l-Hemi-Implicative Semilattices

Abstract

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.