Semigroups of Transformations Preserving an Equivalence Relation and a Cross-Section

Abstract

For a set X, an equivalence relation ρ on X, and a cross-section R of the partition X/ρ induced by ρ, consider the semigroup T(X, ρ, R) consisting of all mappings a from X to X such that a preserves both ρ (if (x, y) ∈ ρ then (xa, ya) ∈ ρ) and R (if r ∈ R then ra ∈ R). The semigroup T(X, ρ, R) is the centralizer of the idempotent transformation with kernel ρ and image R. We determine the structure of T(X, ρ, R) in terms of Green's relations, describe the regular elements of T(X, ρ, R), and determine the following classes of the semigroups T(X, ρ, R): regular, abundant, inverse, and completely regular.

Key Words:

• Transformation
• Equivalence relation
• Idempotent
• Centralizer

2000 Mathematics Subject Classification:

• 20M20

Notes

^#Communicated by P. Higgins.