A combinatorial characterization of normalizations of Boolean algebras |
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Authors: | Adam W. Marczak |
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Affiliation: | (1) Institute of Mathematics and Computer Sciences, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland |
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Abstract: | In this paper we prove that if a groupoid has exactly 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;·) is clone equivalent to a normalization of a nontrivial Boolean algebra if and only if the value of the free spectrum for (G;·) is 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. Received September 15, 2004; accepted in final form October 4, 2005. |
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Keywords: | 06E05 20N02 08A05 08B26 |
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