The algebra of implicit operations

Abstract

For an ordinalα and a class$C$ of topological algebras of a given type (which may be infinite and may contain inflnitary operations), anα-aryimplicit operation on$C$ is any “new”α-ary operation whose introduction does not eliminate any continuous homomorphisms between members of$C$. The set of allα-ary implicit operations on$C$ is denoted by$\bar \Omega _\alpha C$ and forms an algebra of the given type which is endowed with the least topology making continuous all homomorphisms into members of$C$. With this topology,$\bar \Omega _\alpha C$ is a topological algebra in which the subalgebraΩ _α $C$ of allα-ary operations on$C$ which are induced by terms is dense, provided that$C$ is closed under the formation of closed subalgebras and finitary direct products. This is obtained by realizing$\bar \Omega _\alpha C$ as an inverse limit ofα-generated members of$C$. These results are applied to pseudovarieties of topological and finite algebras.