On difunctionality of class relations

• Autor(a) principal: Hoefnagel, Michael
• Data de Publicação: 2020
• Outros Autores: Janelidze, Zurab, Rodelo, Diana
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10316/89450; https://doi.org/10.1007/s00012-020-00651-z
• topic: Class relations, Congruence permutability, Congruence distributivity, Congruence modularity, Directly decomposable congruence classes, Difunctionality, Egg-box property, Mal’tsev condition, Mal’tsev variety, Shifting lemma.

For a given variety V of algebras, we define a class relation to be a binary relation R ⊆ S^2 which is of the form R = S^2 ∩ K for some congruence class K on A^2, where A is an algebra in V such that S ⊆ A. In this paper we study the following property of V: every reflexive class relation is an equivalence relation. In particular, we obtain equivalent characterizations of this property analogous to well-known equivalent characterizations of congruence-permutable varieties. This property determines a Mal’tsev condition on the variety and in a suitable sense, it is a join of Chajda’s egg-box property as well as Duda’s direct decomposability of congruence classes.