Araújo, JoãoKonieczny, Janusz2015-03-242015-03-242004Araújo, João; Konieczny, Janusz - Semigroups of transformations preserving an equivalence relation and a cross-section. "Communications in Algebra" [Em linha]. ISSN 0092-7872 (Print) 1532-4125 (Online). Vol. 32 (2004), p. 1-170092-78721532-4125http://hdl.handle.net/10400.2/3806For 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.engTransformationEquivalence relationIdempotentCentralizer20M20journal article10.1081/AGB-120029913