Implication Zroupoids and Identities of Associative Type

• Autores: Cornejo, Juan Manuel; Sankappanavar, Hanamantagouda P.
• Año de publicación: 2018
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: IMPLICATION ZRUPOID; VARIETY; IDENTITY OF ASSOCIATIVE TYPE
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

An algebra A=⟨A,→,0⟩, where → is binary and 0 is a constant, is called an implication zroupoid (I-zroupoid, for short) if A satisfies the identities: (x→y)→z≈[(z′→x)→(y→z)′]′ and 0′′≈0, where x′:=x→0, and I denotes the variety of all I-zroupoids. An I-zroupoid is symmetric if it satisfies x′′≈x and (x→y′)′≈(y→x′)′. The variety of symmetric I-zroupoids is denoted by S. An identity p≈q, in the groupoid language ⟨→⟩, is called an identity of associative type of length 3 if p and q have exactly 3 (distinct) variables, say x,y,z, and are grouped according to one of the two ways of grouping: (1) ⋆→(⋆→⋆) and (2) (⋆→⋆)→⋆, where ⋆ is a place holder for a variable. A subvariety of I is said to be of associative type of length 3, if it is defined, relative to I, by a single identity of associative type of length 3. In this paper we give a complete analysis of the mutual relationships of all subvarieties of I of associative type of length 3. We prove, in our main theorem, that there are exactly 8 such subvarieties of I that are distinct from each other and describe explicitly the poset formed by them under inclusion. As an application of the main theorem, we derive that there are three distinct subvarieties of the variety S, each defined, relative to S, by a single identity of associative type of length 3.
Fil: Cornejo, Juan Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina
Fil: Sankappanavar, Hanamantagouda P.. State University of New York; Estados Unidos