Generating Sets of Finite Transformation Semigroups PK(n, r) and K (n, r)

Abstract

Let P _n and T _n be the partial transformation and the full transformation semigroups on the set {1,…, n}, respectively. In this paper we find necessary and sufficient conditions for any set of partial transformations of height r in the subsemigroup PK(n, r) = {α ∈P _n : |im (α)| ≤r} of P _n to be a (minimal) generating set of PK(n, r); and similarly, for any set of full transformations of height r in the subsemigroup K(n, r) = {α ∈T _n : |im (α)| ≤r} of T _n to be a (minimal) generating set of K(n, r) for 2 ≤ r ≤ n − 1.

Key Words:

• Full transformation semigroup
• Idempotents
• (Minimal) generating set
• Partial transformation semigroup
• Rank

2010 Mathematics Subject Classification:

• 20M20

Acknowledgments

Dedicated to the memory of John Mackintosh Howie (1936–2011).

Notes

Communicated by V. Gould.