Defining relations for idempotent generators in finite full transformation semigroups

In 1966, John Howie showed that the semigroup $\mathcal{T}_{n}\setminus \mathcal{S}_{n}$ of all singular transformations on a n element set is generated by the set of all idempotent transformations of rank n?1. We give a presentation for $\mathcal{T}_{n}\setminus \mathcal{S}_{n}$ in terms of this generating set.     

Identities in full transformation semigroups

Presented by B. M. Schein.     

Endomorphisms of finite full transformation semigroups

We describe all endomorphisms of finite full transformation semigroups and count their number.

     

Some remarks on finite full transformation semigroups

Communicated by J. M. Howie     

Products of idempotents in finite full transformation semigroups

Communicated by J. M. Howie     

On the representation of abstract semigroups by transformation semigroups: Computer investigations

Communicated by J. S. Ponizovskii     

Defining relations for idempotent generators in finite partial transformation semigroups

We give a presentation for the semigroup of all singular partial transformations on a finite set, in terms of the generating set consisting of all idempotent partial transformations of corank 1.     

On the growth of generating sets for direct powers of semigroups

For a semigroup S its d-sequence is d(S)=(d ₁,d ₂,d ₃,…), where d _i is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups.     

Rank and idempotent rank of finite full transformation semigroups with restricted range

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. In 2011, Sanwong studied the regular part $$F(X,Y)=\bigl\{\alpha\in T(X,Y): X\alpha\subseteq Y\alpha\bigr\}, $$ of T(X,Y) and described its Green’s relations and ideals. In this paper, we compute the rank of F(X,Y) when X is a finite set. Moreover, we obtain the rank and idempotent rank of its ideals.     

On generators and relations for unions of semigroups

Finite generation and presentability of general unions of semigroups, as well as of bands of semigroups, bands of monoids, semilattices of semigroups and strong semilattices of semigroups, are investigated. For instance, it is proved that a band Y of monoids S _α (α∈ Y ) is finitely generated/presented if and only if Y is finite and all S _α are finitely generated/presented. By way of contrast, an example is exhibited of a finitely generated semigroup which is not finitely presented, but which is a disjoint union of two finitely presented subsemigroups. January 21, 2000     

Defining relations of semigroups of all directed transformations of an ordered finite set

The semigroup of all transformations X of a finite (partially) ordered set , such that X for all , is considered. All possible generating sets of a are elucidated. Only one of those sets is irreducible. A system of defining relations is found for that generating set.Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 657–662, June, 1968.     

On the enumeration of the set of saturated numerical semigroups of a given genus

In this paper we present an algorithm for computing the set of saturated numerical semigroups of a given genus. We see how the set of saturated numerical semigroups can be arranged in a tree rooted in \(\mathbb{N}\) and we describe the sons of any vertex of this tree.