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. The 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. The 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. Various dimonoid constructions are presented.
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Translated from Algebra i Logika, Vol. 50, No. 4, pp. 471–496, July-August, 2011.
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Zhuchok, A.V. Dimonoids. Algebra Logic 50, 323–340 (2011). https://doi.org/10.1007/s10469-011-9144-7
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DOI: https://doi.org/10.1007/s10469-011-9144-7