On relative ranks of the semigroup of orientation-preserving transformations on infinite chains

Dimitrova, Ilinka; Koppitz, Jörg

doi:10.1142/s1793557121501461

Cited by 3 publications

(2 citation statements)

References 28 publications

(21 reference statements)

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“…However, this study only features some elements of subsemigroups of order-preserving full mappings in [27]. Therefore, future studies would be to establish some notions to study other elements of subsemigroups in ℘ such as order-preserving partial mappings [1,28,15], order decreasing full mappings [29], symmetric inverse semigroups [5], orientation-preserving mappings [30], fence-preserving mappings [31] among others.…”

Section: Concluding Remarkmentioning

confidence: 99%

On Nonexpansive and Expansive Semigroup of Order-Preserving Total Mappings in Waist Metric Spaces

Wahab

Usamot

Alata³

et al. 2022

J. Nig. Soc. Phys. Sci.

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In this paper, we introduce nonexpansive and expansive semigroup of order-preserving total mappings (ONTn) and (OETn), respectively, to prove some fixed point theorems in waist metric spaces. We examine the existence of mappings that satisfy the conditions ONTn and OETn. We also prove that every semigroup of order-preserving total mappings OTn has fixed point properties and that the set of fixed points is closed and convex. The present study generalised many previous results on semigroup of order-preserving total mappings OTn. Efficacy of the results was justified with some practical examples.

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Section: Concluding Remarkmentioning

confidence: 99%

On Nonexpansive and Expansive Semigroup of Order-Preserving Total Mappings in Waist Metric Spaces

Wahab

Usamot

Alata³

et al. 2022

J. Nig. Soc. Phys. Sci.

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“…It is a proper subsemigroup of OP(X) and OPR(X) where n ≥ 2. The semigroup OP(X) has been widely investigated (see [1], [2], [4], [6], [9], [16]). In particular, the rank of OP(X) is 2 [1], the rank of O(X) is n [6] and the rank of T (X) is 3 [13].…”

Section: Introductionmentioning

confidence: 99%

The Relative Rank of Transformation Semigroups with Restricted Range on a Finite Chain

Tinpun¹

2021

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Let S be a semigroup and let G be a subset of S. A set G is a generating set G of S which is denoted by G = S. The rank of S is the minimal size or the minimal cardinality of a generating set of S, i.e. rankS := min{|G| : G ⊆ S, G = S}. In last twenty years, the rank of semigroups is worldwide studied by many researchers. Then it lead to a new definition o f r ank that is called the relative rank of S modulo U is the minimal size of a subset G ⊆ S such that G ∪ U generates S, i.e. rank(S :The idea of the relative rank was generalized from the concept of the rank of a semigroup and it was firstly i ntroduced b y H owie, R uškuc a nd H iggins i n 1998. Let X be a finite c hain a nd l et Y b e a s ubchain o f X . We denote T (X) the semigroup of full transformations on X under the composition of functions. Let T (X, Y ) be the set of all transformations from X to Y which is so-called the transformation semigroup with restricted range Y . It was firstly i ntroduced a nd s tudied b y S ymons i n 1 975. Many results in T (X) were extended to results in T (X, Y ). In this paper, we focus on the relative rank of semigroup T (X, Y ) and the semigroup OP(X, Y ) of all orientation-preserving transformations in T (X, Y ). In Section 2.1, we determine the relative rank of T (X, Y ) modulo the semigroup OD(X, Y ) of all order-preserving or order-reversing transformations. In Section 2.2, we describe the results of the relative rank of T (X, Y ) modulo the semigroup OP(X, Y ). In Section 2.3, we determine the relative rank of T (X, Y ) modulo the semigroup OPR(X, Y ) of all orientation-preserving or orientation-reversing transformations. Moreover, we obtain that the relative rank T (X, Y ) modulo OP(X, Y ) and modulo OPR(X, Y ) are equal.

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On relative ranks of the semigroup of orientation-preserving transformations on infinite chain with restricted range

Dimitrova

Koppitz

2022

Communications in Algebra

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On relative ranks of the semigroup of orientation-preserving transformations on infinite chains

Abstract: In this paper, we determine the relative rank of the semigroup [Formula: see text] of all orientation-preserving transformations on infinite chains modulo the semigroup [Formula: see text] of all order-preserving transformations.

Cited by 3 publications

References 28 publications

On Nonexpansive and Expansive Semigroup of Order-Preserving Total Mappings in Waist Metric Spaces

On Nonexpansive and Expansive Semigroup of Order-Preserving Total Mappings in Waist Metric Spaces

The Relative Rank of Transformation Semigroups with Restricted Range on a Finite Chain

On relative ranks of the semigroup of orientation-preserving transformations on infinite chain with restricted range

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