# Article

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Keywords:
semigroup variety; band variety; full embedding; $f\!f$-alg-universality; determinacy; $Q$-universality
Summary:
A category $K$ is called $\alpha$-determined if every set of non-isomorphic $K$-objects such that their endomorphism monoids are isomorphic has a cardinality less than $\alpha$. A quasivariety $Q$ is called $Q$-universal if the lattice of all subquasivarieties of any quasivariety of finite type is a homomorphic image of a sublattice of the lattice of all subquasivarieties of $Q$. We say that a variety $V$ is var-relatively alg-universal if there exists a proper subvariety $W$ of $V$ such that homomorphisms of $V$ whose image does not belong to $W$ contains a full subcategory isomorphic to the category of all graphs. A semigroup variety $V$ is nearly $J$-trivial if for every semigroup $S\in V$ any $J$-class containing a group is a singleton. We prove that for a nearly $J$-trivial variety $V$ the following are equivalent: $V$ is $Q$-universal; $V$ is var-relatively alg-universal; $V$ is $\alpha$-determined for no cardinal $\alpha$; $V$ contains at least one of the three specific semigroups. Dually, for a nearly $J$-trivial variety $V$ the following are equivalent: $V$ is $3$-determined; $V$ is not var-relatively alg-universal; the lattice of all subquasivarieties of $V$ is finite; $V$ is a subvariety of one of two special finitely generated varieties.
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