Minimal paths in the commuting graphs of semigroups

Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$, denoted \cg(S), is the graph whose vertices are the non-central elements of $S$ and whose edges are the sets $\{a,b\}$ of vertices such that $a\ne b$ and $ab=ba$. Denote by $T(X)$ the semigroup of full transformations on a finite set $X$. Let $J$ be any ideal of $T(X)$ such that $J$ is different from the ideal of constant transformations on $X$. We prove that if $|X|\geq4$, then, with a few exceptions, the diameter of \cg(J) is 5. On the other hand, we prove that for every positive integer $n$, there exists a semigroup $S$ such that the diameter of \cg(S) is $n$. We also study the left paths in \cg(S), that is, paths $a_1-a_2-...-a_m$ such that $a_1\ne a_m$ and $a_1a_i=a_ma_i$ for all i\in \{1,\ldot, m\}. We prove that for every positive integer $n\geq2$, except $n=3$, there exists a semigroup whose shortest left path has length $n$. As a corollary, we use the previous results to solve a purely algebraic old problem posed by B.M. Schein.Comment: 23 pages; v.2: Lemma 2.1 corrected; v.3: final version to appear in European J. of Combinatoric

The Largest Subsemilattices of the Endomorphism Monoid of an Independence Algebra

An algebra \A is said to be an independence algebra if it is a matroid algebra and every map \al:X\to A, defined on a basis $X$ of \A, can be extended to an endomorphism of \A. These algebras are particularly well behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics such as model theory, group theory, and semigroup theory. It is well known that matroid algebras have a well defined notion of dimension. Let \A be any independence algebra of finite dimension $n$, with at least two elements. Denote by \End(\A) the monoid of endomorphisms of \A. We prove that a largest subsemilattice of \End(\A) has either $2^{n-1}$ elements (if the clone of \A does not contain any constant operations) or $2^n$ elements (if the clone of \A contains constant operations). As corollaries, we obtain formulas for the size of the largest subsemilattices of: some variants of the monoid of linear operators of a finite-dimensional vector space, the monoid of full transformations on a finite set $X$, the monoid of partial transformations on $X$, the monoid of endomorphisms of a free $G$-set with a finite set of free generators, among others. The paper ends with a relatively large number of problems that might attract attention of experts in linear algebra, ring theory, extremal combinatorics, group theory, semigroup theory, universal algebraic geometry, and universal algebra.Comment: To appear in Linear Algebra and its Application

Decidability and Independence of Conjugacy Problems in Finitely Presented Monoids

There have been several attempts to extend the notion of conjugacy from groups to monoids. The aim of this paper is study the decidability and independence of conjugacy problems for three of these notions (which we will denote by $\sim_p$, $\sim_o$, and $\sim_c$) in certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids, $p$-conjugacy is "almost" transitive, $\sim_c$ is strictly included in $\sim_p$, and the $p$- and $c$-conjugacy problems are decidable with linear compexity. For other classes of monoids, the situation is more complicated. We show that there exists a monoid $M$ defined by a finite complete presentation such that the $c$-conjugacy problem for $M$ is undecidable, and that for finitely presented monoids, the $c$-conjugacy problem and the word problem are independent, as are the $c$-conjugacy and $p$-conjugacy problems.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1503.0091

Centralizers in the full transformation semigroup

For an arbitrary set X (finite or infinite), denote by T (X) the semigroup of full transformations on X. For α ∈ T (X), let C(α) = {β ∈ T (X) : αβ = βα} be the centralizer of α in T (X). The aim of this paper is to characterize the elements of C(α). The characterization is obtained by decomposing α as a join of connected partial transformations on X and analyzing the homomorphisms of the directed graphs representing the connected transformations. The paper closes with a number of open problems and suggestions of future investigations

Automorphisms of endomorphism monoids of relatively free bands

For a set X and a variety V of bands, let BV(X) be the relatively free band in V on X. For an arbitrary band variety V and an arbitrary set X, we determine the group of automorphisms of End(BV(X)), the monoid of endomorphisms of BV(X)

Molaie’s generalized groups are completely simple semigroups

In [2] Molaei introduces generalized groups, a class of algebras of interest to physics, and proves some results about them. The aim of this note is to prove that Generalized Groups are the Completely Simple Semigroups. Mathematics subject classification: 20M2

Dense relations are determined by their endomorphism monoids

We introduce the class of dense relations on a set X and prove that for any finitary or infinitary dense relation ρ on X , the relational system (X, ρ) is determined up to semi-isomorphism by the monoid End (X, ρ) of endomorphisms of (X, ρ). In the case of binary relations, a semi-isomorphism is an isomorphism or an anti-isomorphism

Automorphism groups of centralizers of idempotents

For a set X, an equivalence relation Ω on X, and a cross-section R of the partition X/Ω, consider the following subsemigroup of the semigroup T(X) of full transformations on X:T(X, Ω,R) = {a 2 T(X) : Ra μ R and (x, y) 2 Ω ) (xa, ya) 2 Ω}. The semigroup T(X, Ω,R) is the centralizer of the idempotent transformation with kernel Ω and image R. We prove that the automorphisms of T(X, Ω,R) are the inner automorphisms induced by the units of T(X, Ω,R) and that the automorphism group of T(X, Ω,R) is isomorphic to the group of units of T(X, Ω,R)

Semigroups of transformations preserving an equivalence relation and a cross-section

For a set X, an equivalence relation ρ on X, and a cross-section R of the partition X/ρ induced by ρ, consider the semigroup T (X, ρ,R) consisting of all mappings a from X to X such that a preserves both ρ (if (x, y) ∈ ρ then (xa, ya) ∈ ρ) and R (if r ∈ R then ra ∈ R). The semigroup T (X, ρ,R) is the centralizer of the idempotent transformation with kernel ρ and image R. We determine the structure of T (X, ρ,R) in terms of Green’s relations, describe the regular elements of T (X, ρ,R), and determine the following classes of the semigroups T (X, ρ,R): regular, abundant, inverse, and completely regular

Semigroups of partial transformations with kernel and image restricted by an equivalence

For an arbitrary set X and an equivalence relation μ on X, denote by Pμ(X) the semigroup of partial transformations α on X such that xμ⊆x(ker(α)) for every x∈dom(α), and the image of α is a partial transversal of μ. Every transversal K of μ defines a subgroup G=GK of Pμ(X). We study subsemigroups ⟨G,U⟩ of Pμ(X) generated by G∪U, where U is any set of elements of Pμ(X) of rank less than |X/μ|. We show that each ⟨G,U⟩ is a regular semigroup, describe Green’s relations and ideals in ⟨G,U⟩, and determine when ⟨G,U⟩ is an inverse semigroup and when it is a completely regular semigroup. For a finite set X, the top J-class J of Pμ(X) is a right group. We find formulas for the ranks of the semigroups J, G∪I, J∪I, and I, where I is any proper ideal of Pμ(X).authorsversionpublishe