Full Transformation Semigroup Research Articles

Commutative nilpotent transformation semigroups

AbstractCameron et al. determined the maximum size of a null subsemigroup of the full transformation semigroup $$\mathcal {T}(X)$$ T ( X ) on a finite set X and provided a description of the null semigroups that achieve that size. In this paper we extend the results on null semigroups (which are commutative) to commutative nilpotent semigroups. Using a mixture of algebraic and combinatorial techniques, we show that, when X is finite, the maximum order of a commutative nilpotent subsemigroup of $$\mathcal {T}(X)$$ T ( X ) is equal to the maximum order of a null subsemigroup of $$\mathcal {T}(X)$$ T ( X ) and we prove that the largest commutative nilpotent subsemigroups of $$\mathcal {T}(X)$$ T ( X ) are the null semigroups previously characterized by Cameron et al.

Intersecting families of transformations

Consider [Formula: see text] and [Formula: see text] to represent the full transformation semigroup and the symmetric group on the set [Formula: see text], respectively. A subset [Formula: see text] of [Formula: see text] is called an intersecting family if any two transformations [Formula: see text] coincide at some point [Formula: see text]. An intersecting family [Formula: see text] is called maximum if no other intersecting family [Formula: see text] exists with [Formula: see text]. Frankl and Deza [On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22(3) (1977) 352–360] established that the cardinality of a maximum intersecting family of [Formula: see text] equals [Formula: see text]. Subsequent research demonstrated that a maximum intersecting family of [Formula: see text] forms a coset of a stabilizer of one point, as shown by Cameron and Ku [Intersecting families of permutations, European J. Combin. 24(7) (2003) 881–890] and independently by Larose and Malvenuto [Stable sets of maximal size in Kneser-type graphs, European J. Combin. 25(5) (2004) 657–673]. Define [Formula: see text], where [Formula: see text] denotes the image set of [Formula: see text]. This paper presents a formula for the cardinality of a maximum intersecting family of [Formula: see text]. It then characterizes the maximum intersecting families of [Formula: see text] and concludes by determining the number of maximum intersecting families of [Formula: see text].

Fractional domination and fractional total domination on Cayley digraphs of transformation semigroups with fixed sets

<abstract><p>For a set $ X $ and a nonempty subset $ Y $ of $ X $, denote by $ T(X) $ the full transformation semigroup under the composition whose elements are functions on $ X $. Let $ Fix(X, Y) $ be the subsemigroup of $ T(X) $ containing functions $ \alpha\in T(X) $ in which each element in $ Y $ is a fixed point of $ \alpha $. Moreover, let $ A $ be a nonempty subset of $ Fix(X, Y) $. The Cayley digraph of $ Fix(X, Y) $ with respect to a connection set $ A $ is a digraph with vertex set $ Fix(X, Y) $ and two vertices $ \alpha, \beta $ induce an arc $ (\alpha, \beta) $ if $ \beta = \alpha\lambda $ for some $ \lambda\in A $. In this paper, the concepts of fractional dominating and fractional total dominating functions of those Cayley digraphs were investigated. Furthermore, the fractional domination and fractional total domination numbers were determined.</p></abstract>

Minimum degrees of finite rectangular bands, null semigroups, and variants of full transformation semigroups

Minimum degrees of finite rectangular bands, null semigroups, and variants of full transformation semigroups

Magnifiers in semigroups of transformations whose restrictions belong to a given semigroup

Let [Formula: see text] be the full transformation semigroup on a set [Formula: see text]. For a subset [Formula: see text] of [Formula: see text] and a submonoid [Formula: see text] of [Formula: see text], denote by [Formula: see text] the semigroup under composition consisting of all transformations [Formula: see text] such that the restriction [Formula: see text] of [Formula: see text] to [Formula: see text] belongs to [Formula: see text]. We give necessary and sufficient conditions for an element in [Formula: see text] to be left or right magnifier. We apply these descriptions to obtain more concrete results for the semigroups [Formula: see text] and [Formula: see text], where [Formula: see text] (respectively, [Formula: see text]) is the specific submonoid of [Formula: see text] consisting of all injective (respectively, surjective) transformations. The paper also identifies some results on [Formula: see text] that have appeared in the literature.

Endomorphisms of semigroups of monotone transformations

In this paper, we characterize the monoid of endomorphisms of the semigroup of all monotone full transformations of a finite chain, as well as the monoids of endomorphisms of the semigroup of all monotone partial transformations and of the semigroup of all monotone partial permutations of a finite chain.

Regularity of some invariant transformation semigroups

Let [Formula: see text] be a nonempty subset of [Formula: see text]. The invariant subsemigroup of the full transformation semigroup [Formula: see text] is denoted by [Formula: see text] In this paper, we characterize the regularity of the following subsemigroups: [Formula: see text] the complement sets of both [Formula: see text] and [Formula: see text]. These are generalizations of certain subsemigroups of [Formula: see text]. Furthermore, infinitely many regular and nonregular elements of each semigroup are given.

Unit-regular and semi-balanced elements in various semigroups of transformations

Let T(X) be the full transformation semigroup on a set X, and let L(V) be the semigroup under composition of all linear transformations on a vector space V over a field. For a subset Y of X and a subspace W of V, consider the semigroups $${\overline{T}}(X, Y) = \{f\in T(X):Yf \subseteq Y\}$$ and $${\overline{L}}(V, W) = \{f\in L(V):Wf \subseteq W\}$$ under composition. We describe unit-regular elements in $${\overline{T}}(X, Y)$$ and $${\overline{L}}(V, W)$$ . Using these, we determine when $${\overline{T}}(X, Y)$$ and $${\overline{L}}(V, W)$$ are unit-regular. Our results provide an alternative proof that $$f\in L(V)$$ is unit-regular if and only if $${{\,\mathrm{\text{ nullity }}\,}}(f) = {{\,\mathrm{\text{ corank }}\,}}(f)$$ , and L(V) is unit-regular if and only if V is finite-dimensional. A semi-balanced semigroup is a transformation semigroup whose all elements are semi-balanced. We give necessary and sufficient conditions for $${\overline{T}}(X, Y)$$ , $${\overline{L}}(V, W)$$ and L(V) to be semi-balanced.

Regularity and maximal subsemigroups in semigroups of full transformations with fixed permuting sets

Let [Formula: see text] be a nonempty set and [Formula: see text] denote the full transformation semigroup on [Formula: see text]. For a fixed nonempty subset [Formula: see text] of [Formula: see text], let [Formula: see text] where [Formula: see text] is a permutation group on [Formula: see text]. Then [Formula: see text] is a regular submonoid of [Formula: see text]. In this paper, we describe all intra-regular and unit regular elements of [Formula: see text] and give necessary and sufficient conditions for [Formula: see text] to be intra-regular and unit regular. We also count the number of these elements when [Formula: see text] is a finite set. Moreover, we classify the maximal subsemigroups of [Formula: see text] and prove that these maximal subsemigroup coincide with the maximal regular subsemigroups of [Formula: see text] when [Formula: see text] is a finite set.

Magnifiers in various semigroups of transformations that preserve a partition

Let [Formula: see text] be an arbitrary infinite set, and let [Formula: see text] be a partition of [Formula: see text] indexed by [Formula: see text]. Denote by [Formula: see text] the full transformation semigroup on [Formula: see text], and let [Formula: see text]. The character map of an element [Formula: see text] of [Formula: see text] is the self-map [Formula: see text] defined as [Formula: see text] whenever [Formula: see text]. In this paper, we describe magnifiers in [Formula: see text] in terms of their character and restriction maps. We further describe magnifiers in the subsemigroups [Formula: see text], [Formula: see text], and [Formula: see text] of [Formula: see text] that consist of all transformations whose character maps are injective, surjective, and bijective, respectively.

On Presentations of Semigroup of Transformations Restricted by an Equivalence

For a non-empty set X denote the full transformation semigroup of X by T(X). Let \(\sigma\) be an equivalence relation on X and E(X, \(\sigma\)) denotes the semigroup (under composition) of all \(\alpha\) : X \(\mapsto\) X, such that \(\sigma\) \(\subseteq\) ker(\(\alpha\) ). Semigroup of transformations with restricted equivalence occur when we take all transformations whose kernel is contained in some fixed equivalence, E(X, \(\sigma\)). First, we found that E(X, \(\sigma\)) is a disjoint union copies of two generating sets. Next, we discuss the presentations, acts, subacts, direct products and bilateral semidirect product of the semigroup of transformation with restricted equivalence E(X, \(\sigma\)) and its application.

On Regularity of Semigroups of Maps that Preserve a Binary Relation

We consider semigroups of full transformations, partial maps, and multi-value maps from a set X to X, where every map preserves a binary relation σ defined on X. We suggest several definitions for the preservation of σ for partial maps and for multi-value maps. We review results about regularity of the semigroups mentioned above in the cases where σ is a partial order, a quasi-order, an equivalence, or one of some special kinds of binary relations. Also we consider the question about regularity of the semigroup of full transformations that preserve a partial order and an equivalence simultaneously.

Structural properties and isomorphism theorems for Cayley digraphs of full transformation semigroups with respect to Green's equivalence classes

Let T(X) be the full transformation semigroup on a nonempty set X. In this paper, the Cayley digraphs of T(X) with connection sets L and R, the Green's equivalence classes of T(X) according to the Green's relations L and R, are investigated. Furthermore, their connectedness properties are characterized. In addition, the isomorphism theorems for Cayley digraphs of T(X) are also presented.

Quasi-idempotents in finite semigroup of full order-preserving transformations

Let Xn be the finite set {1,2,3· · ·, n} and On defined by On={α∈Tn:(∀x, y∈Xn), x⩽y→xα⩽yα}be the semigroup of full order-preserving mapping on Xn. A transformation α in On is called quasi-idempotent if α=α2=α4. We characterise quasi-idempotent in On and show that the semigroup On is quasi-idempotent generated. Moreover, we obtained an upper bound forquasi-idempotents rank of On, that is, we showed that the cardinality of a minimum quasi-idempotents generating set for On is less than or equal to ⌈3(n−2)2⌉ where ⌈x⌉ denotes the least positive integerm such that x⩽m<x+ 1.

Combinatorial results of collapse for order-preserving and order-decreasing transformations

The full transformation semigroup TnTn is defined to consist of all functions from Xn={1,…,n}Xn={1,…,n} to itself, under the operation of composition. In \cite{JMH1}, for any αα in TnTn, Howie defined and denoted collapse by c(α)=⋃t∈\im(α){tα−1:|tα−1|≥2}c(α)=⋃t∈\im(α){tα−1:|tα−1|≥2}. Let OnOn be the semigroup of all order-preserving transformations and CnCn be the semigroup of all order-preserving and decreasing transformations on XnXn=under its natural order, respectively. Let E(On)E(On) be the set of all idempotent elements of OnOn, E(Cn)E(Cn) and N(Cn)N(Cn) be the sets of all idempotent and nilpotent elements of CnCn, respectively. Let UU be one of {Cn,N(Cn),E(Cn),On,E(On)}{Cn,N(Cn),E(Cn),On,E(On)}. For α∈Uα∈U, we consider the set \imc(α)={t∈\im(α):|tα−1|≥2}\imc(α)={t∈\im(α):|tα−1|≥2}. For positive integers 2≤k≤r≤n2≤k≤r≤n, we define U(k)={α∈U:t∈\imc(α) and |tα−1|=k},U(k,r)={α∈U(k):∣∣⋃t∈\imc(α)tα−1|=r}.U(k)={α∈U:t∈\imc(α) and |tα−1|=k},U(k,r)={α∈U(k):|⋃t∈\imc(α)tα−1|=r}. The main objective of this paper is to determine |U(k,r)||U(k,r)|, and so |U(k)||U(k)| for some values rr and kk.

Ranks of Identity Difference Transformation Semigroup

This study focuses on the ranks of identity difference transformation semigroup. The ideals of all the (sub) semigroups; identity difference full transformation semigroup (IDT_n), identity difference order preserving transformation semigroup, (IDO_n), identity difference symmetric inverse transformation semigroup( IDI_n), identity difference partial order preserving symmetric inverse transformation semigroup( IDPOI_n) and identity difference partial order preserving transformation semigroup ( IDPO_n) were investigated for rank and their combinatorial results obtained respectively.

On certain semigroups of transformations with an invariant set

Let [Formula: see text] be a nonempty set and let [Formula: see text] be the full transformation semigroup on [Formula: see text]. The main objective of this paper is to study the subsemigroup [Formula: see text] of [Formula: see text] defined by [Formula: see text] where [Formula: see text] is a fixed nonempty subset of [Formula: see text]. We describe regular elements in [Formula: see text], and show that [Formula: see text] is regular if and only if [Formula: see text] is finite. We characterize unit-regular elements in [Formula: see text], and prove that [Formula: see text] is unit-regular if and only if [Formula: see text] is finite. We characterize Green’s relations on [Formula: see text], and prove that [Formula: see text] on [Formula: see text] if and only if [Formula: see text] is finite. We also determine ideals of [Formula: see text] and investigate its kernel. This paper extends several results that have appeared in the literature.

Set-theoretic solutions of the Yang–Baxter equation associated to weak braces

We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang–Baxter equation. Specifically, a weak (left) brace is a non-empty set S endowed with two binary operations + and circ such that both (S,+) and (S, circ ) are inverse semigroups and a∘b+c=a∘b-a+a∘canda∘a-=-a+a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} a \\circ \\left( b+c\\right) = \\left( a\\circ b\\right) - a + \\left( a\\circ c\\right) \\qquad \ ext {and} \\qquad a\\circ a^- = - a + a \\end{aligned}$$\\end{document}hold, for all a,b,c in S, where -a and a^- are the inverses of a with respect to + and circ , respectively. In particular, such structures include that of skew braces and form a subclass of inverse semi-braces. Any solution r associated to an arbitrary weak brace S has a behavior close to bijectivity, namely r is a completely regular element in the full transformation semigroup on Stimes S. In addition, we provide some methods to construct weak braces.

Zero-Divisor Graphs of Order-Decreasing Full Transformation Semigroups

Let 𝑛 ∈ ℤ+ and 𝑋𝑛 = {1,2, … , 𝑛} be a finite set. Let 𝐷𝑛 be the order-decreasing full transformation semigroup on 𝑋𝑛. In this paper, we find the left zero-divisors, the right zero-divisors and two sided zerodivisors of 𝐷𝑛. Moreover, for 𝑛 ≥ 4 we define an undirected graph Γ(𝐷𝑛) whose vertices are two-sided zero divisors of 𝐷𝑛 excluding the zero element 𝜃 of 𝐷𝑛. In the graph, distinct two vertices 𝛼 and 𝛽 are joined if and only if 𝛼𝛽 = 𝜃 = 𝛽𝛼. In this paper, we prove that Γ(𝐷𝑛) is a connected graph, and we find diameter, girth, the degrees of all vertices, the maximum degree and the minimum degree in Γ(𝐷𝑛). Moreover, we give lower bounds for clique number and choromatic number of Γ(𝐷𝑛).

Abundance of the ideals of the order-preserving and decreasing full transformation semigroup

Abundance of the ideals of the order-preserving and decreasing full transformation semigroup