João Araújo

Automorphism groups of centralizers of idempotents

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

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

Abstract 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)u2009∈u2009ρ then (xa, ya)u2009∈u2009ρ) and R (if ru2009∈u2009R then rau2009∈u2009R). 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 Greens 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.

Minimal paths in the commuting graphs of semigroups

Let S be a finite non-commutative semigroup. The commuting graph of S, denoted G(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 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|>=4, then, with a few exceptions, the diameter of G(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 G(S) is n. We also study the left paths in G(S), that is, paths a1-a2-...-am such that a1 am and a1ai=amai for all [emailxa0protected]?{1,...,m}. We prove that for every positive integer n>=2, 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.

Groups synchronizing a transformation of non-uniform kernel

Suppose that a deterministic finite automata A=(Q,@S) is such that all but one letters from the alphabet @S act as permutations of the state set Q and the exceptional letter acts as a transformation with non-uniform kernel. Which properties of the permutation group G generated by the letters acting as permutations ensure that A becomes a synchronizing automaton under every possible choice of the exceptional letter (provided the exceptional letter acts as a transformation of non-uniform kernel)? Such permutation groups are called almost synchronizing. It is easy to see that an almost synchronizing group must be primitive; our conjecture is that every primitive group is almost synchronizing. Clearly every synchronizing group is almost synchronizing. In this paper we provide two different methods to find non-synchronizing, but almost synchronizing groups. The infinite families of examples provided by the two different methods have few overlaps. The paper closes with a number of open problems on group theory and combinatorics.

An elementary proof that every singular matrix is a product of idempotent matrices

In this note we give an elementary proof of a theorem first proved by J. A. Erdos [3]. This theorem, which is the main result of [3], states that every noninvertible n × n matrix is a finite product of matrices M with the property that M = M . (These are known as idempotent matrices. Noninvertible matrices are also called singular matrices.) An alternative formulation of this result reads: every noninvertible linear mapping of a finite dimensional vector space is a finite product of idempotent linear mappings α, linear mappings that satisfy α = α. This result was motivated by a result of J. M. Howie asserting that each selfmapping α of a nonempty finite set X with image size at most |X| − 1 (which occurs precisely when α is noninvertible) is a product of idempotent mappings. We shall see that Erdos’s theorem is a consequence of Howie’s result. Together the papers [3] and [4] are cited in over one hundred articles, dealing with subjects including universal algebra, ring theory, topology, and combinatorics. Since its publication, various proofs of the result in [3] have appeared. For example, a semigroup theoretic proof appears in [1, pp. 121-131] and linear operator theory is used to prove the theorem in [2]. Here we give a new proof using a basic combinatorial argument. Unlike the previous proofs our argument involves only elementary results from linear algebra and one basic result concerning permutations. On the way to proving the main result of this note we provide a short proof of Howie’s result. Throughout this paper X signifies an arbitrary nonempty finite set. If α : A → X, where A is a subset of X, then A is the domain of α; we denote this set by dom(α). Naturally, the set α(A) is called the image of α and is denoted by im(α). Recall that a mapping α is injective (or one-to-one) if α(x) 6= α(y) for all x and y in dom(α) with x 6= y. Let TX denote the set of all mappings from X to X with domain X. We note that this set is closed under composition of mappings and that this composition is associative. We now define one of the most important notions we require in the proofs in this note. For a mapping α : dom(α) → X we say that α is a restriction of an element β of TX if β and α agree on the domain of α. In other words, β(x) = α(x) for all x in dom(α). For x and y in X we denote the transposition that fixes every point of X other than x or y and that maps x to y, and vice versa, by (x y).

Semigroups of linear endomorphisms closed under conjugation

Let a,b be singular endomorphisms of a. finite dimensional vector space V and denote by S a the semigroup generated by all the elements g -1ag, where gϵAut(V).The aim of this paper is to prove that bϵS a if and only if rank(b) ≥ rank(a).

Idempotent Generated Endomorphisms of an Independence Algebra

The aim of this note is to give a direct proof for the followingresult proved by Fountain and Lewin: Let A be anindependence algebra of finite rank and let a be a singularendomorphism of A. Then a = e1 ... en where ei2 = eiand rank(a) = rank(ei).

NORMAL SEMIGROUPS OF ENDOMORPHISMS OF PROPER INDEPENDENCE ALGEBRAS ARE IDEMPOTENT GENERATED

LetAbe a proper independence algebra of finite rank, letGbe the group of automorphisms ofA ,l etabe a singular endomorphism and leta G be the semigroup generated by all the elementsg °1 ag ,w here g2G. The aim of this paper is to prove thata G is a semigroup

Between primitive and 2-transitive: synchronization and its friends

The second author was supported by the Fundacao para a Ciencia e Tecnologia (Portuguese Foundation for Science and Technology) through the project CEMAT-CIENCIAS UID/Multi/ 04621/2013

General theorems on automorphisms of semigroups and their applications

We introduce the notion of a strong representation of a semigroup in the monoid of endomorphisms of any mathematical structure, and use this concept to provide a theoretical description of the automorphism group of any semigroup. As an application of our general theorems, we extend to semigroups a well-known result concerning automorphisms of groups, and we determine the automorphism groups of certain transformation semigroups and of the fundamental inverse semigroups.