Endomorphism monoids in varieties of commutative semigroups. (Q382948)

A characterization of universal varieties of semigroups is given by \textit{V. Koubek} and \textit{J. Sichler} [J. Aust. Math. Soc., Ser. A 36, 143-152 (1984; Zbl 0549.20038)]. Since there are categories that are not universal but close to being universal the notion of almost universality is used. A concrete category \(\mathbb K\) is called almost universal if its class of non-constant morphisms contains an isomorphic copy of every category of algebras as a full subcategory. The almost universality of varieties of commutative semigroups and varieties of commutative monoids is investigated. The main result states that the almost universality of a variety \(\mathbb V\) of commutative semigroups is equivalent to each of the following two equivalent conditions: (1) there exists an infinite semigroup \(S\in\mathbb V\) such that \(\mathrm{End}(S)\) consists of the identity mapping and the constant endomorphism whose value is \(0\) of \(S\), (2) \(\mathbb V\) contains the variety of commutative semigroups determined by the identity \(x^2=x^2y\).