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A combinatorial characterization of normalizations of Boolean algebras. (English) Zbl 1096.06011

Summary: In this paper we prove that if a groupoid has exactly \(2^{2^{n}}+n\) distinct \(n\)-ary term operations for \(n=1,2,3\) and the same number of constant unary term operations for \(n=0\), then it is a normalization of a nontrivial Boolean algebra. This, together with some general facts concerning normalizations of algebras, which we recall, yields a clone characterization of normalizations of nontrivial Boolean algebras: A groupoid \((G;\cdot)\) is clone equivalent to a normalization of a nontrivial Boolean algebra if and only if the value of the free spectrum for \((G;\cdot)\) is \(2^{2^{n}}+n\) for \(n=0,1,2,3\). In the last section the Minimal Extension Property for the sequence \((2,3)\) in the class of all groupoids is derived.

MSC:

06E05 Structure theory of Boolean algebras
20N02 Sets with a single binary operation (groupoids)
08A05 Structure theory of algebraic structures
08B26 Subdirect products and subdirect irreducibility
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