Free rectangular dibands and free dimonoids

Abstract

We construct a free rectangular diband and describe its structure.  We also present the least rectangular diband congruence, the least \((r b, rz)\)-congruence,  the least left zero and right zero congruence,  the least rectangular band congruence and the least left zero congruence on free dimonoids and use them to obtain decompositions of free dimonoids.