Artículo
Implication Zroupoids and Birkhoff Systems
Fecha de publicación:
11/2021
Editorial:
University of Hatef
Revista:
Journal of Algebraic Hyperstructures and Logical Algebras
ISSN:
2676-6000
e-ISSN:
2676-6019
Idioma:
Inglés
Tipo de recurso:
Artículo publicado
Clasificación temática:
Resumen
An algebra A = hA, →, 0i, 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 0 → x) → (y → z) 0 ] 0 , where x 0 := x → 0, and 000 ≈ 0. These algebras generalize De Morgan algebras and ∨-semilattices with zero. Let I denote the variety of implication zroupoids. The investigations into the structure of I and of the lattice of subvarieties of I, begun in 2012, have continued in several papers (see the Bibliography at the end of the paper). The present paper is a sequel to that series of papers and is devoted to making further contributions to the theory of implication zroupoids. The main purpose of this paper is to prove that if A is an algebra in the variety I, then the derived algebra Amj := hA; ∧, ∨i, where a ∧ b := (a → b 0 ) 0 and a ∨ b := (a 0 ∧ b 0 ) 0 , satisfies the Birkhoff’s identity (BR): x ∧ (x ∨ y) ≈ x ∨ (x ∧ y). As a consequence, the implication zroupoids A whose derived algebras Amj are Birkhoff systems are characterized. Another interesting consequence of the main result is that there are bisemigroups that are not bisemilattices but satisfy the Birkhoff’s identity, which leads us naturally to define the variety of ”Birkhoff bisemigroups” as bisemigroups satisfying the Birkhoff identity, as a generalization of Birkhoff systems. The paper concludes with some open problems.
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Articulos(INMABB)
Articulos de INST.DE MATEMATICA BAHIA BLANCA (I)
Articulos de INST.DE MATEMATICA BAHIA BLANCA (I)
Citación
Cornejo, Juan Manuel; Sankappanavar, Hanamantagouda P.; Implication Zroupoids and Birkhoff Systems; University of Hatef; Journal of Algebraic Hyperstructures and Logical Algebras; 2; 4; 11-2021; 1-12
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Cornejo, Juan Manuel ; Sankappanavar, Hanamantagouda P. (Springer Verlag Berlín, 2017-12-01)