On Inverse subsemigroups of the semigroup of orientation preserving or orientation-reversing transformations

• Autor(a) principal: Catarino, Paula
• Data de Publicação: 2015
• Outros Autores: Higgins, Peter, Levi, Inessa
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10348/6740
• topic: semigroup; semilattice; inverse subsemigroup; strong inverse; transformation; order-preserving transformation; orientation-preserving transformation; orientation-reversing transformation

It is well-known [16] that the semigroup Tn of all total transformations of a given n-element set Xn is covered by its inverse subsemigroups. This note provides a short and direct proof, based on properties of digraphs of transformations, that every inverse subsemigroup of order-preserving transformations on a finite chain Xn is a semilattice of idempotents, and so the semigroup of all order-preserving transformations of Xn is not covered by its inverse subsemigroups. This result is used to show that the semigroup of all orientation-preserving transformations and the semigroup of all orientation-preserving or orientation-reversing transformations of the chain Xn are covered by their inverse subsemigroups precisely when n less than or equal 3.