Browsing by Author "Konieczny, Janusz"
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- Automorphism groups of centralizers of idempotentsPublication . Araújo, João; Konieczny, JanuszFor 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).
- Automorphisms of endomorphism monoids of 1-Simple free algebrasPublication . Konieczny, Janusz; Araújo, JoãoFor any set X and any variety V of algebras, let A = A_V(X) be the free algebra in V with set X of free generators, and let End(A) be the monoid of endomorphisms of A. We provide a general approach to the description of the automorphism group of End(A), and some subsemigroups of End(A), provided that A is 1-simple. We show that every automorphism of End(A) is a unique extension of an automorphism of the monoid of endomorphisms of rank at most 1.
- Automorphisms of endomorphism monoids of relatively free bandsPublication . Araújo, João; Konieczny, JanuszFor 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).
- Automorphisms of endomorphism semigroups of reflexive digraphsPublication . Araújo, João; Dobson, Edward; Konieczny, JanuszA reflexive digraph is a pair (X,ρ), where X is an arbitrary set and ρ is a reflexive binary relation on X. Let End (X, ρ) be the semigroup of endomorphisms of (X, ρ). We determine the group of automorphisms of End(X,ρ) for: digraphs containing an edge not contained in a cycle, digraphs consisting of arbitrary unions of cycles such that cycles of length ≥ 2 are pairwise disjoint, and some circulant digraphs.
- Centralizers in the full transformation semigroupPublication . Araújo, João; Konieczny, JanuszFor 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.
- Conjugation in semigroupsPublication . Araújo, João; Konieczny, Janusz; Malheiro, AntónioThe action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been several attempts to extend the notion of conjugacy to semigroups. In this paper, we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero. We compare the new notion of conjugacy with existing definitions, characterize the conjugacy in various semigroups of transformations on a set, and count the number of conjugacy classes in these semigroups when the set is infinite.
- Dense relations are determined by their endomorphism monoidsPublication . Araújo, João; Konieczny, JanuszWe 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.
- General theorems on automorphisms of semigroups and ther applicationsPublication . Araújo, João; Konieczny, JanuszWe 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.
- Matrix theory for independence algebrasPublication . Araújo, João; Bentz, Wolfram; Cameron, Peter; Kinyon, Michael; Konieczny, JanuszA universal algebra A with underlying set A is said to be a matroid algebra if (A, 〈·〉), where 〈·〉 denotes the operator subalgebra generated by, is a matroid. A matroid algebra is said to be an independence algebra if every mapping α : X → A defined on a minimal generating 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. In the 1970s, Glazek proposed the problem of extending the matrix theory for vector spaces to a class of universal algebras which included independence algebras. In this paper, we answer that problem by developing a theory of matrices for (almost all) finite-dimensional independence algebras. In the process of solving this, we explain the relation between the classification of inde- pendence algebras obtained by Urbanik in the 1960s, and the classification of finite indepen- dence algebras up to endomorphism-equivalence obtained by Cameron and Szab ́o in 2000. (This answers another question by experts on independence algebras.) We also extend the classification of Cameron and Szab ́o to all independence algebras. The paper closes with a number of questions for experts on matrix theory, groups, semi- groups, universal algebra, set theory or model theory.
- A method for finding new sets of axioms for classes of semigroupsPublication . Araújo, João; Konieczny, JanuszWe introduce a general technique for finding sets of axioms for a given class of semigroups. To illustrate the technique, we provide new sets of defining axioms for groups of exponent n, bands, and semilattices.