Janusz Konieczny

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: T(X,ρ,R)=a∈T(X):Ra⊆Rand(x,y)∈ρ⇒(xa,ya)∈ρ. The 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) ∈ 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 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.

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.

CENTRALIZERS IN THE SEMIGROUP OF INJECTIVE TRANSFORMATIONS ON AN INFINITE SET

For an infinite set X, denote by Γ(X) the semigroup of all injective mappings from X to X. For α∈Γ(X), let C(α)={β∈Γ(X):αβ=βα} be the centralizer of α in Γ(X). For an arbitrary α∈Γ(X), we characterize the elements of C(α) and determine Green’s relations in C(α), including the partial orders of ℒ-, ℛ-, and 𝒥-classes.

Infinite injective transformations whose centralizers have simple structure

AbstractFor an infinite set X, denote by Γ(X) the semigroup of all injective mappings from X to X under function composition. For α ∈ Γ(X), let C(α) = {β ∈ g/g(X): αβ = βα} be the centralizer of α in Γ(X). The aim of this paper is to determine those elements of Γ(X) whose centralizers have simple structure. We find α ∈ (X) such that various Greens relations in C(α) coincide, characterize α ∈ Γ(X) such that the % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX % garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy % Ubqee0evGueE0jxyaibaieYdh9Lrpeeu0dXdh9vqqj-hEeeu0xXdbb % a9frpm0db9Lqpepeea0xd9q8as0-LqLs-Jirpepeea0-as0Fb9pgea % 0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba % Wefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiyaacqWFjeVs % aaa!46C9!

Semigroups of Transformations Commuting with Injective Nilpotents

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Green's equivalences in finite semigroups of binary relations

-classes of C(α) form a chain, and describe Greens relations in C(α) for α with so-called finite ray-cycle decomposition. If α is a permutation, we also find the structure of C(α) in terms of direct and wreath products of familiar semigroups.

Conjugation in semigroups

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). 2000 Mathematics Subject Classification: 20M07, 20M20, 20M15