l-Hemi-Implicative Semilattices

CASTIGLIONI, JOSÉ LUIS; SAN MARTÍN, HERNÁN JAVIER

STUDIA LOGICA

Springer Netherlands

Año: 2017 p. 1 - 16

0039-3215

An l-hemi-implicative semilattice is an algebra (Formula presented.) such that (Formula presented.) is a semilattice with a greatest element 1 and satisfies: (1) for every (Formula presented.), (Formula presented.) implies (Formula presented.) and (2) (Formula presented.). An l-hemi-implicative semilattice is commutative if if it satisfies that (Formula presented.) for every (Formula presented.). 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 (Formula presented.). Endowing (Formula presented.) with the binary operation (Formula presented.) the algebra (Formula presented.) results an l-hemi-implicative semilattice, which also satisfies the identity (Formula presented.). 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.