Varieties Of Orthomodular Lattices Research Articles

The Join of the Variety of MV-Algebras and the Variety of Orthomodular Lattices

We axiomatize the smallest variety that contains both the variety of MV-algebras and the variety (term equivalent to the variety) of orthomodular lattices.

On varieties of basic algebras

It is known that the variety of commutative basic algebras, the variety of MV-algebras, the variety of orthomodular lattices and hence also the variety of Boolean algebras are proper subvarieties of the variety of basic algebras. In the paper several possible axiomatizations of the variety of basic algebras are provided. The variety of symmetric basic algebras is introduced and shown to be a proper subvariety of the variety of commutative basic algebras, and whose members have the underlying lattices distributive. It is proved that commutative basic algebras can be described as residuated $$l$$ l -groupoids.

Logical Approach for Two-Valued States on Quantum Systems

In this paper we develop a logical system associated to two-valued states on orthomodular lattices. An completeness theorem with respect to a variety of orthomodular lattices enriched with an unary operation that represents two-valued states is given.

Orthomodular lattices and permutable congruences

If a variety of ortholattices is congruence-permutable, then we prove that it is a variety of orthomodular lattices.

Self-dual bases for varieties of orthomodular lattices

For any finitely based variety of orthomodular lattices, we determine the sizes of all equational bases that are both irredundant and self-dual.

Orthomodular lattices with rich state spaces

We introduce a new construction technique for orthomodular lattices. In contrast to the preceding constructions, it admits rich spaces of states (= probability measures), i.e., for each pair of incomparable elements a,c there is a state s such that s(a) = 1 > s(c). This allowed a progress in many questions that were open for a long time; among others we prove that there is a continuum of varieties of orthomodular lattices with rich state spaces and solve a problem formulated by R. Mayet in 1985. As a by-product of this research, the uniqueness problem for bounded observables (posed by S. Gudder in 1966) has been solved. As a tool, we introduce also a new construction –identification of atoms in an orthomodular lattice – which may be of separate interest.

Projective orthomodular lattices II

In this paper we continue to investigate projectivity in orthomodular lattices. We prove the somewhat surprising result that no uncountable Boolean algebra is projective in the variety of orthomodular lattices.

Decision problem for orthomodular lattices

This paper answers a question of H. P. Sankappanavar who asked whether the theory of orthomodular lattices is recursively (finitely) inseparable (question 9 in [10]). A very similar question was raised by Stanley Burris at the Oberwolfach meeting on Universal Algebra, July 15–21, 1979, and was later included in G. Kalmbach’s monograph [6] as the problem 42. Actually Burris asked which varieties of orthomodular lattices are finitely decidable. Although we are not able to give a full answer to Burris’ question we have a contribution to the problem. Note here that each finitely generated variety of orthomodular lattices is semisimple arithmetical and therefore directly representable. Consequently each such a variety is finitely decidable. (For a generalization of this, i.e. a characterization of finitely generated congruence modular varieties that are finitely decidable see [5].) In section 3, we give an example of finitely decidable variety of orthomodular lattices that is not finitely generated.

Equational bases for some varieties of orthomodular lattices related to states

Equational bases for some varieties of orthomodular lattices related to states

Varieties of Orthomodular Lattices

In this paper we start investigating the lattice of varieties of orthomodular lattices. The varieties studied here are those generated by orthomodular lattices which are the horizontal sum of Boolean algebras. It turns out that these form a principal ideal in the lattice of all varieties of orthomodular lattices. We give a complete description of this ideal; in particular, we show that each variety in it is generated by its finite members. We furthermore show that each of these varieties is finitely based by exhibiting a (rather complicated) finite equational basis for each variety.Our methods rely heavily on B. Jonsson's fundamental results in [8]. This, however, could be avoided by starting out with the equations given in sections 3 and 4. Some of our arguments were suggested by Baker [1],