On Semigroups of Orientation-preserving Transformations with Restricted Range

Fernandes, Vítor H.; Honyam, Preeyanuch; Quinteiro, Teresa M.; Singha, Boorapa

doi:10.1080/00927872.2014.975345

Cited by 19 publications

(11 citation statements)

References 25 publications

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“…The survey [10] presents these results and similar ones for other classes of transformation monoids, in particular, for monoids of order-preserving transformations and for some of their extensions. More recently, for instance, the papers [1,2,5,[11][12][13][14][15]23,33,34] are dedicated to the computation of the ranks of certain (classes of transformation) semigroups or monoids. Now, let G = (V , E) be a simple graph (i.e.…”

Section: Introductionmentioning

confidence: 99%

Partial automorphisms and injective partial endomorphisms of a finite undirected path

et al. 2021

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In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids IEnd(P n ) and PAut(P n ) of all injective partial endomorphisms and of all partial automorphisms of the undirected path P n with n vertices. We also describe Green's relations of PAut(P n ) and IEnd(P n ) and calculate their cardinals. Keywords Injective partial endomorphismsThis work is funded by national funds through the FCT -Fundação para a Ciência e a Tecnologia, I.P., under the scope of the project UIDB/00297/2020 (Center for Mathematics and Applications).

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Section: Introductionmentioning

confidence: 99%

Partial automorphisms and injective partial endomorphisms of a finite undirected path

et al. 2021

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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%

“…A semigroup T (X, Y ) is called the full transformation semigroup with restricted range and it is defined by Symons [15]. The other semigroups were introduced by Fernandes et al in [8] and [9], respectively. Transformation semigroups with restricted range have been widely investigated (see [7], [8], [10], [14]).…”

Section: Introductionmentioning

confidence: 99%

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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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“…In 1975, Symons [6] introduced the semigroup T (X, Y), and also described all automorphisms of T (X, Y). The study of semigroups T (X, Y) and T (X, Y) [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] includes the aspects of regularity and Green's relations (see [7][8][9]), abundance and starred Green's Relations (see [10,11]), natural partial order (see [11,12]), congruence relation (see [13,14]), (maximal) subsemigroup with some properties (see [15][16][17][18][19][20][21]), and rank (see [22,23]), etc.…”

Section: Introductionmentioning

confidence: 99%