On a symmetrization of hemiimplicative semilattices

Abstract

A hemiimplicative semilattice is a bounded semilattice $(A, \wedge, 1)$ endowed with a binary operation $\to$, satisfying that for every $a, b, c \in A$, $a \leq b \to c$ implies $a \wedge b \leq c$ (that is to say, one of the conditionals satisfied by the residuum of the infimum) and the equation $a \to a = 1$. The class of hemiimplicative semilattices form a variety. These structures provide a general framework for the study of different structures of interest in algebraic logic. In any hemiimplicative semilattice it is possible to define a derived operation by $a \sim b := (a \to b) \wedge (b \to a)$. Endowing $(A, \wedge, 1)$ with the binary operation $\sim$ results again a hemiimplicative semilattice, which also satisfies the identity $a \sim b = b \sim a$. We call the elements of the subvariety of hemiimplicative semilattices satisfying $a \to b = b \to a$, a symmetric hemiimplicative semilattice. In this article, we study the correspondence assigning the symmetric hemiimplicative semilattice $(A, \wedge, \sim , 1)$ to the hemiimplicative semilattice $(A, \wedge, \to, 1)$. In particular, we characterize the image of this correspondence. We also provide many new examples of hemiimplicative semilattice structures on any bounded semillatice (possibly with bottom). Finally, we characterize congruences on the clases of hemiimplicative semilattices introduced as examples and we describe the principal congruences of hemiimplicative semilattices.