Abstract:
It is proved that a system of axioms for a dimonoid is independent and Cayley's theorem for semigroups has an analog in the class of dimonoids. A least separative congruence is constructed on an arbitrary dimonoid endowed with a commutative operation. It is shown that an appropriate quotient dimonoid is a commutative separative semigroup. A least separative congruence on a free commutative dimonoid is characterized. It is stated that each dimonoid with a commutative operation is a semilattice of Archimedean subdimonoids, each dimonoid with a commutative periodic semigroup is a semilattice of unipotent subdimonoids, and each dimonoid with a commutative operation is a semilattice of $a$-connected subdimonoids. Different dimonoid constructions are presented.
Keywords:dimonoid, dimonoid with commutative operation, free commutative dimonoid, semilattice of subdimonoids, semigroup.
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