Green’s Relations in the Commutative Centralizers of Monounary Algebras

Abstract

Abstract. The paper deals with the monounary algebras for which the second centralizer equals the first centralizer. We describe Green's relations on the semigroup C, where C is the centralizer of such algebra. IntroductionIn the present paper, we deal with the semigroup formed as the centralizer of a monounary algebra.For a given (partial) algebra A, its centralizer is defined as the set of those mappings of A into A that commute with all basic operations of A. Further, the second centralizer is the set of all transformations which commute with all elements of the (first) centralizer.Centralizers of transformations appear in several areas of mathematical research. For example, they play a role in finding the group of automorphisms of a general semigroup [1]. They occur naturally in the theory of unary algebras and the knowledge of them is useful in studying homomorphisms of algebraic structures. A monounary algebra is a unary algebra with one operation. Monounary algebras have been investigated by several authors (see, e.g., monographs of B. In semigroup theory, the notion of Green's relations is well known (see [3]). Green's relations provide one of the most important tools in studying semi- The paper [5] contains a description of all monounary algebras with the property that the first centralizer and the second centralizer coincide. Let us remark that this property is equivalent to the property that the (first) centralizer is commutative. Our aim is, applying these results, to characterize Green's relations on the semigroup pC,˝q, where C is the centralizer of a given monounary algebra pA, f q such that C is commutative.The commutativity of C implies that on C all Green relations coincide. Our aim is to describe conditions under which α, β P C are in Green's relation, i.e., α˝C " β˝C (we write αRβ). Also, we will characterize α, β P C such that α˝C Ď β˝C (denoted α ≤ R β).

Centralising Monoids with Low-Arity Witnesses on a Four-Element Set

“…Between 1974 and 1976 Harnau [10][11][12][13][14][15] worked on centralisers of unary operations, which are dual to centralising monoids in terms of the Galois connection induced by commutation of finitary operations (see Section 2). Centralising monoids of single unary operations, i.e., monounary algebras, were investigated in [16][17][18], showing, for example, which centralising monoids of this type are equal to the centralising monoid they describe as a witness [16] (Theorem 4.1, p. 8, Theorem 5.1, p. 10), and which of them have a unique unary operation as their witness [18] (Theorems 3.1 and 3.3, p. 4659 et seq.). Research on centralising monoids was further pushed forward in a series of papers [19][20][21][22][23][24][25] by Machida and Rosenberg, linking in particular maximal centralising monoids to the five types of functions appearing in Rosenberg's Classification Theorem [26] for minimal clones.…”