L-Hemi-Implicative Semilattices

Abstract

An l-hemi-implicative semilattice is an algebra $$\mathbf {A} = (A,\wedge ,\rightarrow ,1)$$ such that $$(A,\wedge ,1)$$ is a semilattice with a greatest element 1 and satisfies: (1) for every $$a,b,c\in A$$ , $$a\le b\rightarrow c$$ implies $$a\wedge b \le c$$ and (2) $$a\rightarrow a = 1$$ . An l-hemi-implicative semilattice is commutative if if it satisfies that $$a\rightarrow b = b\rightarrow a$$ for every $$a,b\in A$$ . 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 $$a \sim b := (a \rightarrow b) \wedge (b \rightarrow a)$$ . Endowing $$(A,\wedge ,1)$$ with the binary operation $$\sim $$ the algebra $$(A,\wedge ,\sim ,1)$$ results an l-hemi-implicative semilattice, which also satisfies the identity $$a \sim b = b \sim a$$ . 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.