Geodesic
Representations of ordered doppelsemigroups by~binary relations
Algebra and discrete mathematics, Tome 27 (2019) no. 1, pp. 144-154.

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We extend the study of doppelsemigroups and introduce the notion of an ordered doppelsemigroup. We construct the ordered doppelsemigroup of binary relations on an arbitrary set and prove that every ordered doppelsemigroup is isomorphic to some ordered doppelsemigroup of binary relations. In particular, we obtain an analogue of Cayley's theorem for semigroups in the class of doppelsemigroups. We also describe the representations of ordered doppelsemigroups by binary transitive relations.
Keywords: doppelsemigroup, ordered doppelsemigroup, semigroup, binary relation, representation. 
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Yurii V. Zhuchok; Jörg Koppitz. Representations of ordered doppelsemigroups by~binary relations. Algebra and discrete mathematics, Tome 27 (2019) no. 1, pp. 144-154. http://geodesic.mathdoc.fr/item/ADM_2019_27_1_a12/

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