John M. Howie

Idempotent rank in finite full transformation semigroups

The subsemigroup Sing n of singular elements of the full transformation semigroup on a finite set is generated by n(n − l)/2 idempotents of defect one. In this paper we extend this result to the subsemigroup K(n, r) consisting of all elements of rank r or less. We prove that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n, r) , the Stirling number of the second kind.

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.

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.

COUNTABLE VERSUS UNCOUNTABLE RANKS IN INFINITE SEMIGROUPS OF TRANSFORMATIONS AND RELATIONS

The relative rank

Nilpotents in finite symmetric inverse semigroups

\rank(S:A)

GENERATORS AND FACTORISATIONS OF TRANSFORMATION SEMIGROUPS

of a subset

A combinatorial property of finite full transformation semigroups

A

Semigroups of high rank

of a semigroup

Rank Properties of the Semigroup of Singular Transformations on a Finite Set

S

A nilpotent generated semigroup associated with a semigroup of full transformations

is the minimum cardinality of a set