Geodesic
Polyabelian loops and Boolean completeness
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 4, pp. 671-686.

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We consider the question of which loops are capable of expressing arbitrary Boolean functions through expressions of constants and variables. We call this property {\it Boolean completeness\/}. It is a generalization of functional completeness, and is intimately connected to the computational complexity of various questions about expressions, circuits, and equations defined over the loop. We say that a loop is {\it polyabelian\/} if it is an iterated affine quasidirect product of Abelian groups; polyabelianness coincides with solvability for groups, and lies properly between nilpotence and solvability for loops. Our main result is that a loop is Boolean-complete if and only if it is not polyabelian. Since groups are Boolean-complete if and only if they are not solvable, this shows that polyabelianness, for these purposes, is the appropriate generalization of solvability to loops.
Classification : 03G05, 06E30, 17A01, 17X08, 20N05, 68Q15, 68W30, 94C10
Keywords: loops; quasigroups; functional closure; solvability; quasidirect products; computational complexity 
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Lemieux, François; Moore, Cristopher; Thérien, Denis. Polyabelian loops and Boolean completeness. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 4, pp. 671-686. http://geodesic.mathdoc.fr/item/CMUC_2000__41_4_a2/