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.