  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.
