Centralizers of a monounary algebra

Abstract

The centralizer of an algebra is defined as the set of all transformations of the algebra which commute with all fundamental operations. Further, the second centralizer is the set of all transformations which commute with all elements of the (first) centralizer. The paper characterizes the monounary algebras having the property that the first and the second centralizers coincide.

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

“…Hence, in these cases { f } * (1) = L (1) fails to be a maximal centralising monoid. 3,4,6,7,8,9,11,12,13,14,16,18,19,24,27,28,30,32,33,35,36,39,44 {g 347 } * (1) = {0, 5, 10, 11, 14, 15, 16, 21, 26, 27, 30, 31, 32, 37, 42, 48, 53, 63, 64, 69, 74, 75, 78, 79, 80, 85, 90, 91, 94, 95, 96, 101, 106, 112, 117, 127, 128, 133, 138, 144, 149, 154, 160, 165, 170, 192, 197, 207, 208, 213, 223, 240, 245, 255} {g 348 } * (1) = {0, 5, 10, 16, 21, 27, 32, 42, 64, 69, 78, 80, 85, 95, 117, 127, 128, 138, 160, 170, 175, 186, 191, 213, 223, 234, 239, 245, 250, 255} {g 351 } * (1) = {0, 5, 10, 15, 16, 21, 27, 30, 32, 42, 47, 48, 58, 63, 64, 69, 80, 85, 128, 138, 143, 160, 170, 175, 176, 186, 191, 192, 202, 207, 224, 234, 239, 240, 250, 255} {g 372 } * (1) = {0, 5, 11, 14, 16, 21, 27, 30, 64, 69, 75, 78, 80, 85, 91, 94, 170, 175, 186, 191, 234, 239, 250, 255} {g 377 } * (1) = {0, 21, 22, 27, 64, 85, 106, 149, 170, 191, 233, 234, 255} {g 396 } * (1) = {0, 21, 23, 27, 39, 42, 43, 61, 62, 63, 64, 85, 106, 128, 149, 170, 192, 255} {g 412 } * (1) = {0, 21, 22, 23, 25, 26, 27, 29, 30, 31, 37, 38, 39, 41, 42, 43, 45, 46, 47, 53, 54, 55, 57, 58, 59, 61, 62, 63, 64, 85, 128, 170, 192, 255} {g 454 } * (1) = {0, 27, 45, 54, 85, 106, 127, 149, 170, 191, 213, 234, 255} {g 563 } * (1) = {0, 5, 10, 27, 78, 80, 85, 95, 160, 170, 175, 177, 228, 245, 250, 255} {g 618 } * (1) = {0, 4, 8, 19, 27, 85, 93, 116, 170, 205, 243, 251, 255} {g 658 } * (1) = {0, 20,23,…”

“…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.…”

Green’s Relations in the Commutative Centralizers of Monounary Algebras

“…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.…”