Peter M. Higgins

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.

Combinatorial results for semigroups of order-preserving mappings

Consider the finite set X n = {1,2, …, n } ordered in the standard way. Let T n denote the full transformation semigroup on X n , that is, the semigroup of all mappings α: X n → X n under composition. We shall call α order-preserving if i ≤ j implies i α ≤ j α for i , j ∈ X n , and α is decreasing if i α ≤ i for all i ∈ X n . This paper investigates combinatorial properties of the semigroup O of all order-preserving mappings on X n , and of its subsemigroup C which consists of all decreasing and order-preserving mappings.

COUNTABLE VERSUS UNCOUNTABLE RANKS IN INFINITE SEMIGROUPS OF TRANSFORMATIONS AND RELATIONS

The relative rank

GENERATORS AND FACTORISATIONS OF TRANSFORMATION SEMIGROUPS

rank(S:A)

Rank Properties of Endomorphisms of Infinite Partially Ordered Sets

of a subset

Generating the full transformation semigroup using order preserving mappings

A

A proof of Simon's theorem on piecewise testable languages

of a semigroup

Finite aperiodic semigroups with commuting idempotents and generalizations

S

Saturated and epimorphically closed varieties of semigroups

is the minimum cardinality of a set

Digraphs and the semigroup of all functions on a finite set

B