Suggestion of a dynamic model of the development of neurodevelopmental disorders and the phenomenon of autism

Bodil Aggernæs; Bodil Aggernæs

doi:10.3934/molsci.2020008

AIMS Molecular Science

2020, Volume 7, Issue 2: 122-182. doi: 10.3934/molsci.2020008

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Research article Special Issues

Suggestion of a dynamic model of the development of neurodevelopmental disorders and the phenomenon of autism

Bodil Aggernæs ^{1,2
,
,}

1.
Department of Child and Adolescent Psychiatry, Psychiatry Region Zealand, 4000 Roskilde, Denmark; Department of Clinical Medicine, Faculty of Medical and Health Sciences, University of Copenhagen, 2200 Copenhagen N, Denmark
2.
Department of Adult Psychiatry, Mental Health Centre Northern Zealand, Mental Health Services of the Capital Region, 3400 Hillerød, Denmark

Received: 14 January 2020 Accepted: 27 May 2020 Published: 29 May 2020

The present article is a theoretical contribution suggesting a simple model of the dynamics involved in the development of neurodevelopmental disorders and the phenomenon of autism offering an explanation of why different individuals have different onset of manifest disease. The model relates to present genetic and epigenetic evidence and previous theoretical models. The dynamic model applies an individualized transdiagnostic and dimensional approach integrating several levels of influence involved in the dynamic interaction between an individual and their environment across time. The dynamic model illustrates the interaction between a basic neurobiological susceptibility, compensating mechanisms, and stress-related releasing mechanisms involved in the development of manifest clinical illness. The model has a particular focus on the dynamics of neurocognitive processes and their relationship to more basic information, psychological and social processes. A basic assumption guiding the model is that even quite normal events related to typical development may increase the risk of enduring stress in cognitively vulnerable individuals, further increasing their risk of developing manifest clinical illness. The model suggests that genetic variation, endogenous epigenetic processes of development, and epigenetic changes influenced by physical/chemical and social environmental risk and resilience factors all may contribute to phenotypical expression. A genetic susceptibility may translate into a biological susceptibility reflected in cognitive impairments. A cognitively vulnerable individual may be at increased risk of misinterpreting sensory inputs, potentially resulting in psychopathological expressions, e.g., autistic symptoms, observed across several neurodevelopmental disorders. The genetically influenced experience of an individual relates to cognitive phenomena occurring at the interfaces between the brain, mind, and society. The severity of clinical illness may differ from the severity of a disorder of reasoning. The dynamic model may help guide future development of personalized medicine in psychiatry and identify relevant points of intervention. A discussion of the implications of the model relating to epigenetic evidence and to previous theoretical models is included at the end of the paper.

Keywords:

Citation: Bodil Aggernæs. Suggestion of a dynamic model of the development of neurodevelopmental disorders and the phenomenon of autism[J]. AIMS Molecular Science, 2020, 7(2): 122-182. doi: 10.3934/molsci.2020008

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Abstract

1. Introduction

Henkin and Skolem introduced Hilbert algebras in the fifties for investigations in intuitionistic and other non-classical logics. Diego ^[4] proved that Hilbert algebras form a variety which is locally finite. Bandaru et al. introduced the notion of GE-algebras which is a generalization of Hilbert algebras, and investigated several properties (see ^[1,2,7,8,9]). The notion of interior operator is introduced by Vorster ^[12] in an arbitrary category, and it is used in ^[3] to study the notions of connectedness and disconnectedness in topology. Interior algebras are a certain type of algebraic structure that encodes the idea of the topological interior of a set, and are a generalization of topological spaces defined by means of topological interior operators. Rachůnek and Svoboda ^[6] studied interior operators on bounded residuated lattices, and Svrcek ^[11] studied multiplicative interior operators on GMV-algebras. Lee et al. ^[5] applied the interior operator theory to GE-algebras, and they introduced the concepts of (commutative, transitive, left exchangeable, belligerent, antisymmetric) interior GE-algebras and bordered interior GE-algebras, and investigated their relations and properties. Later, Song et al. ^[10] introduced the notions of an interior GE-filter, a weak interior GE-filter and a belligerent interior GE-filter, and investigate their relations and properties. They provided relations between a belligerent interior GE-filter and an interior GE-filter and conditions for an interior GE-filter to be a belligerent interior GE-filter is considered. Given a subset and an element, they established an interior GE-filter, and they considered conditions for a subset to be a belligerent interior GE-filter. They studied the extensibility of the belligerent interior GE-filter and established relationships between weak interior GE-filter and belligerent interior GE-filter of type 1, type 2 and type 3. Rezaei et al. ^[7] studied prominent GE-filters in GE-algebras. The purpose of this paper is to study by applying interior operator theory to prominent GE-filters in GE-algebras. We introduce the concept of a prominent interior GE-filter, and investigate their properties. We discuss the relationship between a prominent GE-filter and a prominent interior GE-filter and the relationship between an interior GE-filter and a prominent interior GE-filter. We find and provide examples where any interior GE-filter is not a prominent interior GE-filter and any prominent GE-filter is not a prominent interior GE-filter. We provide conditions for an interior GE-filter to be a prominent interior GE-filter. We provide conditions under which an internal GE-filter larger than a given internal GE filter can become a prominent internal GE-filter, and give an example describing it. We also introduce the concept of a prominent interior GE-filter of type 1 and type 2, and investigate their properties. We discuss the relationship between a prominent interior GE-filter and a prominent interior GE-filter of type 1. We give examples to show that A and B are independent of each other, where A and B are:

$(1)$ $\left\{\begin{array}{ll} \text{ A: prominent interior GE-filter of type 1.}\\ \text{ B: prominent interior GE-filter of type 2.} \end{array}\right.$

$(2)$ $\left\{\begin{array}{ll} \text{ A: prominent interior GE-filter.}\\ \text{ B: prominent interior GE-filter of type 2.} \end{array}\right.$

$(3)$ $\left\{\begin{array}{ll} \text{ A: interior GE-filter.}\\ \text{ B: prominent interior GE-filter of type 1.} \end{array}\right.$

$(4)$ $\left\{\begin{array}{ll} \text{ A: interior GE-filter.}\\ \text{ B: prominent interior GE-filter of type 2.} \end{array}\right.$

2. Preliminaries

Definition 2.1. ^[1] By a GE-algebra we mean a non-empty set ${X}$ with a constant $1$ and a binary operation ${*}$ satisfying the following axioms:

(GE1) ${u}{*} {u} = 1$ ,

(GE2) $1{*}{u} = {u}$ ,

(GE3) ${u}{*} ({v}{*} {w}) = {u}{*} ({v}{*} ({u}{*} {w}))$

for all ${u}, {v}, {w}\in {X}$ .

In a GE-algebra ${X}$ , a binary relation " $\le$ " is defined by

$\begin{align} (\forall {x}, {y}\in {X}) \left({x}\le {y} \; \Leftrightarrow \; {x}{*} {y} = 1\right). \end{align}$

(2.1)

Definition 2.2. ^[1,2,8] A GE-algebra ${X}$ is said to be transitive if it satisfies:

$\begin{align} (\forall {x}, {y}, {z}\in {X})\left({x}{*} {y}\le ({z}{*} {x}){*} ({z}{*} {y})\right). \end{align}$

(2.2)

Proposition 2.3. ^[1] Every GE-algebra ${X}$ satisfies the following items:

$\begin{align} & (\forall {u}\in {X}) \left({u}{*} 1 = 1\right). \end{align}$

(2.3)

$\begin{align} & (\forall {u}, {v}\in {X}) \left({u}{*} ({u}{*} {v}) = {u}{*} {v}\right). \end{align}$

(2.4)

$\begin{align} & (\forall {u}, {v}\in {X}) \left({u} \le {v}{*} {u}\right). \end{align}$

(2.5)

$\begin{align} & (\forall {u}, {v}, {w}\in {X}) \left({u}{*} ({v}{*} {w})\le {v}{*} ({u}{*} {w})\right). \end{align}$

(2.6)

$\begin{align} & (\forall {u}\in {X}) \left({1}\le {u} \; \Rightarrow \; {u} = {1}\right). \end{align}$

(2.7)

$\begin{align} & (\forall {u}, {v}\in {X}) \left({u}\le ({v}{*} {u}){*} {u}\right). \end{align}$

(2.8)

$\begin{align} & (\forall {u}, {v}\in {X}) \left({u}\le ({u}{*} {v}){*} {v}\right). \end{align}$

(2.9)

$\begin{align} & (\forall {u}, {v}, {w}\in {X}) \left({u}\le {v}{*} {w} \; \Leftrightarrow \; {v}\le {u}{*} {w}\right). \end{align}$

(2.10)

If ${X}$ is transitive, then

$\begin{align} & (\forall {u}, {v}, {w}\in {X}) \left({u}\le {v} \; \Rightarrow \; {w}{*} {u}\le {w}{*} {v}, \; {v}{*} {w}\le {u}{*} {w}\right). \end{align}$

(2.11)

$\begin{align} & (\forall {u}, {v}, {w}\in {X}) \left({u}{*} {v}\le ({v}{*} {w}){*} ({u}{*} {w})\right). \end{align}$

(2.12)

Lemma 2.4. ^[1] In a GE-algebra ${X}$ , the following facts are equivalent each other.

$\begin{align} & (\forall {x}, {y}, {z}\in {X}) \left({x}{*} {y}\le ({z}{*} {x}){*} ({z}{*} {y})\right). \end{align}$

(2.13)

$\begin{align} & (\forall {x}, {y}, {z}\in {X}) \left({x}{*} {y}\le ({y}{*} {z}){*} ({x}{*} {z})\right). \end{align}$

(2.14)

Definition 2.5. ^[1] A subset ${F}$ of a GE-algebra ${X}$ is called a GE-filter of ${X}$ if it satisfies:

$\begin{align} & 1\in {F}, \end{align}$

(2.15)

$\begin{align} & (\forall {x}, {y}\in {X}) ({x}{*} {y}\in {F}, \; {x}\in {F} \; \Rightarrow \; {y}\in {F}). \end{align}$

(2.16)

Lemma 2.6. ^[1] In a GE-algebra ${X}$ , every filter ${F}$ of ${X}$ satisfies:

$\begin{align} (\forall {x}, {y}\in {X}) \left({x}\le {y}, \; {x}\in {F} \; \Rightarrow \; {y}\in {F}\right). \end{align}$

(2.17)

Definition 2.7. ^[7] A subset ${F}$ of a GE-algebra ${X}$ is called a prominent GE-filter of ${X}$ if it satisfies (2.15) and

$\begin{align} (\forall {x}, {y}, {z}\in {X}) ({x}{*} ({y}{*} {z})\in {F}, \; {x}\in {F} \; \Rightarrow \; (({z}{*} {y}){*} {y}){*} {z}\in {F}). \end{align}$

(2.18)

Note that every prominent GE-filter is a GE-filter in a GE-algebra (see ^[7]).

Definition 2.8. ^[5] By an interior GE-algebra we mean a pair $({X}, f)$ in which ${X}$ is a GE-algebra and $f :{X} \rightarrow {X}$ is a mapping such that

$\begin{align} & (\forall {x}\in {X}) ({x} \le f({x})), \end{align}$

(2.19)

$\begin{align} & (\forall {x}\in {X}) (( f \circ f)({x}) = f({x})), \end{align}$

(2.20)

$\begin{align} & (\forall {x}, {y}\in {X}) ({x}\le {y} \; \Rightarrow \; f({x})\le f({y})). \end{align}$

(2.21)

Definition 2.9. ^[10] Let $({X}, f)$ be an interior GE-algebra. A GE-filter ${F}$ of ${X}$ is said to be interior if it satisfies:

$\begin{align} (\forall {x}\in {X}) ( f({x})\in {F} \; \Rightarrow \; {x}\in {F}). \end{align}$

(2.22)

3. Prominent interior GE-filters

Definition 3.1. Let $({X}, f)$ be an interior GE-algebra. Then a subset ${F}$ of ${X}$ is called a prominent interior GE-filter in $({X}, f)$ if ${F}$ is a prominent GE-filter of ${X}$ which satisfies the condition (2.22).

Example 3.2. Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in Table 1.

Table 1. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${4}$	${4}$
${3}$	${1}$	${1}$	${1}$	${5}$	${5}$
${4}$	${1}$	${2}$	${3}$	${1}$	${1}$
${5}$	${1}$	${2}$	${2}$	${1}$	${1}$

| Show Table

DownLoad: CSV

Then ${X}$ is a GE-algebra. If we define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} \in \{{1}, {4}, {5}\}, \\ {2} & \text{if } \, {x} \in \{{2}, {3}\}, \end{array}\right. \end{align*}$

then $({X}, f)$ is an interior GE-algebra and ${F} = \{{1}, {4}, {5}\}$ is a prominent interior GE-filter in $({X}, f)$ .

It is clear that every prominent interior GE-filter is a prominent GE-filter. But any prominent GE-filter may not be a prominent interior GE-filter in an interior GE-algebra as seen in the following example.

Example 3.3. Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in Table 2,

Table 2. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${3}$	${4}$	${1}$
${3}$	${1}$	${2}$	${1}$	${4}$	${5}$
${4}$	${1}$	${2}$	${3}$	${1}$	${5}$
${5}$	${1}$	${1}$	${3}$	${4}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} \in \{{1}, {2}, {3}, {5}\}, \\ {4} & \text{if } \, {x} = {4}. \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}\}$ is a prominent GE-filter of ${X}$ . But it is not a prominent interior GE-filter in $({X}, f)$ since $f({2}) = {1}\in {F}$ but ${2}\notin {F}$ .

We discuss relationship between interior GE-filter and prominent interior GE-filter.

Theorem 3.4. In an interior GE-algebra, every prominent interior GE-filter is an interior GE-filter.

Proof. It is straightforward because every prominent GE-filter is a GE-filter in a GE-algebra.

In the next example, we can see that any interior GE-filter is not a prominent interior GE-filter in general.

Example 3.5. Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in Table 3.

Table 3. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${4}$	${4}$
${3}$	${1}$	${2}$	${1}$	${4}$	${4}$
${4}$	${1}$	${1}$	${3}$	${1}$	${1}$
${5}$	${1}$	${1}$	${1}$	${1}$	${1}$

| Show Table

DownLoad: CSV

Then ${X}$ is a GE-algebra. If we define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} = {1}, \\ {2} & \text{if } \, {x} \in \{{2}, {4}, {5}\}, \\ {3} & \text{if } \, {x} = {3}, \end{array}\right. \end{align*}$

then $({X}, f)$ is an interior GE-algebra and ${F} = \{{1}\}$ is an interior GE-filter in $({X}, f)$ . But it is not a prominent interior GE-filter in $({X}, f)$ since ${1} {*} ({2}{*} {3}) = {1} \in {F}$ but $(({3}{*} {2}){*} {2}){*} {3} = {3} \notin {F}$ .

Proposition 3.6. Every prominent interior GE-filter ${F}$ in an interior GE-algebra $({X}, f)$ satisfies:

$\begin{align} (\forall {x}, {y}\in {X}) \left( f({x}{*} {y})\in {F} \; \Rightarrow (({y}{*} {x}){*} {x}){*} {y} \in {F}\right). \end{align}$

(3.1)

Proof. Let ${F}$ be a prominent interior GE-filter in $({X}, f)$ . Let ${x}, {y}\in {X}$ be such that $f({x}{*} {y})\in {F}$ . Then ${x}{*} {y}\in {F}$ by (2.22), and so $1{*} ({x}{*} {y}) = {x}{*} {y}\in {F}$ by (GE2). Since $1\in {F}$ , it follows from (2.18) that $(({y}{*} {x}){*} {x}){*} {y} \in {F}$ .

Corollary 3.7. Every prominent interior GE-filter ${F}$ in an interior GE-algebra $({X}, f)$ satisfies:

$\begin{align} (\forall {x}, {y}\in {X}) \left({x}{*} {y}\in {F} \; \Rightarrow (({y}{*} {x}){*} {x}){*} {y} \in {F}\right). \end{align}$

(3.2)

Proof. Let ${F}$ be a prominent interior GE-filter in $({X}, f)$ . Then ${F}$ is an interior GE-filter in $({X}, f)$ by Theorem 3.4. Let ${x}, {y}\in {X}$ be such that ${x}{*} {y}\in {F}$ . Since ${x}{*} {y} \le f({x}{*} {y})$ by (2.19), it follows from Lemma 2.6 that $f({x}{*} {y})\in {F}$ . Hence $(({y}{*} {x}){*} {x}){*} {y} \in {F}$ by Proposition 3.6.

Corollary 3.8. Every prominent interior GE-filter ${F}$ in an interior GE-algebra $({X}, f)$ satisfies:

$\begin{align*} (\forall {x}, {y}\in {X}) \left({x}{*} {y}\in {F} \; \Rightarrow \; f((({y}{*} {x}){*} {x}){*} {y}) \in {F}\right). \end{align*}$

Proof. Straightforward.

Corollary 3.9. Every prominent interior GE-filter ${F}$ in an interior GE-algebra $({X}, f)$ satisfies:

$\begin{align*} (\forall {x}, {y}\in {X}) \left( f({x}{*} {y})\in {F} \; \Rightarrow \; f((({y}{*} {x}){*} {x}){*} {y}) \in {F}\right). \end{align*}$

Proof. Straightforward.

In the following example, we can see that any interior GE-filter ${F}$ in an interior GE-algebra $({X}, f)$ does not satisfy the conditions (3.1) and (3.2).

Example 3.10. Consider the interior GE-algebra $({X}, f)$ in Example 3.4. The interior GE-filter ${F} : = \{{1}\}$ does not satisfy conditions (3.1) and (3.2) since $f({2}{*} {3}) = f({1}) = {1}\in {F}$ and ${2}{*} {3} = {1}\in {F}$ but $(({3}{*} {2}){*} {2}){*} {3} = {3} \notin {F}$ .

We provide conditions for an interior GE-filter to be a prominent interior GE-filter.

Theorem 3.11. If an interior GE-filter ${F}$ in an interior GE-algebra $({X}, f)$ satisfies the condition $(3.1)$ , then ${F}$ is a prominent interior GE-filter in $({X}, f)$ .

Proof. Let ${F}$ be an interior GE-filter in $({X}, f)$ that satisfies the condition (3.1). Let ${x}, {y}, {z}\in {X}$ be such that ${x}{*} ({y}{*} {z})\in {F}$ and ${x}\in {F}$ . Then ${y}{*} {z}\in {F}$ . Since ${y}{*} {z} \le f({y}{*} {z})$ by (2.19), it follows from Lemma 2.6 that $f({y}{*} {z})\in {F}$ . Hence $(({z}{*} {y}){*} {y}){*} {z} \in {F}$ by (3.1), and therefore ${F}$ is a prominent interior GE-filter in $({X}, f)$ .

Lemma 3.12. ^[10] In an interior GE-algebra, the intersection of interior GE-filters is also an interior GE-filter.

Theorem 3.13. In an interior GE-algebra, the intersection of prominent interior GE-filters is also a prominent interior GE-filter.

Proof. Let $\{{F}_i \mid i\in \Lambda \}$ be a set of prominent interior GE-filters in an interior GE-algebra $({X}, f)$ where $\Lambda$ is an index set. Then $\{{F}_i \mid i\in \Lambda \}$ is a set of interior GE-filters in $({X}, f)$ , and so $\cap \{{F}_i \mid i\in \Lambda \}$ is an interior GE-filter in $({X}, f)$ by Lemma 3.12. Let ${x}, {y}\in {X}$ be such that $f({x}{*} {y})\in \cap \{{F}_i \mid i\in \Lambda \}$ . Then $f({x}{*} {y})\in {F}_i$ for all $i\in \Lambda$ . It follows from Proposition 3.6 that $(({y}{*} {x}){*} {x}){*} {y}\in {F}_i$ for all $i\in \Lambda$ . Hence $(({y}{*} {x}){*} {x}){*} {y}\in \cap \{{F}_i \mid i\in \Lambda \}$ and therefore $\cap \{{F}_i \mid i\in \Lambda \}$ is a prominent interior GE-filter in $({X}, f)$ by Theorem 3.11.

Theorem 3.14. If an interior GE-filter ${F}$ in an interior GE-algebra $({X}, f)$ satisfies the condition $(3.2)$ , then ${F}$ is a prominent interior GE-filter in $({X}, f)$ .

Proof. Let ${F}$ be an interior GE-filter in $({X}, f)$ that satisfies the condition (3.2). Let ${x}, {y}, {z}\in {X}$ be such that ${x}{*} ({y}{*} {z})\in {F}$ and ${x}\in {F}$ . Then ${y}{*} {z}\in {F}$ and thus $(({z}{*} {y}){*} {y}){*} {z} \in {F}$ . Therefore ${F}$ is a prominent interior GE-filter in $({X}, f)$ .

Given an interior GE-filter ${F}$ in an interior GE-algebra $({X}, f)$ , we consider an interior GE-filter ${G}$ which is greater than ${F}$ in $({X}, f)$ , that is, we take two interior GE-filters ${F}$ and ${G}$ such that ${F} \subseteq {G}$ in an interior GE-algebra $({X}, f)$ . We are now trying to find the condition that ${G}$ can be a prominent interior GE-filter in $({X}, f)$ .

Theorem 3.15. Let $({X}, f)$ be an interior GE-algebra in which ${X}$ is transitive. Let ${F}$ and ${G}$ be interior GE-filters in $({X}, f)$ . If ${F} \subseteq {G}$ and ${F}$ is a prominent interior GE-filter in $({X}, f)$ , then ${G}$ is also a prominent interior GE-filter in $({X}, f)$ .

Proof. Assume that ${F}$ is a prominent interior GE-filter in $({X}, f)$ . Then it is an interior GE-filter in $({X}, f)$ by Theorem 3.4. Let ${x}, {y}\in {X}$ be such that $f({x}{*} {y})\in {G}$ . Then ${x}{*} {y}\in {G}$ by (2.22), and so $1 = ({x}{*} {y}){*} ({x}{*} {y})\le {x}{*} (({x}{*} {y}){*} {y})$ by (GE1) and (2.6). Since $1\in {F}$ , it follows from Lemma 2.6 that ${x}{*} (({x}{*} {y}){*} {y})\in {F}$ . Hence $(((({x}{*} {y}){*} {y}){*} {x}){*} {x}){*} (({x}{*} {y}){*} {y})\in {F} \subseteq {G}$ by Corollary 3.7. Since

$\begin{align*} (((({x}{*} {y}){*} {y}){*} {x}){*} {x}){*} (({x}{*} {y}){*} {y})\le ({x}{*} {y}){*} ((((({x}{*} {y}){*} {y}){*} {x}){*} {x}) {*} {y}) \end{align*}$

by (2.6), we have $({x}{*} {y}){*} ((((({x}{*} {y}){*} {y}){*} {x}){*} {x}) {*} {y})\in {G}$ by Lemma 2.6. Hence

$(((({x}{*} {y}){*} {y}){*} {x}){*} {x}) {*} {y}\in {G}.$

Since ${y}\le ({x}{*} {y}){*} {y}$ , it follows from (2.11) that

$\begin{align*} (((({x}{*} {y}){*} {y}){*} {x}){*} {x}){*} {y}\le (({y}{*} {x}){*} {x}){*} {y}. \end{align*}$

Thus $(({y}{*} {x}){*} {x}){*} {y}\in {G}$ by Lemma 2.6. Therefore ${G}$ is a prominent interior GE-filter in $({X}, f)$ . by Theorem 3.11.

The following example describes Theorem 3.15.

Example 3.16. Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in Table 4,

Table 4. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${5}$	${5}$
${3}$	${1}$	${1}$	${1}$	${5}$	${5}$
${4}$	${1}$	${3}$	${3}$	${1}$	${1}$
${5}$	${1}$	${3}$	${3}$	${1}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} = {1}, \\ {3} & \text{if } \, {x} \in \{{2}, {3}\}, \\ {5} & \text{if } \, {x} \in \{{4}, {5}\}. \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra in which ${X}$ is transitive, and ${F} : = \{{1}\}$ and ${G} : = \{{1}, {4}, {5}\}$ are interior GE-filters in $({X}, f)$ with ${F}\subseteq {G}$ . Also we can observe that ${F}$ and ${G}$ are prominent interior GE-filters in $({X}, f)$ .

In Theorem 3.15, if ${F}$ is an interior GE-filter which is not prominent, then ${G}$ is also not a prominent interior GE-filter in $({X}, f)$ as shown in the next example.

Example 3.17. Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in Table 5,

Table 5. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${4}$	${1}$
${3}$	${1}$	${5}$	${1}$	${4}$	${5}$
${4}$	${1}$	${1}$	${1}$	${1}$	${1}$
${5}$	${1}$	${1}$	${1}$	${4}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} = {1}, \\ {3} & \text{if } \, {x} = {3}, \\ {4} & \text{if } \, {x} = {4}, \\ {2} & \text{if } \, {x} \in \{{2}, {5}\}. \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra in which ${X}$ is transitive, and ${F} : = \{{1}\}$ and ${G} : = \{{1}, {3}\}$ are interior GE-filters in $({X}, f)$ with ${F}\subseteq {G}$ . We can observe that ${F}$ and ${G}$ are not prominent interior GE-filters in $({X}, f)$ since ${2} {*} {3} = 1\in {F}$ but $(({3} {*} {2}){*} {2}) {*} {3} = ({5} {*} {2}) {*} {3} = {1} {*} {3} = {3}\notin {F}$ , and ${4} {*} {2} = {1} \in {G}$ but $(({2} {*} {4}){*} {4}) {*} {2} = ({4} {*} {4}) {*} {2} = {1} {*} {2} = {2}\notin {G}.$

In Theorem 3.15, if ${X}$ is not transitive, then Theorem 3.15 is false as seen in the following example.

Example 3.18. Let ${X} = \{{1}, {2}, {3}, {4}, {5}, {6}\}$ be a set with the Cayley table which is given in Table 6.

Table 6. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$	${6}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$	${6}$
${2}$	${1}$	${1}$	${1}$	${6}$	${6}$	${6}$
${3}$	${1}$	${1}$	${1}$	${5}$	${5}$	${5}$
${4}$	${1}$	${1}$	${3}$	${1}$	${1}$	${1}$
${5}$	${1}$	${2}$	${3}$	${2}$	${1}$	${1}$
${6}$	${1}$	${2}$	${3}$	${2}$	${1}$	${1}$

| Show Table

DownLoad: CSV

If we define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} = {1}, \\ {4} & \text{if } \, {x} = {4}, \\ {5} & \text{if } \, {x} = {5}, \\ {6} & \text{if } \, {x} = {6}, \\ {2} & \text{if } \, {x} \in \{{2}, {3}\}, \\ \end{array}\right. \end{align*}$

then $({X}, f)$ is an interior GE-algebra in which ${X}$ is not transitive. Let ${F} : = \{{1}\}$ and ${G} : = \{{1}, {5}, {6}\}$ . Then ${F}$ is a prominent interior GE-filter in $({X}, f)$ and ${G}$ is an interior GE-filter in $({X}, f)$ with ${F}\subseteq {G}$ . But ${G}$ is not prominent interior GE-filter since ${5} {*} ({3} {*} {4}) = {5} {*} {5} = {1}\in {G}$ and ${5}\in {G}$ but $(({4} {*} {3}) {*} {3}){*} {4} = ({3}{*} {3}) {*} {4} = {1} {*} {4} = {4} \notin {G}$ .

Definition 3.19. Let $({X}, f)$ be an interior GE-algebra and let ${F}$ be a subset of ${X}$ which satisfies (2.15). Then ${F}$ is called:

$\bullet$ A prominent interior GE-filter of type 1 in $({X}, f)$ if it satisfies:

$\begin{align} (\forall {x}, {y}, {z}\in {X}) \left({x}{*} ({y}{*} f({z}))\in {F}, \, f({x})\in {F} \; \Rightarrow (( f({z}){*} {y}){*} {y}){*} f({z})\in {F}\right). \end{align}$

(3.3)

$\bullet$ A prominent interior GE-filter of type 2 in $({X}, f)$ if it satisfies:

$\begin{align} (\forall {x}, {y}, {z}\in {X}) \left({x}{*} ({y}{*} f({z}))\in {F}, \, f({x})\in {F} \; \Rightarrow (({z}{*} f({y})){*} f({y})){*} {z}\in {F}\right). \end{align}$

(3.4)

Example 3.20. (1). Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in Table 7,

Table 7. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${1}$	${1}$
${3}$	${1}$	${2}$	${1}$	${2}$	${2}$
${4}$	${1}$	${1}$	${1}$	${1}$	${1}$
${5}$	${1}$	${1}$	${1}$	${1}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} \in \{{1}, {3}\} \\ {2} & \text{if } \, {x} = {2}, \\ {4} & \text{if } \, {x} = {4}, \\ {5} & \text{if } \, {x} = {5}. \\ \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}, {3}\}$ is a prominent interior GE-filter of type 1 in $({X}, f)$ .

(2). Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in Table 8,

Table 8. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${1}$	${1}$
${3}$	${1}$	${1}$	${1}$	${4}$	${1}$
${4}$	${1}$	${1}$	${1}$	${1}$	${5}$
${5}$	${1}$	${1}$	${3}$	${4}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} = {1}, \\ {2} & \text{if } \, {x} \in \{{2}, {3}, {4}, {5}\}. \\ \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}, {3}\}$ is a prominent interior GE-filter of type 2 in $({X}, f)$ .

Theorem 3.21. In an interior GE-algebra, every prominent interior GE-filter is of type 1.

Proof. Let ${F}$ be a prominent interior GE-filter in an interior GE-algebra $({X}, f)$ . Let ${x}, {y}, {z}\in {X}$ be such that ${x}{*} ({y}{*} f({z}))\in {F}$ and $f({x})\in {F}$ . Then ${x}\in {F}$ by (2.22). It follows from (2.18) that $((f({z}){*} {y}){*} {y}){*} f({z})\in {F}$ . Hence ${F}$ is a prominent interior GE-filter of type 1 in $({X}, f)$ .

The following example shows that the converse of Theorem 3.21 may not be true.

Example 3.22. Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in Table 9,

Table 9. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${1}$	${1}$
${3}$	${1}$	${1}$	${1}$	${1}$	${5}$
${4}$	${1}$	${1}$	${3}$	${1}$	${1}$
${5}$	${1}$	${1}$	${1}$	${1}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} = {1}, \\ {2} & \text{if } \, {x} \in \{{2}, {3}\}, \\ {5} & \text{if } \, {x} \in \{{4}, {5}\}. \\ \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}\}$ is a prominent interior GE-filter of type 1 in $({X}, f)$ . But it is not a prominent interior GE-filter in $({X}, f)$ since ${1}{*} ({3}{*} {4}) = {1}\in {F}$ but $({4}{*} {3}){*} {3}){*} {4} = {4} \notin {F}$ .

The following example shows that prominent interior GE-filter and prominent interior GE-filter of type 2 are independent of each other, i.e., prominent interior GE-filter is not prominent interior GE-filter of type 2 and neither is the inverse.

Example 3.23. (1). Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in the following Table 10,

Table 10. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${1}$	${1}$
${3}$	${1}$	${5}$	${1}$	${1}$	${5}$
${4}$	${1}$	${1}$	${1}$	${1}$	${1}$
${5}$	${1}$	${3}$	${3}$	${1}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} = {1}, \\ {4} & \text{if } \, {x} \in \{{3}, {4}\} \\ {5} & \text{if } \, {x} \in \{{2}, {5}\}. \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}\}$ ${F}$ is a prominent interior GE-filter in $({X}, f)$ . But it is not a prominent interior GE-filter of type 2 since ${1}{*} ({5} {*} f({2})) = {5} {*} {5} = {1}\in {F}$ and $f({1}) = {1} \in {F}$ but $(({2} {*} f({5})){*} f(5)) {*} {2} = (({2} {*} {5}) {*} {5}) {*} {2} = ({1} {*} {5}) {*} {2} = {5} {*} {2} = {3}\notin {F}$ .

(2). Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in the following Table 11,

Table 11. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${1}$	${1}$
${3}$	${1}$	${2}$	${1}$	${1}$	${1}$
${4}$	${1}$	${1}$	${1}$	${1}$	${1}$
${5}$	${1}$	${1}$	${1}$	${1}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} = {1}, \\ {5} & \text{if } \, {x} \in \{{2}, {3}, {4}, {5}\}. \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}\}$ is a prominent interior GE-filter of type2 in $({X}, f)$ . But it is not a prominent interior GE-filter in $({X}, f)$ since ${1}{*} ({2} {*} {3}) = {1} {*} {1} = {1}\in {F}$ and ${1}\in {F}$ but $(({3} {*} {2}){*} {2}) {*} {3} = ({2} {*} {2}) {*} {3} = {1} {*}{3} = {3}\notin {F}$ .

The following example shows that prominent interior GE-filter of type 1 and prominent interior GE-filter of type 2 are independent of each other.

Example 3.24. (1). Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in the following Table 12,

Table 12. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${5}$	${5}$
${3}$	${1}$	${1}$	${1}$	${1}$	${1}$
${4}$	${1}$	${1}$	${1}$	${1}$	${1}$
${5}$	${1}$	${1}$	${1}$	${1}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}, {2}, {4}\}$ is a prominent interior GE-filter of type 1 in $({X}, f)$ . But it is not a prominent interior GE-filter of type 2 since ${1}{*} ({5} {*} f({2})) = {1} {*} ({5} {*} {3}) = {1} {*} {1} = {1}\in {F}$ and $f({1}) = {1} \in {F}$ but $(({2} {*} f({5})){*} f(5)) {*} {2} = (({2} {*} {5}) {*} {5}) {*} {2} = ({5} {*} {5}) {*} {2} = {1} {*} {2} = {2}\notin {F}$ .

(2). Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in the following Table 13,

Table 13. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${4}$	${4}$	${5}$
${3}$	${1}$	${1}$	${1}$	${1}$	${1}$
${4}$	${1}$	${2}$	${2}$	${1}$	${5}$
${5}$	${1}$	${1}$	${1}$	${1}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} = {1}, \\ {2} & \text{if } \, {x} = {2}, \\ {4} & \text{if } \, {x} = {4}, \\ {3} & \text{if } \, {x} \in \{{3}, {5}\}. \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}\}$ is a prominent interior GE-filter of type 2 in $({X}, f)$ . But it is not a prominent interior GE-filter of type 1 in $({X}, f)$ since ${1}{*} ({5} {*} f(2)) = {1} {*} ({5} {*} {2}) = {1} {*} {1} = {1}\in {F}$ and $f({1})\in {F}$ but $((f(2) {*} {5}){*} {5}) {*} f(2) = (({2} {*} {5}){*} {5}) {*} {2} = ({5} {*} {5}) {*} {2} = {1} {*} {2} = {2}\notin {F}$ .

The following example shows that interior GE-filter and prominent interior GE-filter of type 1 are independent of each other.

Example 3.25. (1). Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in the following Table 14,

Table 14. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${5}$	${5}$	${5}$
${3}$	${1}$	${1}$	${1}$	${1}$	${1}$
${4}$	${1}$	${1}$	${1}$	${1}$	${1}$
${5}$	${1}$	${1}$	${1}$	${1}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} = {1}, \\ {2} & \text{if } \, {x} = {2}, \\ {5} & \text{if } \, {x} \in \{{3}, {4}, {5}\}. \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}\}$ is an interior GE-filter in $({X}, f)$ . But ${F}$ is not prominent interior GE-filter of type 1 since ${1}{*} ({5} {*} f({2})) = {1} {*} ({5} {*} {2}) = {1} {*} {1} = {1}\in {F}$ and $f({1}) = {1} \in {F}$ but $((f({2}) {*} {5}){*} {5}) {*} {2} = (({2} {*} {5}) {*} {5}) {*} {2} = ({5} {*} {5}) {*} {2} = {1} {*} {2} = {2}\notin {F}$ .

(2). Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in the following Table 15,

Table 15. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${5}$	${1}$	${5}$
${3}$	${1}$	${2}$	${1}$	${1}$	${1}$
${4}$	${1}$	${1}$	${3}$	${1}$	${5}$
${5}$	${1}$	${1}$	${1}$	${1}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} \in \{{1}, {2}, {4}\}, \\ {5} & \text{if } \, {x} \in \{{3}, {5}\}. \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}, {2} \}$ is a prominent interior GE-filter of type 1 in $({X}, f)$ . But it is not an interior GE-filter in $({X}, f)$ since ${2} {*} {4} = {1}$ and ${2} \in {F}$ but ${4} \notin {F}$ .

The following example shows that interior GE-filter and prominent interior GE-filter of type 2 are independent of each other.

Example 3.26. (1). Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in the following Table 16,

Table 16. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${1}$	${1}$
${3}$	${1}$	${2}$	${1}$	${1}$	${2}$
${4}$	${1}$	${2}$	${3}$	${1}$	${5}$
${5}$	${1}$	${1}$	${1}$	${1}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} \in \{{1}, {4}\} \\ {2} & \text{if } \, {x} = {2}, \\ {3} & \text{if } \, {x} = {3}, \\ {5} & \text{if } \, {x} = {5}. \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}, {4}\}$ is an interior GE-filter in $({X}, f)$ . But ${F}$ is not prominent interior GE-filter of type 2 since ${4}{*} ({2} {*} f({3})) = {4} {*} ({2} {*} {3}) = {4} {*} {1} = {1}\in {F}$ and $f({4}) = {1} \in {F}$ but $((3 {*} f({2})) {*} f(2)){*} {3} = (({3} {*} {2}) {*} {2}) {*} {3} = ({2} {*} {2}) {*} {3} = {1} {*} {3} = {3}\notin {F}$ .

(2). Let ${X} = \{{1}, {2}, {3}, {4}, {5}\}$ be a set with the Cayley table which is given in the following Table 17,

Table 17. Cayley table for the binary operation "

${*}$ ".

${*}$	${1}$	${2}$	${3}$	${4}$	${5}$
${1}$	${1}$	${2}$	${3}$	${4}$	${5}$
${2}$	${1}$	${1}$	${1}$	${1}$	${5}$
${3}$	${1}$	${1}$	${1}$	${1}$	${1}$
${4}$	${1}$	${1}$	${1}$	${1}$	${5}$
${5}$	${1}$	${1}$	${1}$	${1}$	${1}$

| Show Table

DownLoad: CSV

and define a mapping $f$ on ${X}$ as follows:

$\begin{align*} f : {X} \rightarrow {X}, \; {x}\mapsto \left\{\begin{array}{ll} {1} & \text{if } \, {x} = {1}, \\ {3} & \text{if } \, {x} \in \{{2}, {3}, {4}, {5}\}. \end{array}\right. \end{align*}$

Then $({X}, f)$ is an interior GE-algebra and ${F} : = \{{1}, {2}, {5} \}$ is a prominent interior GE-filter of type 2 in $({X}, f)$ . But it is not an interior GE-filter in $({X}, f)$ since ${5} {*} {4} = {1}$ and ${5} \in {F}$ but ${4} \notin {F}$ .

Before we conclude this paper, we raise the following question.

Question. Let $({X}, f)$ be an interior GE-algebra. Let ${F}$ and ${G}$ be interior GE-filters in $({X}, f)$ . If ${F} \subseteq {G}$ and ${F}$ is a prominent interior GE-filter of type 1 (resp., type 2) in $({X}, f)$ , then is ${G}$ also a prominent interior GE-filter of type 1 (resp., type 2) in $({X}, f)$ ?

4. Conclusions

We have introduced the concept of a prominent interior GE-filter (of type 1 and type 2), and have investigated their properties. We have discussed the relationship between a prominent GE-filter and a prominent interior GE-filter and the relationship between an interior GE-filter and a prominent interior GE-filter. We have found and provide examples where any interior GE-filter is not a prominent interior GE-filter and any prominent GE-filter is not a prominent interior GE-filter. We have provided conditions for an interior GE-filter to be a prominent interior GE-filter. We have given conditions under which an internal GE-filter larger than a given internal GE filter can become a prominent internal GE-filter, and have provided an example describing it. We have considered the relationship between a prominent interior GE-filter and a prominent interior GE-filter of type 1. We have found and provide examples to verify that a prominent interior GE-filter of type 1 and a prominent interior GE-filter of type 2, a prominent interior GE-filter and a prominent interior GE-filter of type 2, an interior GE-filter and a prominent interior GE-filter of type 1, and an interior GE-filter and a prominent interior GE-filter of type 2 are independent each other. In future, we will study the prime and maximal prominent interior GE-filters and their topological properties. Moreover, based on the ideas and results obtained in this paper, we will study the interior operator theory in related algebraic systems such as MV-algebra, BL-algebra, EQ-algebra, etc. It will also be used for pseudo algebra systems and further to conduct research related to the very true operator theory.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B02006812).

The authors wish to thank the anonymous reviewers for their valuable suggestions.

Conflict of interest

All authors declare no conflicts of interest in this paper.

Abbreviation ASD: Autism Spectrum Disorders; CNV: Copy Number Variants; DNA: Deoxyribonucleic Acid; ESSENCE: Early Symptomatic Syndromes Eliciting Neurodevelopmental Clinical Examinations; HPA axis: Hypothalamic-Pituitary-Adrenal Axis; IQ: Intelligence Quotient; LoF: Loss-of-Function; MAPP: Maltreatment-Associated Psychiatric Problems; OCD: Obsessive Compulsive Disorder; PTSD: Post Traumatic Stress Disorder; QoL: Quality of Life; RNA: Ribonucleic Acid (RNA); SNP: Single-Nucleotide Polymorphism; WGS: Whole-Genome Sequencing;
Acknowledgments

The author is very grateful to all the children, adolescents and their families who across the past years have shared their experiences as part of my clinical practice. Their experiences have inspired my theoretical research, thereby contributing to the development of the dynamic model, including the hypotheses guiding the model.

Conflict of interest

The author declares no conflicts of interest.

References

[1]	Baird G, Simonoff E, Pickles A, et al. (2006) Prevalence of disorders of the autism spectrum in a population cohort of children in South Thames: the Special Needs and Autism Project (SNAP). Lancet 368: 210-215. doi: 10.1016/S0140-6736(06)69041-7
[2]	Baxter AJ, Brugha TS, Erskine HE, et al. (2015) The epidemiology and global burden of autism spectrum disorders. Psychol Med 45: 601-613. doi: 10.1017/S003329171400172X
[3]	Brugha TS, Spiers N, Bankart J, et al. (2016) Epidemiology of autism in adults across age groups and ability levels. Br J Psychiatry 209: 498-503. doi: 10.1192/bjp.bp.115.174649
[4]	Elsabbagh M, Divan G, Koh Y-J, et al. (2012) Global prevalence of autism and other pervasive developmental disorders. Autism Res 5: 160-179. doi: 10.1002/aur.239
[5]	Atladottir HO, Gyllenberg D, Langridge A, et al. (2015) The increasing prevalence of reported diagnoses of childhood psychiatric disorders: a descriptive multinational comparison. Eur Child Adolesc Psychiatry 24: 173-183. doi: 10.1007/s00787-014-0553-8
[6]	Hansen SN, Schendel DE, Parner ET (2015) Explaining the increase in the prevalence of autism spectrum disorders. The proportion attributable to changes in reporting practices. JAMA Pediatr 169: 56-62. doi: 10.1001/jamapediatrics.2014.1893
[7]	Simonoff E (2012) Autism spectrum disorder: prevalence and cause may be bound together. Br J Psych 201: 88-89. doi: 10.1192/bjp.bp.111.104703
[8]	Gillberg C (2010) The ESSENCE in child psychiatry: Early Symptomatic Syndromes Eliciting Neurodevelopmental Clinical Examinations. Res Dev Dis 31: 1543-1551. doi: 10.1016/j.ridd.2010.06.002
[9]	Grove J, Ripke S, Als TD, et al. (2019) Identification of common genetic risk variants for autism spectrum disorder. Nat Genet 51: 431-444. doi: 10.1038/s41588-019-0344-8
[10]	Simonoff E, Pickles A, Charman, et al. (2008) Psychiatric disorders in children with autism spectrum disorders: Prevalence, comorbidity, and associated factors in a population-derived sample. J Am Acad Child Adolesc Psychiatry 47: 921-929. doi: 10.1097/CHI.0b013e318179964f
[11]	Wender CLA, Veenstra-VanderWeele J (2017) Challenge and Potential for Research on Gene-Environment Interactions in Autism Spectrum Disorder. Gene-Environment Transactions in Developmental Psychopathology, Advances in Development and Psychopathology: Brain Research Foundation Symposium Series 2 Springer International Publishing AG, 157-176. doi: 10.1007/978-3-319-49227-8_9
[12]	Sandin S, Lichtenstein P, Kuja-Halkola R, et al. (2014) The familial risk of autism. JAMA 311: 1770-1777. doi: 10.1001/jama.2014.4144
[13]	Robinson EB, St Pourcain B, Anttila V, et al. (2016) Genetic risk for autism spectrum disorders and neuropsychiatric variation in the general population. Nat Genet 48: 552-555. doi: 10.1038/ng.3529
[14]	Craddock N, Owen M (2010) The Kraepelinian dichotomy – going, going…but still not gone. Br J Psychiatry 196: 92-95. doi: 10.1192/bjp.bp.109.073429
[15]	Cross-Disorder Group of the Psychiatric Genomics Consortium (2013) Identification of risk loci with shared effects on five major psychiatric disorders: a genome-wide analysis. Lancet 381: 1371-1379.
[16]	Aggernæs B (2018) Autism: A transdiagnostic, dimensional construct of reasoning? Eur J Neurosci 47: 515-533. doi: 10.1111/ejn.13599
[17]	Aggernæs B (2016) Rethinking the concept of psychosis and the link between autism and Schizophrenia. Scand J Child Adol Psychiat Psychol 4: 4-11.
[18]	Bleuler E (1911) Dementia Praecox oder Gruppe der Schizophrenien Leipzig: Deuticke.
[19]	Piaget J (1967) La psychologie de l'intelligence Paris: A. Colin.
[20]	Alberoni F (1984) Movement and Institution New York: Columbia University Press.
[21]	Aggernæs A, Haugsted R (1976) Experienced reality in three to six year old children: A study of direct reality testing. J Child Psychol Psychiat 17: 323-335. doi: 10.1111/j.1469-7610.1976.tb00407.x
[22]	Aggernæs A, Paikin H, Vitger J (1981) Experienced reality in schizophrenia: How diffuse is the defect in reality testing? Indian J Psychol Med 4: 1-13. doi: 10.1177/0975156419810101
[23]	Clemmensen L, van Os J, Skovgaard AM, et al. (2014) Hyper-Theory-of-Mind in children with psychotic experiences. PLoS ONE 9: e113082. doi: 10.1371/journal.pone.0113082
[24]	Hallmayer J, Cleveland S, Torres A, et al. (2011) Genetic heritability and shared environmental factors among twin pairs with autism. Arch Gen Psychiatry 68: 1095-1102. doi: 10.1001/archgenpsychiatry.2011.76
[25]	Latham KE, Sapienza C, Engel N (2012) The epigenetic lorax: gene–environment interactions in human health. Epigenomics 4: 383-402. doi: 10.2217/epi.12.31
[26]	Miake K, Hirasawa T, Koide T, et al. (2012) Epigenetics in autism and other neurodevelopmental diseases. Adv Exp Med Biol 724: 91-98. doi: 10.1007/978-1-4614-0653-2_7
[27]	Shanen NC (2006) Epigenetics of autism spectrum disorders. Human Molecular Genetics 15: R138-R150. doi: 10.1093/hmg/ddl213
[28]	Siniscalco D, Cirillo A, Bradstreet JJ, et al. (2013) Epigenetic findings in autism: New perspectives for therapy. Int J Environ Res Public Health 10: 4261-4273. doi: 10.3390/ijerph10094261
[29]	Belmonte MK, Cook EH, Anderson GM, et al. (2004) Autism as a disorder of neural information processing: directions for research and targets for therapy. Mol Psychiatry 9: 646-663. doi: 10.1038/sj.mp.4001499
[30]	Courchesne E (2002) Abnormal early brain development in autism. Mol Psychiatry 7: S21-S23. doi: 10.1038/sj.mp.4001169
[31]	Menon V (2011) Large-scale brain networks and psychopathology: A unifying triple network model. Trends Cogn Sci 15: 483-506. doi: 10.1016/j.tics.2011.08.003
[32]	Szatmari P (2000) The classification of autism, Asperger's syndrome, and pervasive developmental disorder. Can J Psychiat 45: 731-738. doi: 10.1177/070674370004500806
[33]	Szyf M (2009) Epigenetics, DNA methylation, and chromatin modifying drugs. Annu Rev Pharmacol Toxicol 49: 243-263. doi: 10.1146/annurev-pharmtox-061008-103102
[34]	Plomin R (2011) Commentary: Why are children in the same family so different? Non-shared environment three decades later. Int J Epidemiol 40: 582-592. doi: 10.1093/ije/dyq144
[35]	Larsson HJ, Eaton WW, Madsen KM, et al. (2005) Risk factors for autism: perinatal factors, parental psychiatric history, and socioeconomic status. Am J Epidemiol 161: 916-925. doi: 10.1093/aje/kwi123
[36]	Lauritsen MB, Pedersen CB, Mortensen PB (2005) Effects of familial risk factors and place of birth on the risk of autism: a nationwide registerbased study. J Child Psychol Psyc 46: 963-971. doi: 10.1111/j.1469-7610.2004.00391.x
[37]	Gardener H, Spiegelman D, Buka SL (2009) Prenatal risk factors for autism: comprehensive meta-analysis. Br J Psychiatry 195: 7-14. doi: 10.1192/bjp.bp.108.051672
[38]	Ramaswami G, Geschwind DH (2018) Chapter 21 Genetics of autism spectrum disorder. Handbook of Clinical Neurology, Vol. 147 (3rd series) Neurogenetics, Part I Elsevier, 321-329.
[39]	Betancur C (2011) Etiological heterogeneity in autism spectrum disorders: More than 100 genetic and genomic disorders and still counting. Brain Res 1380: 42-77. doi: 10.1016/j.brainres.2010.11.078
[40]	Nava C, Keren B, Mignot C, et al. (2014) Prospective diagnostic analysis of copy number variants using SNP microarrays in individuals with autism spectrum disorders. Eur J Hum Genet 22: 71-78. doi: 10.1038/ejhg.2013.88
[41]	The Autism Genome Project Consortium (2007) Mapping autism risk loci using genetic linkage and chromosomal rearrangements. Nat Genet 39: 319-328.
[42]	Weiner DJ, Wigdor EM, Ripke S, et al. (2017) Polygenic transmission disequilibrium confirms that common and rare variation act additively to create risk for autism spectrum disorders. Nat Genet 49: 978-985. doi: 10.1038/ng.3863
[43]	Gaugler T, Klei L, Sanders SJ, et al. (2014) Most genetic risk for autism resides with common variation. Nat Genet 46: 881-885. doi: 10.1038/ng.3039
[44]	Guilmatre A, Dubourg C, Mosca AL, et al. (2009) Recurrent rearrangements in synaptic and neurodevelopmental genes and shared biologic pathways in schizophrenia, autism, and mental retardation. Arch Gen Psychiat 66: 947-956. doi: 10.1001/archgenpsychiatry.2009.80
[45]	McCarthy SE, Gillis J, Kramer M, et al. (2014) De novo mutations in schizophrenia implicate chromatin remodeling and support a genetic overlap with autism and intellectual disability. Mol Psychiatry 19: 652-658. doi: 10.1038/mp.2014.29
[46]	Hannon E, Schendel D, Ladd-Acosta C, et al. (2018) Elevated polygenic burden for autism is associated with differential DNA methylation at birth. Genome Med 10: 19. doi: 10.1186/s13073-018-0527-4
[47]	Tran NQV, Miyake K (2017) Neurodevelopmental disorders and environmental toxicants: epigenetics as an underlying mechanism. Int J Genomics 2017: 7526592.
[48]	Waddington CH (2012) The epigenotype. Int J Epidemiol 41: 10-13. doi: 10.1093/ije/dyr184
[49]	Fraga MF, Ballestar E, Paz MF, et al. (2005) Epigenetic differences arise during the lifetime of monozygotic twins. Proc Natl Acad Sci USA 102: 10604-10609. doi: 10.1073/pnas.0500398102
[50]	Lee R, Avramopoulos D (2014) Introduction to Epigenetics in Psychiatry. Epigenetics in Psychiatry Elsevier Inc., 3-25. doi: 10.1016/B978-0-12-417114-5.00001-2
[51]	Bale TL (2014) Lifetime stress experience: transgenerational epigenetics and germ cell programming. Dialogues Clin Neurosci 16: 297-305. doi: 10.31887/DCNS.2014.16.3/tbale
[52]	Gos M (2013) Epigenetic mechanisms of gene expression regulation in neurological diseases. Acta Neurobiol Exp 73: 19-37.
[53]	James SJ, Melnyk S, Jernigan S, et al. (2006) Metabolic endophenotype and related genotypes are associated with oxidative stress in children with autism. Am J Med Genet Part B 141B: 947-956. doi: 10.1002/ajmg.b.30366
[54]	Deth R, Muratore C, Benzecry J, et al. (2008) How environmental and genetic factors combine to cause autism: A redox/methylation hypothesis. Neurotoxicology 29: 190-201. doi: 10.1016/j.neuro.2007.09.010
[55]	Homs A, Codina-Solà M, Rodríguez-Santiago B, et al. (2016) Genetic and epigenetic methylation defects and implication of the ERMN gene in autism spectrum disorders. Transl Psychiatry 6: e855. doi: 10.1038/tp.2016.120
[56]	Ginsberg MR, Rubin RA, Falcone T, et al. (2012) Brain transcriptional and epigenetic associations with autism. PLoS ONE 7: e44736. doi: 10.1371/journal.pone.0044736
[57]	Wong CCY, Meaburn EL, Ronald A, et al. (2014) Methylomic analysis of monozygotic twins discordant for autism spectrum disorder and related behavioural traits. Mol Psychiatry 19: 495-503. doi: 10.1038/mp.2013.41
[58]	Minnis H (2013) Maltreatment-Associated Psychiatric Problems: An Example of Environmentally Triggered ESSENCE? Sci World J 2013: 148468. doi: 10.1155/2013/148468
[59]	Pritchett R, Pritchett J, Marshall E, et al. (2013) Reactive attachment disorder in the general population: a hidden ESSENCE disorder. Sci World J 2013: 81815.
[60]	Dinkler L, Lundström S, Gajwani R, et al. (2017) Maltreatment-associated neurodevelopmental disorders: a co-twin control analysis. J Child Psychol Psychiatr 58: 691-701. doi: 10.1111/jcpp.12682
[61]	McEwen BS (2008) Central effects of stress hormones in health and disease: understanding the protective and damaging effects of stress and stress mediators. Eur J Pharmacol 583: 174-185. doi: 10.1016/j.ejphar.2007.11.071
[62]	Rostène W, Sarrieau A, Nicot A, et al. (1995) Steroid effects on brain functions: An example of the action of glucocorticoids on central dopaminergic and neurotensinergic systems. J Psychiatry Neurosci 20: 349-356.
[63]	Meaney MJ, Szyf M (2005) Environmental programming of stress responses through DNA methylation: life at the interface between a dynamic environment and a fixed genome. Dialogues Clin Neurosci 7: 103-123. doi: 10.31887/DCNS.2005.7.2/mmeaney
[64]	Weaver ICG, Champagne FA, Brown SE, et al. (2005) Reversal of maternal programming of stress responses in adult offspring through methyl supplementation: Altering epigenetic marking later in life. J Neurosci 25: 11045-11054. doi: 10.1523/JNEUROSCI.3652-05.2005
[65]	McGowan PO, Matthews SG (2018) Prenatal stress, glucocorticoids, and developmental programming of the stress response. Endocrinology 159: 69-82. doi: 10.1210/en.2017-00896
[66]	Stenz L, Schechter DS, Serpa SR, et al. (2018) Intergenerational transmission of DNA methylation signatures associated with early life stress. Curr Genomics 19: 665-675. doi: 10.2174/1389202919666171229145656
[67]	Oberlander TF, Weinberg J, Papsdorf M, et al. (2008) Prenatal exposure to maternal depression, neonatal methylation of human glucocorticoid receptor gene (NR3C1) and infant cortisol stress responses. Epigenetics 3: 97-106. doi: 10.4161/epi.3.2.6034
[68]	Corbett BA, Mendoza S, Wegelin JA, et al. (2008) Variable cortisol circadian rhythms in children with autism and anticipatory stress. J Psychiatry Neurosci 33: 227-234.
[69]	Corbett BA, Schupp CW, Levine S, et al. (2009) Comparing cortisol, stress, and sensory sensitivity in children with autism. Autism Res 2: 39-49. doi: 10.1002/aur.64
[70]	Corbett BA, Schupp CW, Lanni KE (2012) Comparing biobehavioral profiles across two social stress paradigms in children with and without autism spectrum disorders. Mol Autism 3: 13. doi: 10.1186/2040-2392-3-13
[71]	Corbett BA, Muscatello RA, Blain SD (2016) Impact of sensory sensitivity on physiological stress response and novel peer interaction in children with and without autism spectrum disorder. Front Neurosci 10: 278. doi: 10.3389/fnins.2016.00278
[72]	Tordjman S, Anderson GM, Kermarrec S, et al. (2014) Altered circadian patterns of salivary cortisol in low-functioning children and adolescents with autism. Psychoneuroendocrinology 50: 227-245. doi: 10.1016/j.psyneuen.2014.08.010
[73]	Bishop-Fitzpatrick L, Mazefsky CA, Minshew NJ, et al. (2015) The relationship between stress and social functioning in adults with autism spectrum disorder and without intellectual disability. Autism Res 8: 164-173. doi: 10.1002/aur.1433
[74]	Bishop-Fitzpatrick L, Minshew NJ, Mazefsky CA, et al. (2017) Perception of life as stressful, not biological response to stress, is associated with greater social disability in adults with autism spectrum disorder. Autism Dev Disord 47: 1-16. doi: 10.1007/s10803-016-2910-6
[75]	Sønderby IE, Gústafsson Ó, Doan NT, et al. (2018) Dose response of the 16p11.2 distal copy number variant on intracranial volume and basal ganglia. Mol Psychiatry 25: 584-602. doi: 10.1038/s41380-018-0118-1
[76]	Turner TN, Coe BP, Dickel DE, et al. (2017) Genomic patterns of de novo mutation in simplex autism. Cell 171: 710-722. doi: 10.1016/j.cell.2017.08.047
[77]	Schork AJ, Won H, Appadurai V, et al. (2019) A genome-wide association study of shared risk across psychiatric disorders implicates gene regulation during fetal neurodevelopment. Nat Neurosci 22: 353-361. doi: 10.1038/s41593-018-0320-0
[78]	Gandal MJ, Haney JR, Parikshak NN, et al. (2018) Shared molecular neuropathology across major psychiatric disorders parallels polygenic overlap. Science 359: 693-697. doi: 10.1126/science.aad6469
[79]	Shulha HP, Cheung I, Guo Y, et al. (2013) Coordinated cell type–specific epigenetic remodeling in prefrontal cortex begins before birth and continues into early adulthood. PLoS Genet 9: e1003433. doi: 10.1371/journal.pgen.1003433
[80]	Belmonte MK, Yurgelun-Todd DA (2003) Functional anatomy of impaired selective attention and compensatory processing in autism. Brain Res Cogn Res 17: 651-664. doi: 10.1016/S0926-6410(03)00189-7
[81]	Livingston LA, Happé F (2017) Conceptualising compensation in neurodevelopmental disorders: Reflections from autism spectrum disorders. Neurosci Biobehav Rev 80: 729-742. doi: 10.1016/j.neubiorev.2017.06.005
[82]	Weems CF (2015) Biological correlates of child and adolescent responses to disaster exposure: A bio-ecological model. Curr Psychiatry Rep 17: 1-7. doi: 10.1007/s11920-015-0588-7
[83]	Weems CF, Russel JD, Neill EL, et al. (2019) Annual Research Review: Pediatric posttraumatic stress disorder from a neurodevelopmental network perspective. J Child Psychol Psychiatr 60: 395-408. doi: 10.1111/jcpp.12996
[84]	Ouss-Ryngaert L, Golse B (2010) Linking neuroscience and psychoanalysis from a developmental perspective: Why and how? J Physiol 104: 303-308.
[85]	Hosman CMH, van Doesum KTM, van Santvoort F (2009) Prevention of emotional problems and psychiatric risks in children of parents with a mental illness in the Netherlands: I. The scientific basis to a comprehensive approach. Aust e-J Advancement Ment Health 8: 250-263. doi: 10.5172/jamh.8.3.250
[86]	Sarovic D (2018) A framework for neurodevelopmental disorders: Operationalization of a pathogenetic triad for clinical and research use. Gillberg Neuropsychiatry Centre, Institute of Neuroscience and Physiology, Sahlgrenska Academy University of Gothenburg Available from: https://psyarxiv.com/mbeqh/.
[87]	Caspi A, Houts RM, Belsky DW, et al. (2014) The p factor: One general psychopathology factor in the structure of psychiatric disorders? Clin Psychol Sci 2: 119-137. doi: 10.1177/2167702613497473
[88]	Sameroff AJ, Mackenzie MJ (2003) Research strategies for capturing transactional models of development: The limits of the possible. Dev Psychopathol 15: 613-640. doi: 10.1017/S0954579403000312
[89]	Bishop-Fitzpatrick L, Mazefsky CA, Eack SM (2018) The combined impact of social support and perceived stress on quality of life in adults with autism spectrum disorder and without intellectual disability. Autism 22: 703-711. doi: 10.1177/1362361317703090
[90]	Bishop-Fitzpatrick L, Smith LE, Greenberg JS, et al. (2017) Participation in recreational activities buffers the impact of perceived stress on quality of life in adults with autism spectrum disorder. Autism Res 10: 973-982. doi: 10.1002/aur.1753
[91]	Yehuda R, Lehrner A, Bierer LM (2018) The public reception of putative epigenetic mechanisms in the transgenerational effects of trauma. Environ Epigenet 4: 1-7. doi: 10.1093/eep/dvy018
[92]	Clarke T-K, Lupton MK, Fernandez-Pujals AM, et al. (2016) Common polygenic risk for autism spectrum disorder (ASD) is associated with cognitive ability in the general population. Mol Psychiatry 21: 419-425. doi: 10.1038/mp.2015.12
[93]	Robinson EB, Samocha KE, Kosmicki JA, et al. (2014) Autism spectrum disorder severity reflects the average contribution of de novo and familial influences. Proc Natl Acad Sci USA 111: 15161-15165. doi: 10.1073/pnas.1409204111
[94]	Sanders SJ, He X, Willsey AJ, et al. (2015) Insights into autism spectrum disorder genomic architecture and biology from 71 risk loci. Neuron 87: 1215-1233. doi: 10.1016/j.neuron.2015.09.016
[95]	Bakermans-Kranenburg MJ, van Ijzendoorn MH (2006) Gene-environment interaction of the dopamine D4 receptor (DRD4) and observed maternal insensitivity predicting externalizing behavior in preschoolers. Dev Psychobiol 48: 406-409. doi: 10.1002/dev.20152
[96]	Kumsta R, Stevens S, Brookes K, et al. (2010) 5HTT genotype moderates the influence of early institutional deprivation on emotional problems in adolescence: evidence from the English and Romanian Adoptee (ERA) study. J Child Psychol Psychiatry 51: 755-762. doi: 10.1111/j.1469-7610.2010.02249.x
[97]	Yang C-J, Tan H-P, Yang F-Y (2015) The cortisol, serotonin and oxytocin are associated with repetitive behavior in autism spectrum disorder. Res Autism Spectr Disord 18: 12-20. doi: 10.1016/j.rasd.2015.07.002
[98]	Meier SM, Petersen L, Schendel DE, et al. (2015) Obsessive-compulsive disorder and autism spectrum disorders: Longitudinal and offspring risk. PLoS ONE 10: e0141703. doi: 10.1371/journal.pone.0141703
[99]	Valla JM, Belmonte MK (2013) Detail-oriented cognitive style and social communicative deficits, within and beyond the autism spectrum: independent traits that grow into developmental interdependence. Dev Rev 33: 371-398. doi: 10.1016/j.dr.2013.08.004
[100]	Lai MC, Lombardo MV, Ruigrok AN, et al. (2017) Quantifying and exploring camouflaging in men and women with autism. Autism 21: 690-702. doi: 10.1177/1362361316671012
[101]	Atladóttir HO, Thorsen P, Schendel DE, et al. (2010) Association of hospitalization for infection in childhood with diagnosis of autism spectrum disorders. Arch Pediatr Adolesc Med 164: 470-477. doi: 10.1001/archpediatrics.2010.9
[102]	Nudel R, Wang Y, Appadurai V, et al. (2019) A large-scale genomic investigation of susceptibility to infection and its association with mental disorders in the Danish population. Transl Psychiatry 9: 283. doi: 10.1038/s41398-019-0622-3
[103]	Danese A, Moffitt TE, Arseneault L, et al. (2017) The origins of cognitive deficits in victimized children: Implications for neuroscientists and clinicians. Am J Psychiatry 174: 349-361. doi: 10.1176/appi.ajp.2016.16030333
[104]	Kim SY, Bottema-Beutel K (2019) A meta regression analysis of quality of life correlates in adults with ASD. Res Autism Spectr Disord 63: 23-33. doi: 10.1016/j.rasd.2018.11.004
[105]	Plana-Ripoll O, Pedersen CB, Holtz Y, et al. (2019) Exploring comorbidity within mental disorders among a danish national population. JAMA Psychiatry 76: 259-270. doi: 10.1001/jamapsychiatry.2018.3658
[106]	Cuthbert N, Insel R (2013) Toward the future of psychiatric diagnosis: the seven pillars of RDoC. BMC Med 11: 126. doi: 10.1186/1741-7015-11-126
[107]	(2013) American Psychiatric AssociationDiagnostic and Statistical Manual of Mental Disorders. American Psychiatric Publishing.
[108]	(1993) WHOThe ICD-10 Classification of Mental and Behavioural Disorders: Diagnostic Criteria for Research. Geneva: World Health Organization.
[109]	Polanczyk GV, Casella C, Jaffee SR (2019) Commentary: ADHD lifetime trajectories and the relevance of the developmental perspective to Psychiatry: reflections on Asherson and Agnew-Blais. J Child Psychol Psychiatry 60: 353-355. doi: 10.1111/jcpp.13050
[110]	Narusyte J, Neiderhiser JM, D'Onofrio BM, et al. (2008) Testing different types of genotype–environment correlation: An extended children-of-twins model. Dev Psychol 44: 1591-1603. doi: 10.1037/a0013911
[111]	Sykes NH, Lamb JA (2007) Autism: The quest for the genes. Expert Rev Mol Med 9: 1-15. doi: 10.1017/S1462399407000452

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