On connected unars with regular endomorphism monoids
Jan Chvalina · Czech digital mathematics library · 1980
A monounary algebra, i.e. a pair (A 9 f) 9 where A is a non-void set andfa self-map of the set A 9 is briefly called a unar.This paper aims to give some conditions of the topological and algebraic character equivalent to the regularity of the endomorphism monoid of a connected unar.There are used the descriptions of unars with regular and inverse endomorphism monoids obtained by L. A. Skornjakov in [12] and results of papers [4], [6].In the below stated characterizations we consider mostly endomorphism monoids which are not groups.For the characterization of unars whose endomorphism monoids are automorphism groups see [12] Theorem 3. Fundamental used notions concerning monounary algebras can be found e.g. in papers [5], [8], [11], [12].Let (A 9 f) be a connected unar.The set of all cyclic elements of (A 9 f) (i.e.such elements as A that f H (a) «• a for some integer n *z 1) will be denoted in regard with [8] by A* 2 and further A 001 = {x e A \ A mi : there is a sequence {xt} i€m such that x 0 » x and f(x t + t ) = x t for each ieco} 9 A 0 ** =* {x e A:f~l(x) = 0}.A unar is called a cycle if A « A mi .The upper cone of an element a, i.e. the set {/"(a): n = 0,1,2,...} will be denoted by [a)/, the lower cone {xeA: f n (x) = a for some neco} by (a] f .We agree on denoting the cardinality of a set A by | A |.A connected unar (A 9 f) with | A | = K 0 andf -a permutation of A is called a line.A connected unar (A 9 f) is said to be a cycle with short tails or a line with short tails if it contains a cycle or a line C such thatf(*) e C for every x e A. If | B mi | £ 1 for each component (B 9 f B ) of a unar (A 9 f) we put a jg ft for a 9 b e A if there exists n e a> withf*(a) == b and a < f b if a jg f b 9 a # b.Further, we denote by (A, f) the factor-unar (i.e. the factor-algebra of a monounary algebra (A 9 f)) corresponding to the congruence m f on (A 9 f) defined by a zs f b if a » b or a.beA* 32 .The monoid of all endomorphisms of (A 9 f) is denoted by E(A 9 f).For the definition of a regular and inverse semigroup see [3] § 1.9.A certain strengthening of the notion of a regular semigroup is the notion of an anti-regular semigroup (cf.[10]) called in [1] an anti-inverse semipoup* Let us recall the necessary definitions (see [1] and [10]): A semigroup S is said to be anti-inverse if for each element aeS there is an element be S such that aba = b and bob = a.The elements a and b are then called anti-inverses.