Nik Ruskuc

Constructions and presentations for monoids

Presentations are found for the wreath product of two monoids, the Schutzenberger product of two monoids, the Bruck-Reilly extension of a monoid, strong semilattices of monoids and Rees matrix semigroups of monoids.

Automatic monoids and change of generators

The main result of this paper establishes invariance under change of generators for automatic structures for monoids (a property that is well known to hold for automatic groups but fails for semigroups). This result is then applied to show that if a free product of two monoids is automatic, then so are both the free factors. Finally, the difference between automatic structures, in terms of monoid generating sets and semigroup generating sets, is discussed.

On relative ranks of full transformation semigroups

For a semigroup S and a set the relative rank of S modulo A is the minimal cardinality of a setB such that generates S. We show that the relative rank of an infinite full transformation semigroup modulo the symmetric group, and also modulo the set of all idempotent mappings, is equal to 2. We also characterise all pairs of mappings which, together with the symmetric group or the set of all idempotents, generate the full transformation semigroup.

Generators and relations of direct products of semigroups

The purpose of this paper is to give necessary and sufficient; conditions for the direct product of two semigroups to be finitely generated, and also for the direct product to be finitely presented. As a consequence we construct a semigroup S of order 11 such that S x T is finitely generated but not finitely presented for every finitely generated infinite semigroup T. By way of contrast we show that, if S and T belong to a wide class of semigroups, then S x T is finitely presented if and only if both S and T are finitely presented, exactly as in the case of groups and monoids.

COUNTABLE VERSUS UNCOUNTABLE RANKS IN INFINITE SEMIGROUPS OF TRANSFORMATIONS AND RELATIONS

The relative rank

On the rank of completely 0-simple semigroups

\rank(S:A)

Generators and relations of Rees matrix semigroups

of a subset

Maximal subgroups of free idempotent-generated semigroups over the full transformation monoid

A

Rank Properties of Endomorphisms of Infinite Partially Ordered Sets

of a semigroup

GROUPS AND ACTIONS IN TRANSFORMATION SEMIGROUPS

S