Geodesic
On characteristic properties of semigroups
Algebra and discrete mathematics, Tome 20 (2015) no. 1, pp. 32-39.

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Let $\mathcal{K}$ be a class of semigroups and $\mathcal{P}$ be a set of general properties of semigroups. We call a subset $Q$ of $\mathcal{P}$ characteristic for a semigroup $S\in\mathcal{K}$ if, up to isomorphism and anti-isomorphism, $S$ is the only semigroup in $\mathcal{K}$, which satisfies all the properties from $Q$. The set of properties $\mathcal{P}$ is called char-complete for $\mathcal{K}$ if for any $S\in \mathcal{K}$ the set of all properties $P\in\mathcal{P}$, which hold for the semigroup $S$, is characteristic for $S$. We indicate a 7-element set of properties of semigroups which is a minimal char-complete set for the class of semigroups of order $3$.
Keywords: semigroup, idempotent, Cayley table, characteristic property, char-complete set.
Mots-clés : anti-isomorphism 
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Vitaliy M. Bondarenko; Yaroslav V. Zaciha. On characteristic properties of semigroups. Algebra and discrete mathematics, Tome 20 (2015) no. 1, pp. 32-39. http://geodesic.mathdoc.fr/item/ADM_2015_20_1_a3/

[1] V. M. Bondarenko, Y. V. Zaciha, “On defining relations for minimal generator systems of three-order semigroups”, Science Journal of National Pedagogical Dragomanov University, Series 1: Physics and Mathematics, 2013, no. 14, 62–67 (Ukrainian)