The number of non-isomorphic models in quasi-varieties of semigroups

Abstract

Define\( n_K (\lambda )\) tobe eitherω, or the number of non-isomorphic algebras in\(K\)] having cardinality λ, whichever cardinal is larger. It is proved here that if\(K\)] is a quasi-variety (universal Horn class) of semigroups, then\(n_K\) is one of four functions. Each of these functions satisfies:\(n_K (\omega ) = \omega\) or\(n_K (\omega ) = 2^\omega\). If\(n_K (\lambda )< 2^\lambda\) for some infinite λ then\(K\)] is a residually finitevariety.