Abstract
In this paper, we give correspondences between unary algebras, semigroups and congruences on free semigroups. We establish isomorphisms between the complete lattice of varieties of semigroups and the complete lattices of families of varieties of unary algebras, and families of filters of congruences on free semigroups. Similar correspondences between generalized varieties and pseudovarieties of semigroups and corresponding families of algebras and congruences are also established.
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