l-hemi-implicative semilattices

JOSÉ LUIS CASTIGLIONI; HERNÁN JAVIER SAN MARTÍN

STUDIA LOGICA

Springer

Año: 2018 vol. 106 p. 675 - 690

0039-3215

An $l$-hemi-implicative semilattice is an algebra $mathbf{A} = (A,wedge,ightarrow,1)$ such that $(A,wedge,1)$ is a semilattice with a greatest element $1$ and satisfies: for every $a,b,cin A$, $aleq bightarrow c$ implies $awedge b leq c$ and 2) $aightarrow a = 1$. A $l$-hemi-implicative semilattice is commutative if if it satisfies that $aightarrow b = bightarrow a$ for every $a,bin A$. 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 $a s b := (a ightarrow b) wedge (b ightarrow a)$. Endowing $(A,wedge,1)$ with the binary operation $s$ the algebra $(A,wedge,s,1)$ results an $l$-hemi-implicative semilattice, which also satisfies the identity $a s b = b s a$. 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.