S_n-normal semigroups

Abstract

Certain subsemigroups of the full transformation semigroup T n on a finite set of cardinality n are investigated, namely those subsemigroups S of T n which are normalised by the symmetric group on n elements, the group of units of T n . The S n -normal closure of an element of T n is determined, and the structure o f the S.-normal ideals consisting of the members of T n whose image contains at most r elements is studied.1991 Mathematics subject classification: 20M20.Let T n denote the full transformation semigroup on a set of finite cardinality n, and let S n denote the symmetric group on n elements, the group of units of T n . A subsemigroup S of Tj, is defined to be S n -normal if for each a in S and for each h in S n , the element h~lah is in S. Both T n and S n themselves are S n -normal; so are the ideals K(n,r) = Given aeT n , denote by <a:S n > the smallest S n -normal subsemigroup of T n containing a. Thus <a:S n > is the subsemigroup S of T n generated by {g~lag:geS n }. If a is a permutation then <a:S n > is a normal subgroup of S n and we know what that is. Assuming for the rest of this paper that a is not a permutation, associate with a the partition n(a) of X such that x and y are in the same class of n(a) if and only if xa = ya. Partitions P, Q of X are said to be of the same type (denoted by P = Q) if they have the same number of classes of each size. We show that (a:S n ) is idem potent-generated and consists of all transformations b in T n for which n(b) contains a partition of the same type as n(a).The idempotent rank of an idempotent-generated semigroup S is the cardinality of a minimal generating set of idempotents of S [2]. It was shown in [2] that the idempotent rank of the S B -normal semigroup K(n,r), consisting of all transformations a with |im(a)|^r, is S(n,r), the Stirling number of the second kind. We define the S n -idempotent rank of an S n -normal semigroup S to be the cardinality of a minimal generating set A of idempotents of S such that S = </1:S B > ( = <{g~lag:aeA,geS n }y). Given l g r^n , let T(n,r) denote the number of different types of partitions of an n-element set into r subsets. We present a recursive formula for T(n,r) and show that the S n -idempotent rank of K(n,r) is T(n,r). Moreover, we can choose a minimal S n -generating set of idempotents in a single L-class of both T n and S.For each r such that 2 g r g « , the principal factor K(n,r)/K(n,r-1) of T n is denoted by P r in [2]. Each P r is a completely 0-simple semigroup whose non-zero elements may be thought of as the elements a of T n having |im(a)| = r. Then P r is a band of T (n,r) 471

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