u+v_{j}, v\approx v+u_i, i\in \underline{k}, j\in \underline{\ell}$ can be derived from $\Sigma$ (see ^[9]).

To solve the word problem for the variety ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$, the following notions and notations are needed. Let $q$ be an element of $X^+$. Then

● the head of $q$, denoted by $h(q)$, is the first variable occurring in $q$;

● the tail of $q$, denoted by $t(q)$, is the last variable occurring in $q$;

● the content of $q$, denoted by $c(q)$, is the set of variables occurring in $q$;

● the length of $q$, denoted by $|q|$, is the number of variables occurring in $q$ counting multiplicities;

● the initial part of $q$, denoted by $i(q)$, is the word obtained from $q$ by retaining only the first occurrence of each variable;

● the final part of $q$, denoted by $f(q)$, is the word obtained from $q$ by retaining only the last occurrence of each variable;

● $r(q)$ denotes set $\{x\in X\, |\, \mbox{the number of occurrences of $x$ in $q$ is odd}\}$.

By [13,Lemma 1.2], ${\bf{Sr}}(3, 1)$ satisfies the identity $p\approx q$ if and only if $(i(p), f(p), r(p)) = (i(q), f(q),$ $r(q))$. This result will be used later without any further notice. The basis for each one of $N_2, T_2, Z_2, W_2$ can be found from ^[2] (See Table 1).

By [15,Lemma 1.1] and the Table 1, we have

Lemma 2.1. Let $u\approx v$ be a nontrivial ${\bf{SR}}$-identity, where $u = u_1+u_2+\cdots+u_m$, $v = v_1+v_2+\cdots+v_n$, $u_i, v_j\in X^+$, $i\in\underline{m}, j\in\underline{n}$. Then

$\textrm{(i)}$ $N_2\models u\approx v\; {if\; and\; only\; if}\; \{u_{i}\in u\, |\, |u_{i}| = 1\} = \{v_{i}\in v\, |\, |v_{i}| = 1\}$;

$\textrm{(ii)}$ $T_2\models u\approx v\; {if\; and\; only\; if}\; \{u_{i}\in u\, |\, |u_{i}|\geq 2\}\neq\phi, \, \{v_{i}\in v\, |\, |v_{i}|\geq 2\}\neq\phi$;

$\textrm{(iii)}$ $Z_2\models u\approx v\; {if\; and\; only\; if}\; (\forall x\in X)\, u\neq x, \, v\neq x$;

$\textrm{(iv)}$ $W_2\models u\approx v\; {if\; and\; only\; if}\; m = n = 1, c(u_{1}) = c(v_{1})\; {or}\; m, n\geq2$.

Suppose that $u = u_1+\cdots+u_m, u_i\in X^+, i\in \underline{m}$. Let $1$ be a symbol which is not in $X$ and $Y$ an arbitrary subset of $\bigcup^{i = m}_{i = 1}c(u_1)$. For any $u_i$ in $u$, if $c(u_i)\subseteq Y$, put $h_Y(u_i) = 1$. Otherwise, we shall denote by $h_Y(u_i)$ the first variable occurring in the word obtained from $u_i$ by deleting all variables in $Y$. The set $\{h_{Y}(u_i)|u_i\in u\}$ is written $H_Y(u)$. Dually, we have the notations $t_{Y}(u_i)$ and $T_{Y}(u_i)$. In particular, if $Y = \emptyset$, then $h_{Y}(u_i) = h(u_i)$ and $t_{Y}(u_i) = t(u_i)$. Moreover, if $c(u_i)\cap Y\neq\emptyset$ for every $u_i$ in $u$, then we write $D_Y(u) = \emptyset$. Otherwise, $D_Y(u)$ is the sum of all terms $u_i$ in $u$ such that $c(u_i)\cap Y = \emptyset$. By [13,Lemma 2.3 and 2.11] and [4,Lemma 2.4 and its dual,Lemma 2.5 and 2.6], we have

Lemma 2.2. Let $u\approx u+q$ be an AI-identity, where $u = u_{1}+\cdots+u_{m}, u_{i}, q\in X^{+}, i\in \underline{m}$. If $u\approx u+q$ holds in ${\bf{Sr}}(3, 1)$, then

$\textrm{(i)}$ for every $Z\subseteq \bigcup^{i = m}_{i = 1}c(u_i)\backslash c(q)$, there exists $p_1$ in $X^+$ with $r(p_1) = r(q)$ and $c(q)\subseteq c(p_1)\subseteq \bigcup^{i = k}_{i = 1}c(u_i)$ such that $D_Z(u)\approx D_Z(u)+p_1$ holds in ${\bf{Sr}}(3, 1)$, where $D_Z(u) = u_1+\cdots +u_k$.

$\textrm{(ii)}$ for every $Y\subseteq Z = \bigcup^{i = m}_{i = 1}c(u_i)\backslash c(q)$, $H_Y(D_Z(u)) = H_Y(D_Z(u)+p_1)$ and $T_Y(D_Z(u)) = T_Y(D_Z(u)+p_1)$.

Throughout this paper, $u\overset{(3.1), (3.2), \cdots}\approx v$ denotes the identity $u\approx v$ can be derived from the identities $(3.1), (3.2), \cdots$ and the identities determining ${\bf{SR}}$. For other notations and terminology used in this paper, the reader is referred to ^{[1,4,7,13,15]}.

3.   Equational basis of ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$

In this section, we shall show that the variety ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$ is finitely based. Indeed, we have

Theorem 3.1. The semiring variety ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$ is determined by (3.1)–(3.12),

Proof. From ^[13] and Lemma 2.1, we know that both ${\bf{Sr}}{(3, 1)}$ and ${\bf{HSP}}(N_2, T_2, Z_{2}, W_{2})$ satisfy identities (3.1)–(3.12) and so does ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$.

Next, we shall show that every identity that holds in ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$ can be derived from (3.1)–(3.12) and the identities determining $\textbf{SR}$. Let $u\approx v$ be such an identity, where $u = u_1+u_2+\cdots+u_m$, $v = v_1+v_2+\cdots+v_n$, $u_i, v_j\in X^+$, $1\leq i\leq m, 1\leq j\leq n.$ By Lemma 2.1 (ⅳ), we only need to consider the following two cases:

Case 1. $m = n = 1$ and $c(u_{1}) = c(v_{1})$. From ${\bf{Sr}}(3, 1), T_2, Z_2\models u_{1}\approx v_{1},$ it follows that $(i(u_{1}), f(u_{1}), r(u_{1})) = (i(v_{1}), f(v_{1}), r(v_{1}))$, $|u_{1}|\geq 2$ and $|v_{1}|\geq2$. Hence $u_{1}\overset{(3.1)\sim(3.6)}\approx v_{1}$.

Case 2. $m, n\geq2.$ It is easy to verify that $u\approx v$ and the identity $(3.12)$ imply the identities $u\approx u+v_j,$ $v\approx v+u_i$ for all $i, j$ such that $1\leq i\leq m, 1\leq j\leq n.$ Conversely, the latter $m+n$ identities imply $u\approx u+v\approx v$. Thus, to show that $u\approx v$ is derivable from (3.1)–(3.12) and the identities determining $\textbf{SR}$, we need only show that the simpler identities $u\approx u+v_j,$ $v\approx v+u_i$ for all $i, j$ such that $1\leq i\leq m, 1\leq j\leq n.$ Hence we need to consider the following two cases:

Case 2.1. $u\approx u+q,$ where $|q| = 1$. Since $N_2\models u\approx u+q,$ there exists $u_s = q$. Thus $u+q\approx u'+u_s+q\approx u'+u_s+u_s\overset{(3.12)}\approx u'+u_s\approx u.$

Case 2.2. $u\approx u+q,$ where $|q|\geq 2$. Since $u\approx u+q$ holds in $T_{2}$, it follows from Lemma 2.1 (ⅱ) that there exists $u_{i}$ in $u$ such that $u_{i} > 1$. Put $Z = (\bigcup_{i = 1}^{i = m}c(u_{i}))\backslash c(q)$. Assume that $D_{Z}(u) = u_{1}+\cdots+u_{k}$. Then $\bigcup_{i = 1}^{i = k}c(u_{i}) = c(q)$. By Lemma 2.2 (ⅰ), there exists $p_1\in X^+$ such that $r(p_1) = r(q)$ and $c(q)\subseteq c(p_1)\subseteq \bigcup_{i = 1}^{i = k}c(u_{i})$. Moreover,

Write $p = p^{3}_{1}u^{2}_{1}u^{2}_{2}\cdots u^{2}_{k}$. Thus $c(p) = c(q)$, $r(p) = r(q)$ and we have derived the identity

Due to $|p| > 1$, it follows that (3.4) implies the identity

Suppose that $i(q) = x_{1}x_{2}\cdots x_{\ell}$. We shall show by induction on $j$ that for every $1\leq j\leq\ell$, $u\approx u+x^{2}_{1}x^{2}_{2}\cdots x^{2}_{\ell}p$ is derivable from (3.1)–(3.11) and the identities defining SR.

From Lemma 2.1 (ⅱ), there exists $u_{i_{1}}$ in $D_Z(u)$ with $c(u_{i_{1}})\subseteq c(q)$ such that $h(u_{i_{1}}) = h(q) = x_1$. Furthermore,

Therefore

Assume that for some $1 < j\leq\ell$,

is derivable from (3.1–3.12) and the identities defining SR. By Lemma 2.1 (ⅱ), there exists $u_i$ in $D_Z(u)$ with $c(u_i)\subseteq c(q)$ such that $u_{i} = u_{i_{1}}x_{j}u_{i_{2}}$ and $c(u_{i_{1}})\subseteq\{x_1, x_2, \ldots, x_{j-1}\}.$ It follows that

Consequently

Moreover, we have

Hence $u\approx u+x^{2}_1x^{2}_2\cdots x^{2}_{j-1}x^{2}_{j}p$. Using induction we have

Dually,

Thus

It follows that $u\approx u+q$.

4.   The lattice ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2))$

In this section we characterize the lattice ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2))$. Throughout this section, $t(x_1, \ldots, x_n)$ denotes the term $t$ which contains no other variables than $x_1, \ldots, x_n$ (but not necessarily all of them). Let $S \in{\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$ and let $E^{+}(S)$ denote the set $\{a\in S\, |\, a+a = a\}$, where any element of $E^{+}(S)$ is said to be an additive idempotent of $(S, +)$. Notice that ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$ satisfies the identities

By (4.1) and (4.2), it is easy to verify that $E^{+}(S) = \{a+a\, |\, a\in S\}$ forms a subsemiring of $S$. To characterize the lattice ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2))$, we need to consider the following mapping

It is easy to prove that $\varphi({\bf{W}}) = \{E^{+}(S)\, |\, S\in {\bf{W}}\}$ for each member ${\bf{W}}$ of ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2},$ $Z_2, W_2))$. If ${\bf{W}}$ is the subvariety of ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2})$ determined by the identities

then ${\widehat{\bf{W}}}$ denotes the subvariety of ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$ determined by the identities

Lemma 4.1. ^[16] The ai-semiring variety ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2})$ is determined by the identities (3.1)–(3.11) and ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}))$ is a distributive lattice of order 716.

Lemma 4.2. Let ${\bf{W}}$ be a member of ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}))$. Then, ${\widehat{\bf{W}}} = {\bf{W}}\vee{\bf{HSP}}(Z_2, W_2).$

Proof. Since ${\bf{W}}$ satisfies the identities (4.4), it follows that ${\bf{W}}$ is a subvariety of ${\widehat{\bf{W}}}$. Both $Z_2$ and $W_2$ are members of ${\widehat{\bf{W}}}$ and so ${\bf{W}}\vee{\bf{HSP}}(Z_2, W_2)\subseteq{\widehat{\bf{W}}}$. To show the converse inclusion, it suffices to show that every identity that is satisfied by ${\bf{W}}\vee{\bf{HSP}}(Z_2, W_2)$ can be derived by the identities holding in ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$ and

if ${\bf{W}}$ is the subvariety of ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}))$ determined by $u_i(x_{i_1}, \ldots, x_{i_{n}})\approx v_i(x_{i_1}, \ldots, x_{i_{n}}),$ $i\in \underline{k}$. Let $u\approx v$ be such an identity, where $u = u_{1}+u_{2}+\cdots+u_{m}, v = v_{1}+v_{2}+\cdots+v_{n}, u_{i}, v_{j}\in X^{+}, 1\leq i\leq m, 1\leq j\leq n$. By Lemma 2.1 (8), we only need to consider the following two cases.

Case 1. $m, n\geq 2$. By identity (3.12), ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$ satisfies the identities

Since $u\approx v$ holds in ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2})$, we have that it is derivable from the collection $\Sigma$ of $u_i\approx v_i, i\in \underline{k}$ and the identities determining ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2})$. From [1,Exercise Ⅱ.14.11], it follows that there exist $t_1, t_2, \ldots, t_\ell\in P_{f}(X^+)$ such that

● $t_1 = u, t_\ell = v$;

● For any $i = 1, 2, \ldots, \ell-1$, there exist $p_i, q_i, r_i\in P_{f}(X^+)$ (where $p_i$, $q_i$ and $r_i$ may be empty words), a semiring substitution $\varphi_{i}$ and an identity $u'_{i}\approx v'_{i}\in \Sigma$ such that

Let $\Sigma'$ denote the set $\{u+u\approx v+v\, |\, u\approx v\in \Sigma\}$. For any $i = 1, 2, \ldots, \ell-1$, we shall show that $t_{i}+t_i\approx t_{i+1}+t_{i+1}$ is derivable from $\Sigma'$ and the identities holding in ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$. Indeed, we have

Further,

This implies the identity

We now have

Case 2. $m = n = 1$ and $c(u) = c(v)$. Since $Z_2\models u_{1}\approx v_{1}$, $u_{1}\neq x, v_{1}\neq x$, for every $x\in X$. Since $u_{1}\approx v_{1}$ holds in ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2})$, we have that it is derivable from the collection $\Sigma$ of $u_i\approx v_i, i\in \underline{k}$ and the identities defining ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2})$. From [1,Exercise Ⅱ.14.11], it follows that there exist $t_1, t_2, \ldots, t_\ell\in P_{f}(X^+)$ such that

● $t_1 = u_1, t_\ell = v_1$;

● For any $i = 1, 2, \ldots, \ell-1$, there exist $p_i, q_i\in P_{f}(X^+)$ (where $p_i$ and $q_i$ may be empty words), a semiring substitution $\varphi_{i}$ and an identity $u'_{i}\approx v'_{i}\in \Sigma$ (where $u'_{i}$ and $v'_{i}$ are words) such that

By Lemma 4.1, we have that $u_{1}\approx v_{1}$ can be derived from (3.1)–(3.6), so, by Theorem 3.1, it can be derived from monomial identities holding in ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$. This completes the proof.

Lemma 4.3. The following equality holds

There are $716$ intervals in ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2))$, and each interval is a congruence class of the kernel of the complete epimorphism $\varphi$ in $\textrm{(4.3)}$.

Proof. Firstly, we shall show that equality (4.9) holds. It is easy to see that

So it suffices to show that

for each member ${\bf{W}}$ of ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}))$. If ${\bf{W}}_1$ is a member of $[{\bf{W}}, \widehat{{\bf{W}}}]$, then it is routine to verify that ${\bf{W}}\subseteq \{E^+(S)\, |\, S\in {\bf{W}}_1\}\subseteq {\bf{W}}$. This implies that $\{E^+(S)\, |\, S\in {\bf{W}}_1\} = {\bf{W}}$ and so $\varphi({\bf{W}}_1) = {\bf{W}}$. Hence, ${\bf{W}}_1$ is a member of $\varphi^{-1}({\bf{W}})$ and so $[{\bf{W}}, \widehat{{\bf{W}}}]\subseteq \varphi^{-1}({\bf{W}})$. Conversely, if ${\bf{W}}_1$ is a member of $\varphi^{-1}({\bf{W}})$, then ${\bf{W}} = \varphi({\bf{W}}_1) = \{E^+(S)\, |\, S\in {\bf{W}}_1\}$ and so $\varphi^{-1}({\bf{W}})\subseteq[{\bf{W}}, \widehat{{\bf{W}}}]$. This shows that (4.9) holds.

From Lemma 4.1, we know that ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}))$ is a lattice of order $716$. So there are $716$ intervals in ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2))$. Next, we show that $\varphi$ a complete epimorphism. On one hand, it is easy to see that $\varphi$ is a complete $\wedge$-epimorphism. On the other hand, let $({\bf{W}}_i)_{i\in I}$ be a family of members of ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2,$ $W_2))$. Then, by (4.3), we have that $\varphi({\bf{W}}_i)\subseteq {\bf{W}}_i\subseteq \widehat{\varphi({\bf{W}}_i)}$ for each $i\in I$. Further,

This implies that $\varphi(\underset{i\in I}\bigvee{\bf{W}}_i) = \underset{i\in I}\bigvee\varphi({\bf{W}}_i).$ Thus, $\varphi$ is a complete $\vee$-homomorphism and so $\varphi$ is a complete $\vee$-epimorphism. By (4.10), we deduce that each interval in (4.3) is a congruence class of the kernel of the complete epimorphism $\varphi$.

In order to characterize the lattice ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2))$, by Lemma 4.3, we only need to describe the interval $[{\bf{W}}, {\widehat{\bf{W}}}]$ for each member ${\bf{W}}$ of ${\cal L}({\bf{HSP}}(B^{0},$ $(B^{0})^{\ast}, A^{0}, N_{2}, T_{2}))$. Next, we have

Lemma 4.4. Let ${\bf{W}}$ be a member of ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}))$. Then, ${\bf{W}}\vee{\bf{HSP}}(Z_2)$ is the subvariety of $\widehat{{\bf{W}}}$ determined by the identity

Proof. It is easy to see that both, ${\bf{W}}$ and ${\bf{HSP}}(Z_2)$ satisfy the identity (4.11) and so does ${\bf{W}}\vee{\bf{HSP}}(Z_2)$. In the following we prove that every identity that is satisfied by ${\bf{W}}\vee{\bf{HSP}}(Z_2)$ is derivable from (4.11) and the identities holding in $\widehat{{\bf{W}}}$. Let $u\approx v$ be such an identity, where $u = u_{1}+u_{2}+\cdots+u_{m}, v = v_{1}+v_{2}+\cdots+v_{n}, u_{i}, v_{j}\in X^{+}, 1\leq i\leq m, 1\leq j\leq n$. We only need to consider the following cases.

Case 1. $m = n = 1$. Since $Z_2$ satisfies $u_{1}\approx v_{1}$, it follows that $|u_{1}|\neq 1$ and $|v_{1}|\neq1$. By Lemma 4.2, $\widehat{{\bf{W}}}$ satisfies the identity $u_1^3+u_1^3\approx v_1^3+v_1^3$. Hence $u_1\overset{(3.4)}\approx u_1^3\overset{(4.11)}\approx u_1^3+u_1^3\approx v_1^3+v_1^3\overset{(4.11)}\approx v_1^3\overset{(3.4)}\approx v_1$.

Case 2. $m = 1$, $n\geq 2$. Since $Z_2$ satisfies $u_{1}\approx v$, it follows that $|u_{1}|\neq 1$. By Lemma 4.2, $\widehat{{\bf{W}}}$ satisfies the identity $u_1^3+u_1^3\approx v+v$. Hence $u_1\overset{(3.4)}\approx u^3_1\overset{(4.11)}\approx u^3_1+u^3_1\approx v+v\overset{(3.11)}\approx v$.

Case 3. $m\geq 2$, $n = 1$. Similar to Case 2.

Case 4. $m, n\geq 2$. By Lemma 4.2, $\widehat{{\bf{W}}}$ satisfies the identity $u+u\approx v+v$. Hence $u\overset{(3.11)}\approx u+u\approx v+v\overset{(3.11)}\approx v$.

Lemma 4.5. Let ${\bf{W}}$ be a member of ${\cal L}({\bf{Sr}}(3, 1))$. Then ${\bf{W}}\vee{\bf{HSP}}(W_2)$ is the subvariety of $\widehat{{\bf{W}}}$ determined by the identities

Proof. It is easy to see that both, ${\bf{W}}$ and ${\bf{HSP}}(W_2)$ satisfy the identity (4.12) and so does ${\bf{W}}\vee{\bf{HSP}}(W_2)$. So it suffices to show that every identity that is satisfied by ${\bf{W}}\vee{\bf{HSP}}(W_2)$ is derivable from (4.12) and the identities holding in $\widehat{{\bf{W}}}$. Let $u\approx v$ be such an identity, where $u = u_{1}+u_{2}+\cdots+u_{m}, v = v_{1}+v_{2}+\cdots+v_{n}, u_{i}, v_{j}\in X^{+}, 1\leq i\leq m, 1\leq j\leq n$. By Lemma 4.2, $\widehat{{\bf{W}}}$ satisfies the identity $u^{3}\approx v^{3}$. Hence, $u\overset{(4.12)}\approx u^{3}\approx v^{3}\overset{(4.12)}\approx v$.

Lemma 4.6. Let ${\bf{W}}$ be a member of ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}))$. Then the interval $[{\bf{W}}, \widehat{{\bf{W}}}]$ of ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2))$ is given in Figure 1.

Proof. Suppose that ${\bf{W}}_1$ is a member of $[{\bf{W}}, \widehat{{\bf{W}}}]$ such that ${\bf{W}}_1\neq \widehat{{\bf{W}}}$ and ${\bf{W}}_1\neq {\bf{W}}$. Then, there exists a nontrivial identity $u\approx v$ holding in ${\bf{W}}_1$ such that it is not satisfied by $\widehat{{\bf{W}}}$. Also, we have that ${\bf{W}}_1$ does not satisfy the identity $x+x\approx x$. By Lemma 4.2, we only need to consider the following two cases.

Case 1. ${\bf{HSP}}(Z_2)\models u\approx v, {\bf{HSP}}(W_2) \not\models u\approx v$. Then, $u\approx v$ satisfies one of the following three cases:

● $m = n = 1$, $c(u_1)\neq c(v_1)$, $|u_1|\neq1$ and $|v_1|\neq1$;

● $m = 1, n > 1$ and $|u_1|\neq1$;

● $m > 1, n = 1$ and $|v_1|\neq1$.

It is easy to see that, in each of the above cases, $u\approx v$ can imply the identity $x^3\approx x^3+x^3$. By Lemma 4.4, we have that ${\bf{W}}_1$ is a subvariety of ${\bf{W}}\vee {\bf{HSP}}(Z_2)$. On the other hand, since ${\bf{W}}_{1}\models x^3\approx x^3+x^3$ and ${\bf{W}}_{1}\not\models x+x\approx x$, it follows that $Z_2$ is a member of ${\bf{W}}_1$ and so ${\bf{W}}\vee {\bf{HSP}}(Z_2)$ is a subvariety of ${\bf{W}}_1$. Thus, ${\bf{W}}_1 = {\bf{W}}\vee {\bf{HSP}}(Z_2)$.

Case 2. ${\bf{HSP}}(Z_2)\not\models u\approx v, {\bf{HSP}}(W_2) \models u\approx v$. Then, $u\approx v$ satisfies one of the following two cases:

● $m = n = 1$, $c(u_1) = c(v_1)$ and $|u_1| = 1$;

● $m = n = 1$, $c(u_1) = c(v_1)$ and $|v_1| = 1$.

If $N_2, T_2\not\in{\bf{W}}$, then, in each of the above cases, $u\approx v$ can imply the identity $x\approx x^3$. By Lemma 4.5, ${\bf{W}}_1$ is a subvariety of ${\bf{W}}\vee {\bf{HSP}}(W_2)$. On the other hand, since ${\bf{W}}_{1}\models x\approx x^3$ and ${\bf{W}}_{1}\not\models x\approx x+x$, it follows that $W_2$ is a member of ${\bf{W}}_1$ and so ${\bf{W}}\vee {\bf{HSP}}(W_2)$ is a subvariety of ${\bf{W}}_1$. Thus, ${\bf{W}}_1 = {\bf{W}}\vee {\bf{HSP}}(W_2)$.

If $N_2\in{\bf{W}}$, then, by Lemma 2.1 (ⅰ), $|u_1| = |v_1| = 1$, a contradiction. Thus, ${\bf{V}}_1 = \widehat{{\bf{V}}}$.

If $T_2\in{\bf{W}}$, then, by Lemma 2.1 (ⅱ), $|u_1|\geq2, |v_1|\geq2$, a contradiction. Thus, ${\bf{V}}_1 = \widehat{{\bf{V}}}$.

By Lemma 4.3 and 4.6, we can show that the lattice ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2,$ $W_2))$ of subvarieties of the variety ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$ contains 2327 elements. In fact, we have

Theorem 4.7. ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2))$ is a distributive lattice of order $\textrm{2327}$.

Proof. We recall from ^[16] that ${\bf{Sr}}(3, 1)\vee T_2$ [${\bf{Sr}}(3, 1)\vee N_2$] contains 358 subvarieties since ${\bf{Sr}}(3, 1)$ contains 179 subvarieties. By Lemma 4.3 and 4.6, we can show that ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2,$ $W_2))$ has exactly 2327 (where $2327 = 179\times4+358\times3\times2-179\times3$) elements. Suppose that ${\bf{W}}_1, {\bf{W}}_2$ and ${\bf{W}}_3$ are members of ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0},$ $N_{2}, T_{2}, Z_2, W_2))$ such that ${\bf{W}}_1\vee {\bf{W}}_2 = {\bf{W}}_1\vee {\bf{W}}_3$ and ${\bf{W}}_1\wedge {\bf{W}}_2 = {\bf{W}}_1\wedge{\bf{W}}_3$. Then, by Lemma 4.3

and

Since ${\cal L}({\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2})$ is distributive, it follows that $\varphi({\bf{W}}_2) = \varphi({\bf{W}}_3)$. Write ${\bf{W}}$ for $\varphi({\bf{W}}_2)$. Then both ${\bf{W}}_{2}, {\bf{W}}_{3}$ are members of $[{\bf{W}}, \widehat{{\bf{W}}}]$. Suppose that ${\bf{W}}_2\neq{\bf{W}}_3$. Then, by Lemma 4.6, ${\bf{W}}_1\vee {\bf{W}}_2 = {\bf{W}}_1\vee {\bf{W}}_3$ and ${\bf{W}}_1\wedge {\bf{W}}_2 = {\bf{W}}_1\wedge{\bf{W}}_3$ can not hold at the same time. This implies that ${\bf{W}}_2 = {\bf{W}}_3$.

By Theorem 4.1, 4.7 and [14,Corollary 1.2], we now immediately deduce

Corollary 4.8. ${\bf{HSP}}(B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2)$ is hereditarily finitely based.

5.   Conclusions

This article considers a semiring variety generated by $B^{0}, (B^{0})^{\ast}, A^{0}, N_{2}, T_{2}, Z_2, W_2$. The finite basis problem for semirings is an interesting developing topic, with plenty of evidence of a high level of complexity along the lines of the more well-developed area of semigroup varieties. This article is primarily a contribution toward the property of being hereditarily finite based, meaning that all subvarieties are finitely based. This property is of course useful because it guarantees the finite basis property of a large number of examples.

Acknowledgments

This work was supported by the Natural Science Foundation of Chongqing (cstc2019jcyj-msxmX0156, cstc2020jcyj-msxmX0272, cstc2021jcyj-msxmX0436), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJQN202001107, KJQN202101130) and the Scientific Research Starting Foundation of Chongqing University of Technology (2019ZD68).

Conflict of interest

The authors declare that they do not have any conflict of interests regarding this paper.