Weaker universalities in semigroup varieties

M. Demlová, V. Koubek

A variety $\mathbb V$ has an alg-universal $n$-expansion if the addition of $n$ nullary operations to algebras from $\mathbb V$ produces an alg-universal category. It is proved that any semigroup variety $\mathbb V$ containing a semigroup that is neither an inflation of a completely simple semigroup nor an inflation of a semilattice of groups has an alg-universal 3-expansion. We say that a variety $\mathbb V$ is var-relatively alg-universal if for some proper subvariety $\mathbb W$ of $\mathbb V$ there is a faithful functor $F$ from the category of all digraphs and compatible mappings into $\mathbb V$ such that $\operatorname{Im}(Ff)$ belongs to $\mathbb W$ for no compatible mapping $f$ and if $f\colon F\mathbf G\to F\mathbf G'$ is a homomorphism where $\mathbf G$ and $\mathbf G'$ are digraphs then either $\operatorname{Im}(f)$ belongs to $\mathbb W$ or $f=Fg$ for a compatible mapping $g\colon\mathbf G\to\mathbf G'$. For a cardinal $\alpha\geq2$, a variety $\mathbb V$ is $\alpha$-determined if any set $\mathcal A$ of $\mathbb V$-algebras of cardinality a such that the endomorphism monoids of $A$ and $B$ are isomorphic for all $A,B\in\mathcal A$ contains distinct isomorphic algebras. Similar sufficient conditions for a semigroup variety $\mathbb V$ to be $\alpha$-determined for no cardinal a or var-relatively alg-universal are given. These results are proved by an analysis of three specific semigroup varieties.