Isomorphism types of infinite algebras
Stephen D. Comer, J. J. Le Tourneau · Proceedings of the American Mathematical Society · 1969
A basic problem in the general theory of algebras is the determination of the number of nonisomorphic algebras with a given (finitary) similarity type and infinite power. Our main result is to show that in every case the number of such algebras actually equals the most naturally computed upper bound. In particular we show that if r is any similarity type containing n symbols, at least one of which is positive rank, and m is any infinite cardinal then there exist exactly 2m nonisomorphic algebras with similarity type r and power m. The corresponding problem for finite algebras was considered in Harrison [1]. In ?0 we introduce definitions and accompanying notation; in ?1 we consider the special problem of determining the number of nonisomorphic unary algebras (our work is based on the construction presented before Lemma 1); and in ?2 we consider this determination problem for arbitrary algebraic similarity types.