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Abstract

In this paper, we investigate and explore the properties of quasi-topological loops with respect to irresoluteness. Moreover, we construct an example of a quasi-irresolute topological inverse property-loop by using a zero-dimensional additive metrizable perfect topological inverse property-loop $L^{*}$ with relative topology $τ_{L^{'}}$ .

Keywords

Semi-open set pre-semi-open mapping irresolute mapping semi-compactness quasi-irresolute topological inverse property-loop

Introduction

It is always captivating to dig into the relationship of certain topological spaces over various algebraic structures. Mostly, it requires the continuity of algebraic operations. By and large, to generalize these structures, an enfeeble form of continuity is discussed. With the instauration of a semi-open set by N. Levine, many of the mathematicians examined several results by using semi-continuity and semi-open sets.^1–9 A number of new results are obtained when an open set is replaced by a semi-open set and continuity is replaced by irresoluteness.^10–13

Levine¹⁴ defined a semi-open set as a subset $M$ of a topological space $X$ is semi-open, if there is an open set $O$ in $X$ such that

O \subseteq M \subseteq C l (O)

M \subseteq C l (I n t (M))

Here,

C l (O)

is the closure of the open set

O

. The class of all semi-open sets in

X

is denoted by

S O (X)

. If

M_{1} \subseteq X

and

M_{2} \subseteq Y

are semi-open sets then

M_{1} \times M_{2}

is semi-open in

X \times Y

. A set

M_{2} \subseteq X

is a semi-open neighbourhood of a point

t \in X

, if there exists

M_{1} \in S O (X)

such that

t \in M_{1} \subseteq M_{2}

A point

t \in X

is a semi-interior point of

M

, if there exists a semi-open set

M^{'}

such that

t \in M^{'} \subseteq M

s I n t (M)

is the set of all semi-interior points of

M

s C l (M)

is the intersection of all the semi-closed super sets of

M

. For any semi-open set

M_{t}

t \in s C l (M^{'})

if and only if

M_{t} \cap M^{'} \neq ϕ

A mapping $f : X \to Y$ is said to be

Irresolute, if for each semi-open set $M_{2} \subseteq Y$ the set $f^{- 1} (M_{2})$ is semi-open in $X$ or Bosan.¹⁵

Pre-semi-open, if for each semi-open set $M$ in $X$ , $f (M)$ is semi-open in.

Semi-homeomorphism, if $f$ is bijective, pre-semi-open, and irresolute.

A subset

M^{'}

of a topological space

X

is said to be semi-compact (semi-Lindelof), if there exists a finite (countable) subcover for any semi-open cover of

M^{'}

X

.^16,17 A space

(L, τ)

is called s-regular, if for every

t \in L

and every closed set

Q \subseteq L

, there exists semi-open neighbourhoods

M

containing

t

such that

t \notin Q

and

N

containing

Q

such that

M \cap N = ϕ

. A class of subsets of a space

X

is said to be semi-discrete if each

t \in X

has a semi-open neighbourhood intersecting at most one member of collection. A space

(L, τ)

is said to be semi-homogeneous, if there is a semi-homeomorphism

f

L

onto itself such that for every

r, s \in L

implies

f (r) = s

.¹⁸ Let

M \in S O (L, e)

in a quasi-irresolute topological loop

(L, *, τ)

. A subset

S

L

is said to be

M

-semi disjoint, if for every disjoint

r, s \in S

implies

r \notin s * M

A groupoid $(L, *)$ is a loop if the conditions given below are satisfied:

$L$ contains an identity element.

For every $t_{1} \in L$ , the mappings $l_{t_{1}} : L \to L$ and $r_{t_{1}} : L \to L$ are bijective, where $l_{t_{1}} (t_{2}) = t_{1} * t_{2}$ and $r_{t_{1}} (t_{2}) = t_{2} * t_{1}$ for all $t_{2} \in L$ .

An inverse property loop (IP-loop)

L

is a loop having two sided inverse

t^{- 1}

such that

(r * t) * t^{- 1} = r = t^{- 1} * (t * r)

for all

r, t \in L

Left (right) translations and left (right) inverse mappings are defined on a loop $(L, *)$ as follows:

Left translation $l_{t_{1}} : L \to L$ is given by $l_{t_{1}} (t_{2}) = t_{1} * t_{2}$ .

Right translation $r_{t_{1}} : L \to L$ is given as $r_{t_{1}} (t_{2}) = t_{2} * t_{1}$ .

Left inverse mapping $i_{L} : L \to L$ is defined by $i_{L} (t) = (t)_{L}^{- 1}$ .

Right inverse mapping $i_{R} : L \to L$ is defined as $i_{R} (t) = (t)_{R}^{- 1}$ .

where

t, t_{1}, t_{2} \in L

This article is devoted to investigate how separately irresolute multiplication and irresolute inverse mapping of topological spaces are defined over loops, in particular, over IP-loops. In addition, we have provided some examples of quasi-topological loops and quasi-topological IP-loops with respect to irresoluteness.

Some significant results on quasi-irresolute topological IP-loop

Definition 1

A triplet $(L, *, τ)$ is said to be a quasi-irresolute topological IP-loop, if the conditions given below are satisfied:

$(L, *)$ is an IP-loop.

$(L, τ)$ is a topological space.

Multiplication mapping is separately irresolute in $(L, *, τ)$ .

Inverse mapping is irresolute in $(L, *, τ)$ .

Example 1

Consider a zero-dimensional additive metrizable perfect topological IP-loop $L^{*}$ having a commutative property with an infinite null sequence ${N_{0}}$ . Here, we fix a sequence $S_{n}$ of clopen subsets in $L^{*}$ such that: for $n^{*} \neq n$ , separate unions with their additive inverses, respectively, have empty intersection, and any open set $M$ containing identity contains all the terms of $S_{n}$ after some fixed terms. Assume a base $M_{n}$ of clopen sets containing identity in $L^{*}$ with $S_{n} \subseteq M_{n} \subseteq M_{n} + M_{n} \subseteq M_{n - 1} = - M_{n - 1}$ . Consider $μ^{*}$ containing clopen sets $M$ in $(L^{*} ∖ {0}) \times L^{*}$ with $M = - M$ and $\cup {((M_{m} ∖ {0}) \times {0}) \cup (M_{2 n} \times M_{n + 1})} \subseteq M$ for all $n > m - 2$ . Now, set $μ$ consisting of ${t} \cup (M + t)$ , where $t \in L^{*} \times L^{*}$ and $M \in μ^{*}$ form a base for a topology $τ$ on $L^{*} \times L^{*}$ . Fix two dense-in-itself sub-loops $L_{1}, L_{2}$ of $L^{*}$ such that ${N_{0}} \subseteq C l_{L^{*}} L_{1}$ . For $L^{'} = L_{1} \times L_{2}$ as subspace of $τ$ form quasi-irresolute topological IP-loop with relative topology $τ_{L^{'}}$ .

Example 2

The smallest IP loop $L = {1, 2, 3, 4, 5, 6, 7}$ of order $7$ with topology is not a quasi-irresolute topological IP-loop. Here, inverse mapping is irresolute but left as well as right translations are not irresolute.

Theorem 1

For $M, N \in S O (L, e)$ in quasi-irresolute topological IP-loop $(L, *, τ)$ such that $N^{4} \subseteq M$ and $N^{- 1} = N$ . If $S \subseteq L$ is $M$ -semi-disjoint, then ${s * N : s \in S}$ being the family of semi-open sets is semi-discrete in $L$ .

Proof

To prove semi-discreteness of ${s * N : s \in S}$ , it is sufficient to prove that for every $t \in L$ , $t * N$ intersects at most one of ${s * N : s \in S}$ . Assume contrarily that, for some $t \in L$ , there exists $r, s \in S$ such that $t * N \cap s * N \neq ϕ$ and $t * N \cap r * N \neq ϕ$ , then $t^{- 1} * s \in N^{2}$ and $t \in r * N^{2}$ . Here $s = t * (t^{- 1} * s) \in r * N^{4} \subseteq r * M$ . Hence, $s \in r * M$ . A contradiction to the fact that $S$ is $M$ -semi-disjoint. So ${s * N : s \in S}$ being the family of semi-open sets is semi-discrete in $L$ . □

Theorem 2

If a homomorphism $f$ from $L$ to $L^{'}$ (quasi-irresolute topological IP-loops) is irresolute at $e_{L}$ , then is on $L$ .

Proof

Let for all $t_{1} \in L$ , suppose $M_{2}^{'} \in S O (L^{'}, t_{2})$ , where $t_{2} = f (t_{1})$ . Then, there is $M_{1}^{'} \in S O (L^{'}, e_{L^{'}})$ such that $l_{t_{2}}^{'} (M_{1}^{'}) = t_{2} * M_{1}^{'} \subseteq M_{2}^{'}$ . This implies $f (M_{1}) \subseteq M_{1}^{'}$ for some $M_{1} \in S O (L, e_{L})$ . So for $t_{1} * M_{1} \in S O (L, t_{1}),$ we have $f (t_{1} * M_{1}) = f (t_{1}) * f (M_{1}) = t_{2} * f (M_{1}) \subseteq t_{2} * M_{1}^{'} \subseteq M_{2}^{'}$ . Hence, $f$ is irresolute on $L$ . □

Theorem 3

A free product of a semi-open subset with any subset in a quasi-irresolute topological IP-loop is semi-open.

Proof

Let $(L, *, τ)$ be a quasi-irresolute topological IP-loop, $M \in S O (L)$ and $N \subseteq L$ . If $n \in N$ and $t \in M * n$ then $t = m * n$ for some $m \in M$ . This implies that $m = t * n^{- 1} = r_{n^{- 1}} (t)$ for fixed $n^{- 1} \in L$ , therefore, there is a semi-open set $S_{t}$ such that $r_{n^{- 1}} (S_{t}) \subseteq M$ , $S_{t} * n^{- 1} \subseteq M$ , $S_{t} \subseteq M * n$ . This implies that $t$ is the semi-interior point of $M * n$ . Hence $M * n \in S O (L)$ . Therefore, $M * N = \cup_{n \in N} M * n$ is semi-open. □

Corollary 1

In quasi-irresolute topological IP-loop $(L, *, τ)$ . If $M \in S O (L)$ then $\cup_{n = 1}^{\infty} M^{n} \in S O (L)$ .

Corollary 2

Left and right translations are semi-homeomorphism in a quasi-irresolute topological IP-loop.

Theorem 4

A quasi-irresolute topological IP-loop is a semi-homogeneous space.

Proof

Let $(L, *, τ)$ be a quasi-irresolute topological loop. Suppose $n = t^{- 1} * m$ , where $m, n, t \in L$ . So $r_{n} (t) = t * n = t * (t^{- 1} * m) = m$ . □

Theorem 5

Multiplication mapping is jointly pre-semi-open in a quasi-irresolute topological IP-loop.

Proof

Let $(L, *, τ)$ be a quasi-irresolute topological IP-loop and $f : L \times L \to L$ be a multiplication mapping defined by $f (M \times N) = M N$ . If $M, N \in S O (L)$ , then $M \times N$ is semi-open in $L \times L$ . So $f (M \times N) = M N$ is semi-open in $L$ by Theorem 3. Hence, $f$ is pre-semi-open. □

Theorem 6

Every sub-loop of a quasi-irresolute topological IP-loop containing a non-empty semi-open subset is semi-open.

Proof

Let $(L, *, τ)$ be a quasi-irresolute topological IP-loop and $N$ is its sub-loop. $M$ is a non-empty semi-open set in $L$ with $M \subseteq N$ . Then by Theorem 2, for every $n \in N$ the set $r_{n} (M) = M * n$ is semi-open in $(L, *, τ)$ . Therefore, $N = \cup_{n \in N} (M * n)$ is semi-open in $L$ . □

Remark 1

Every semi-open sub-loop in a quasi-irresolute topological IP-loop is also semi-closed.

Theorem 7

A semi-interior of a non-empty sub-loop $L^{'}$ of a quasi-irresolute topological IP-loop $(L, *, τ)$ is non-empty if and only if it is semi-open.

Proof

Suppose, $s I n t (L^{'}) \neq ϕ$ and $t \in s I n t (L^{'})$ then there exists $M \in S O (L)$ such that $t \in M \subseteq L^{'}$ . So, for all $s \in L^{'}$ , we have $l_{s} (M) = s * M \subseteq s * L^{'} = L^{'}$ . Additionally, $s * M$ is semi-open, therefore $L^{'} = \cup {s * M : s \in L^{'}}$ is semi-open. The converse is trivial. □

Theorem 8

Let $(L, *, τ)$ be a quasi-irresolute topological IP-loop. Then for each semi-open neighbourhood $M$ of $e$ and for each $S \subseteq L$ gives $s C l (S) \subseteq S * M$ .

Proof

Consider $t \in s C l (S)$ and for a semi-open neighbourhood $N$ of $e$ , $t * N$ is semi-open neighbourhood of $t$ . So, there exists $s \in S \cap t * N$ , that is, $s \in t * N$ . It gives $s = t * n$ for some $n \in N$ . Therefore, $t = s * n^{- 1} \in s * N^{- 1} \subseteq S * M$ by Theorem $2.10$ ¹⁰ for $M \in S O (L, e)$ . □

Theorem 9

Let $μ_{e}$ be a class of semi-open sets containing $e$ in quasi-irresolute topological IP-loop $(L, *, τ)$ . Then for each $S \subseteq L$ , $s C l (S) = \cap {S * M : M \in μ_{e}}$ .

Proof

For $t \notin s C l (S)$ implies $t \notin S$ , and there exists $N \in μ_{e}$ such that $t * N \cap S = ϕ$ . As $M \in μ_{e}$ satisfies $M^{- 1} \subseteq N$ . Therefore, $t * M^{- 1} \cap S = ϕ$ , ${t} \cap S * M = ϕ$ . So $t \notin S * M$ . Moreover, by Theorem 8 $s C l (S) \subseteq S * M$ . Thereupon, $s C l (S) = \cap {S * M : M \in μ_{e}}$ . □

Theorem 10

Semi-closure of every semi-open neighbourhood of identity element in a quasi-irresolute topological IP-loop contain in its free product with itself.

Proof

Let $M$ is semi-open neighbourhood of $e$ and $t \in s C l (M)$ . As $t * M^{- 1}$ is semi-open neighbourhood of $t$ so it meets $M$ . Furthermore, there exists $r \in M$ such that $r = t * s^{- 1}$ for $s \in M$ . Then $t = r * s \in M * M$ . □

Some properties of a quasi-irresolute topological loop

Definition 2

A triplet $(L, *, τ)$ is said to be a quasi-irresolute topological loop, if the conditions given below are satisfied:

$(L, *)$ is a loop.

$(L, τ)$ is a topological space.

Multiplication mapping is separately irresolute in $(L, *, τ)$ .

Left and right inverse mappings are irresolute in $(L, *, τ)$ .

Example 3

$L_{5} (2)$ with discrete and indiscrete topologies is a quasi-irresolute topological loop.

Example 4

Let $L = L_{5} (3)$ be a loop of order $6$ and a topology $τ^{'} = {ϕ, {e}, {t_{1}, t_{2}}, {e, t_{1}, t_{2}}, L}$ on $L$ . For ${e, t_{1}, t_{2}} \in S O (L)$ , $l_{t_{2}}^{- 1} {e, t_{1}, t_{2}} = {t_{2}, t_{5}, e} \notin S O (L)$ . Hence, $(L, *, τ)$ is not a quasi-irresolute topological loop.

Lemma 1

Left and right inverse mappings are semi-homeomorphism in a quasi-irresolute topological loop.

Corollary 3

In a loop with topology, if left (right) inverse mapping is pre-semi-open, then right (left) inverse mapping is irresolute.

Theorem 11

By taking semi-closure and semi-interior, every set in a quasi-irresolute topological loop preserves its symmetric property.

Proof

Let $(L, *, τ)$ be a quasi-irresolute topological loop and $K$ is a symmetric subset of $L$ . Then $i_{L} (s C l (K)) = (s C l (K))_{L}^{- 1} = s C l (K_{L}^{- 1}) = s C l K)$ .¹⁹ Similarly, $(s C l (K))_{R}^{- 1} = s C l (K)$ and for semi-interior. □

Theorem 12

In a quasi-irresolute topological loop, openness is a sufficient condition for a sub-loop is to be a quasi-irresolute topological loop.

Proof

Let $(L, *, τ)$ be a quasi-irresolute topological loop and $L^{'}$ be an open sub-loop of $L$ . So for every semi-open neighbourhood $M_{1}$ in $L$ with $t \in L^{'}$ and $M_{1} \subseteq L^{'}$ , $l_{t}^{- 1} (M_{1}) \in S O (L)$ and $l_{t}^{- 1} (M_{1}) \subseteq L^{'}$ , gives $l_{t}^{- 1} (M_{1}) \in S O (L^{'})$ .²⁰ Clearly, $l_{t}$ is irresolute in $L^{'}$ . In the same way, $r_{t}, i_{L} a n d i_{R}$ are irresolute in $L^{'}$ . □

Note that a non-empty subset $L^{'}$ of a loop $L$ is said to be a sub-loop of $L$ if and only if for all $t_{1}, t_{2} \in L^{'}$ , $t_{1} * (t_{2})_{L}^{- 1} \in L^{'}$ and $t_{1} * (t_{2})_{R}^{- 1} \in L^{'}$ .

Theorem 13

For a semi-discrete sub-loop $L^{* *}$ , $s C l (L^{* *}) = L^{*}$ , where $L^{*}$ is a sub-loop, and $(L, *, τ)$ is a quasi-irresolute topological loop having pre-semi-open left translation.

Proof

Let $L^{'}$ be a discrete sub-loop of a quasi-irresolute topological loop $(L, *, τ)$ . For $t_{1}, t_{2} \in s C l (L^{'})$ , if $M_{1}$ and $M_{2}$ are semi-open neighbourhoods of $t_{1}$ and $t_{2}$ , then for $(t_{1})_{L}^{- 1}, (t_{2})_{L}^{- 1} \in L$ , $l_{{(t_{1})}_{L}^{- 1}} (M_{1}) = (t_{1})_{L}^{- 1} * M_{1}$ , and $l_{{(t_{2})}_{L}^{- 1}} (M_{2}) = (t_{2})_{L}^{- 1} * M_{2}$ are semi-open neighbourhoods of $e$ . So $((t_{1})_{L}^{- 1} * M_{1}) \cap L^{'} \neq ϕ$ and $((t_{2})_{L}^{- 1} * M_{2}) \cap L^{'} \neq ϕ$ . Therefore, $((t_{1} * (t_{2})_{L}^{- 1}) * ((t_{1})_{L}^{- 1} * M_{1}) \cap (t_{1} * (t_{2})_{L}^{- 1}) * L^{'}) \cup ((t_{1} * (t_{2})_{L}^{- 1}) * ((t_{2})_{L}^{- 1} * M_{2}) \cap (t_{1} * (t_{2})_{L}^{- 1}) * L^{'}) \neq ϕ$ . By using distributive law, $((t_{1} * (t_{2})_{L}^{- 1}) * ((t_{1})_{L}^{- 1} * M_{1}) \cup ((t_{1} * (t_{2})_{L}^{- 1}) * ((t_{2})_{L}^{- 1} * M_{2}))) \cap ((t_{1} * (t_{2})_{L}^{- 1}) * L^{'}) \neq ϕ$ . That is, $H \cap ((t_{1} * (t_{2})_{L}^{- 1}) * L^{'}) \neq ϕ$ , where $H = (t_{1} * (t_{2})_{L}^{- 1}) * ((t_{1})_{L}^{- 1} * M_{1}) \cup ((t_{1} * (t_{2})_{L}^{- 1}) * ((t_{2})_{L}^{- 1} * M_{2}))$ is a semi-open neighbourhood of $t_{1} * (t_{2})_{L}^{- 1}$ , implies $t_{1} * (t_{2})_{L}^{- 1} \in s C l (L^{'})$ . Accordingly, $t_{1} * (t_{2})_{R}^{- 1} \in s C l (L^{'})$ . □

Remark 2

The sufficient condition for a sub-loop of a quasi-irresolute topological loop with pre-semi-open left translation to be semi-clopen is its semi-openness.

The theorem given below is about semi-compactness and semi-Lindelof.

Theorem 14

In a quasi-irresolute topological loop, semi-compactness (semi-Lindelofness) is preserved in inverses and free products with a finite (countable) set.

Proof

Let $M$ be a semi-compact (semi-Lindelof), this implies $i_{L} (M) = M_{L}^{- 1}$ and $i_{R} (M) = M_{R}^{- 1}$ are semi-compact (semi-Lindelof).^16,17 Moreover, let $N$ be a finite (countable) and for all $n \in N$ , $N \cdot M$ as a finite union of the sets $n M = l_{n} (M)$ is semi-compact (semi-Lindelof).^16,17 □

Conclusion

In this study, we have investigated that by what means separately irresolute multiplication and irresolute inverse mappings of topological spaces are defined over loops. We also bring into light some properties of quasi-irresolute topological IP-loops. We illustrate a quasi-irresolute topological IP-loop with an interesting example. We have constructed a zero-dimensional additive metrizable perfect topological IP-loop $L^{*}$ having a commutative property with an infinite null sequence ${N_{0}}$ which forms a quasi-irresolute topological IP-loop with a relative topology $τ_{L^{'}}$ . Factually, these results have a useful contribution in the field of topological algebra.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: This project is supported by the Researchers Supporting Project Number (RSP-2021/317), King Saud University, Riyadh, Saudi Arabia.

Ethics approval

Not applicable, because this article does not contain any studies with human or animal subjects.

ORCID iD

Kashif Maqbool

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On quasi-irresolute characterization of topological loops

Abstract

Keywords

Introduction

Some significant results on quasi-irresolute topological IP-loop

Some properties of a quasi-irresolute topological loop

Conclusion

Footnotes

Declaration of conflicting interests

Funding

Ethics approval

ORCID iD

References