On Closedness of Some Homotypical Varieties of Semigroups

It is well known that every subvariety of the variety of semigroups need not be absolutely closed. Thus, identifying those subvarieties that are closed in themselves or in larger varieties is an interesting open problem in the theory of semigroup dominions. The significance of finding such varieties lies in the fact that, in these varieties, epimorphisms are precisely the surjective morphisms. In this paper, we address this problem and demonstrate that for each , the varieties of semigroups , where i ∈ {1, 2, 3} and j is any nontrivial permutation of {1, 2, 3} not fixing 1 when i = 2, are closed.