Lattice Laws Forcing Distributivity Under Unique Complementation
نویسندگان
چکیده
In this paper, we give several new lattice identities valid in nonmodular lattices such that a uniquely complemented lattice satisfying any of these identities is necessarily Boolean. Since some of these identities are consequences of modularity as well, these results generalize the classical result of Birkhoff and von Neumann that every uniquely complemented modular lattice is Boolean. In particular, every uniquely complemented lattice in M ∨N5, the least non-modular variety, is Boolean. ∗Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38. †Supported by an operating grant from NSERC of Canada (OGP8215).
منابع مشابه
Lattices with Unique Complements
Introduction. For'several years one of the outstanding problems of lattice theory has been the following: Is every lattice with unique complements a Boolean algebra? Any number of weak additional restrictions are sufficient for an affirmative answer. For example, if a lattice is modular (G. Bergman [l](1)) or ortho-complemented (G. Birkhoff [l]) or atomic (G. Birkhoff and M. Ward [l]), then uni...
متن کاملFirst Steps in Relational Lattice
Relational lattice reduces the set of six classic relational algebra operators to two binary lattice operations: natural join and inner union. We give an introduction to this theory with emphasis on formal algebraic laws. New results include Spight distributivity criteria and its applications to query transformations.
متن کاملThe distributivity numbers of of P(ω)/fin and its square
We show that in a model obtained by forcing with a countable support iteration of Mathias forcing of length ω2, the distributivity number of P(ω)/fin is ω2, whereas the distributivity number of r.o.(P(ω)/fin) 2 is ω1. This answers an old problem of Balcar, Pelant and Simon, and others.
متن کاملAn Analogue of Distributivity for Ungraded Lattices
In this paper, we define a property, trimness, for lattices. Trimness is a not-necessarily-graded generalization of distributivity; in particular, if a lattice is trim and graded, it is distributive. Trimness is preserved under taking intervals and suitable sublattices. Trim lattices satisfy a weakened form of modularity. The order complex of a trim lattice is contractible or homotopic to a sph...
متن کاملGeneralized Fuzzy Compactness in L-Topological Spaces
In this paper, we shall introduce generalized fuzzy compactness in L-spaces where L is a complete de Morgan algebra. This definition does not rely on the structure of basis lattice L and no distributivity is required. The intersection of a generalized fuzzy compact L-set and a generalized closed Lset is a generalized fuzzy compact L-set. The generalized irresolute image of a generalized fuzzy c...
متن کامل