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A new product of algebras and a type reduction theorem

Authors:	Ralph McKenzie

Institution:	(1) University of California, Berkeley, California, USA

Abstract:	A construction is defined which associates, to every algebra $\mathfrak{A}$ of a fixed but arbitrary finite similarity type, a groupoidF $\mathfrak{A}$ . The identities ofF $\mathfrak{A}$ are finitely based if and only if those of $\mathfrak{A}$ are, andF $\mathfrak{A}$ is finite if and only if $\mathfrak{A}$ is finite. Up to isomorphism,F $\mathfrak{A}$ has the same endomorphism monoid and subalgebra lattice as $\mathfrak{A}$ , but the congruence lattice ofF $\mathfrak{A}$ is the result of adjoining a new 1 to the congruence lattice of $\mathfrak{A}$ .F is functorial, preserves the satisfaction (and the non-satisfaction) of most Mal'cev conditions, and produces, by composition with the operation of forming the generated variety, an isomorphism of the lattice of varieties of fixed type to an interval in the lattice of varieties of groupoids.The construction makes use of a new product operation, applicable to two algebras of differing similarity types, which is introduced and studied in this paper.Research supported by National Science Foundation grant MCS-8103455.Presented by K. A. Baker.

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