Abstract
A doppelsemigroup is a nonempty set equipped with two binary associative operations satisfying certain identities. In this paper, we consider the variety of rectangular doppelsemigroups which are analogs of rectangular semigroups. We construct the free rectangular doppelsemigroup and characterize the least rectangular congruence on the free doppelsemigroup. As a consequence, the free rectangular semigroup is presented. We also describe all (maximal) subdoppelsemigroups, all idempotents and all endomorphisms of the free rectangular doppelsemigroup, and give a criterion for an isomorphism of endomorphism semigroups of free rectangular doppelsemigroups. In addition, we show that the endomorphism semigroup of the free rectangular doppelsemigroup is not regular in general.
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