The Gumm level equals the alvin level in congruence distributive varieties
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Archiv der Mathematik
The Gumm level equals the alvin level in congruence distributive varieties Paolo Lipparini
Abstract. Congruence modular and congruence distributive varieties can be characterized by the existence of sequences of Gumm and J´ onsson terms, respectively. Such sequences have variable lengths, in general. It is immediate from the above paragraph that there is a variety with Gumm terms but without J´ onsson terms. We prove the unexpected result that, on the other hand, if some variety has both kinds of terms, then the minimal lengths of the sequences differ at most by 1. It follows that every onsson congruence distributive variety with r+1 Day terms has r2 −r+3 J´ terms. Mathematics Subject Classification. Primary 08B10. Keywords. Gumm terms, Alvin terms, J´ onsson terms, Congruence distributive variety, Congruence modular variety, Congruence identity.
An algebra (short for algebraic system) is a nonempty set endowed with a family of operations. A variety is a class of algebras of the same type which is definable by a set of equations. A congruence on some algebra is the kernel of some homomorphism, equivalently, a compatible equivalence relation. The set of congruences on some algebra has a lattice structure. An algebra is congruence modular if its lattice of congruences is modular and a variety is congruence modular if all of its members are. Congruence distributivity is defined in a similar manner. Congruence modular varieties include the varieties of groups, of rings, of quasigroups, as well as all congruence distributive varieties. Congruence distributive varieties include the varieties of lattices and of Boolean algebras. Work performed under the auspices of G.N.S.A.G.A. Work partially supported by PRIN 2012 “Logica, Modelli e Insiemi”. The author acknowledges the MIUR Survival Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
P. Lipparini
Arch. Math.
Of course, there have been interactions between the general theory of algebras and more specific kinds of algebraic systems [19]. Interesting connections recently emerged with the theory of computational complexity, in particular, the algebraic approach to the constraint satisfaction problem. In a nutshell, well-behaved classes of algebras first studied only for their algebraic properties—and the identities such classes satisfy—turned out to correspond to algorithmic results for CSP over constraint languages. See, e.g., [1,10] for a survey. Congruence distributivity and congruence modularity are among the first studied and most important conditions providing ‘good algebraic structure’. We show that two well-known and intensively studied characterizations of the above conditions are equivalent even as far as length is concerned, provided they are both applicable. In more detail, a term is, roughly, a word obtained by composition from the basic operations of a variety. Both congruence distributivity and congruence modularity can be characterized by the existence of sequences of terms
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