On the Maximal Subsemigroups of Some Transformation Semigroups

Abstract

Let Singn be the semigroup of all singular transformations on an n-element set. We consider two subsemigroups of Singn: the semigroup On of all isotone singular transformations and the semigroup Mn of all monotone singular transformations. We describe the maximal subsemigroups of these two semigroups, and study the connections between them.

Maximal subsemigroups of finite transformation and diagram monoids

“…From this, the description of the maximal subsemigroups of type (M3) follows from Proposition 2.10, and the description of the maximal subsemigroups of type (M4) follows from Proposition 2. 11. In particular, the total number of both types of maximal subsemigroups is two less than the number of vertices of ∆(S).…”

“…The maximal subsemigroups of the singular ideal of O n were found in [11], and those of the singular ideal of PO n in [14]. Additionally, the maximal subsemigroups of the singular ideal of OD n were described in [11] and [28]. However, since the group of units of OD n is non-trivial, this is a fundamentally different problem than finding the maximal subsemigroups of OD n .…”

“…However, the group of units of OD n is non-trivial, and the units act on the monoid in such a way as to break the correspondence between the maximal subsemigroups of the singular part, and the maximal subsemigroups of OD n itself. Thus [11,28] solve an essentially different problem than the description of the maximal subsemigroups of OD n .…”

The maximal subsemigroups of semigroups of transformations preserving or reversing the orientation on a finite chain

“…Before we give the description of the maximal subsemigroups of the ideals of the semigroup OP n , we recall the following result presented by the first and third author in [5] (see also [12]). Theorem 1.10 ( [5]) Let n ≥ 3 and 2 ≤ k ≤ n − 1. Then a subsemigroup of O(n, k) is maximal if and only if it belongs to one of the following types:…”

“…As usual, O n denotes the submonoid of T n of all order-preserving transformations of X n . This monoid has been largely studied, for instance in [1,5,8,9,12].…”