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The algebra of implicit operations
Authors:Jorge Almeida
Institution:(1) Centro de Matemática Faculdade de Ciências, Universidade do Porto, Porto, Portugal
Abstract:For an ordinalagr and a class 
$$C$$
of topological algebras of a given type (which may be infinite and may contain inflnitary operations), anagr-aryimplicit operation on 
$$C$$
is any ldquonewrdquoagr-ary operation whose introduction does not eliminate any continuous homomorphisms between members of 
$$C$$
. The set of allagr-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 subalgebraOHgr agr 
$$C$$
of allagr-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 ofagr-generated members of 
$$C$$
. These results are applied to pseudovarieties of topological and finite algebras.This work was supported, in part, by INIC grant 85/CEX/4. This paper was written while the author was a faculty member at the Universidade do Minho.Presented by Ralph McKenzie.
Keywords:
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