On congruences in weak implicative semi-lattices (Q1701818)

The weak implicative semi-lattices are obtained from Heyting algebras \((H,\wedge,\vee,\rightarrow,0,1)\) by replacing the residuation property: \[ z\leq x\rightarrow y\text{ if and only if }x\wedge z\leq y \leqno{(R)} \] by its weaker form: \[ \text{if }z\leq x\rightarrow y,\text{ then }x\wedge z\leq y. \leqno{(R^{\prime})} \] The paper presents some properties of weak implicative semi-lattices and gives characterizations of congruences in this class of algebras, including varieties of interest for the logic.