l-Hemi-Implicative Semilattices

Autor: Hernán Javier San Martín, José Luis Castiglioni
Rok vydání: 2017
Předmět:
Zdroj: Studia Logica. 106:675-690
ISSN: 1572-8730
0039-3215
DOI: 10.1007/s11225-017-9759-3
Popis: An l-hemi-implicative semilattice is an algebra A=(A,∧,→,1) such that (A,∧,1) is a semilattice with a greatest element 1 and satisfies: (1) for every a,b,c∈A , a≤b→c implies a∧b≤c and (2) a→a=1 . An l-hemi-implicative semilattice is commutative if if it satisfies that a→b=b→a for every a,b∈A . It is shown that the class of l-hemi-implicative semilattices is a variety. These algebras provide a general framework for the study of different algebras of interest in algebraic logic. In any l-hemi-implicative semilattice it is possible to define an derived operation by a∼b:=(a→b)∧(b→a) . Endowing (A,∧,1) with the binary operation ∼ the algebra (A,∧,∼,1) results an l-hemi-implicative semilattice, which also satisfies the identity a∼b=b∼a . In this article, we characterize the (derived) commutative l-hemi-implicative semilattices. We also provide many new examples of l-hemi-implicative semilattice on any semillatice with greatest element (possibly with bottom). Finally, we characterize congruences on the classes of l-hemi-implicative semilattices introduced earlier and we characterize the principal congruences of l-hemi-implicative semilattices. Fil: Castiglioni, José Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina Fil: San Martín, Hernán Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina
Databáze: OpenAIRE
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