Semigroups of partial transformations with kernel and image restricted by an equivalence

• Autor(a) principal: André, Jorge M.
• Data de Publicação: 2020
• Outros Autores: Konieczny, Janusz
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10362/101775
• topic: Partial transformation semigroups; Equivalence relations; Green’s relations; Regular semigroups; Ideals; Rank

For an arbitrary set X and an equivalence relation μ on X, denote by Pμ(X) the semigroup of partial transformations α on X such that xμ⊆x(ker(α)) for every x∈dom(α), and the image of α is a partial transversal of μ. Every transversal K of μ defines a subgroup G=GK of Pμ(X). We study subsemigroups ⟨G,U⟩ of Pμ(X) generated by G∪U, where U is any set of elements of Pμ(X) of rank less than |X/μ|. We show that each ⟨G,U⟩ is a regular semigroup, describe Green’s relations and ideals in ⟨G,U⟩, and determine when ⟨G,U⟩ is an inverse semigroup and when it is a completely regular semigroup. For a finite set X, the top J-class J of Pμ(X) is a right group. We find formulas for the ranks of the semigroups J, G∪I, J∪I, and I, where I is any proper ideal of Pμ(X).