On characteristic properties of semigroups

Abstract

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 \(\mathcal{P}\)   cha\-racteristic 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 \(3\).