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.