Marczak, Adam W. A combinatorial characterization of normalizations of Boolean algebras. (English) Zbl 1096.06011 Algebra Univers. 55, No. 1, 57-66 (2006). 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. Cited in 1 Document 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 Keywords:normal identity; normalization of an algebra; nilpotent extension of an algebra; nilpotent shift of a variety; Boolean algebra; groupoid; Sheffer stroke; MEP for a finite sequence PDFBibTeX XMLCite \textit{A. W. Marczak}, Algebra Univers. 55, No. 1, 57--66 (2006; Zbl 1096.06011) Full Text: DOI