Semitransitive subsemigroups of the symmetric inverse semigroups

Abstract

We classify minimal transitive subsemigroups of the finitary inverse symmetric semigroup modulo the classification of minimal transitive subgroups of finite symmetric groups; and semitransitive subsemigroups of the finite inverse symmetric semigroup of the minimal cardinality modulo the classification of transitive subgroups of the minimal cardinality of finite symmetric groups.

Semitransitive subsemigroups of the singular part of the finite symmetric inverse semigroup

“…In this paper we continue the research started in [4]. There we described the semitransitive subsemigroups of I n of the minimal cardinality.…”

“…8, 9, 10}. Let G be the four-element cyclic group acting on X 2 generated by the cycle (3,5,4,6). The elements of the semigroup (G × T )/I as the semigroup of partial permutation on X are as follows:…”

“…The restriction |(gSg) \ {0}| + |(hSh) \ {0}| = n, together with the result from [4] (recalled in Theorem 2 above), provides us with the information about the structure of gSg and hSh. We can therefore assume that for i ∈ A we have |X i | = p and for i ∈ B we also have |X i | = kp for some k ≥ 1.…”

“…A number of papers has been published which are devoted to the study of semitransitive actions of spaces of linear operators [1,2,3,8,9,10]. In [4] we initiated the study of semitransitive semigroups of transformations.…”

On a stronger reconstruction notion for monoids and clones

“…The technique to prove it is inspired by [28] and uses the assumption of being weakly directed (this notion has appeared in [28, 10.1, p. 60] as semi-transitive, but the latter term has been introduced in [26] to designate a different semigroup property recurring in a number of articles, e.g. [4,5,6,7,14,15]). Hence, we say that an action of a semigroup S on a set A is weakly directed if for all a, b ∈ A there are f, g ∈ S and c ∈ A such that (f, c) → a and (g, c) → b.…”