Orthomodular lattices with state-separated noncompatible pairs

In the logico-algebraic foundation of quantum mechanics one often deals with the orthomodular lattices (OML) which enjoy state-separating properties of noncompatible pairs (see e.g. [18], [9] and [15]). These properties usually guarantee reasonable richness of the state space—an assumption needed in developing the theory of quantum logics. In this note we consider these classes of OMLs from the universal algebra standpoint, showing, as the main result, that these classes form quasivarieties. We also illustrate by examples that these classes may (and need not) be varieties. The results supplement the research carried on in [1], [3], [4], [5], [6], [11], [12], [13] and [16].     

Orthomodular lattices in ordered vector spaces

In a partly ordered space the orthogonality relation is defined by incomparability. We define integrally open and integrally semi-open ordered real vector spaces. We prove: if an ordered real vector space is integrally semi-open, then a complete lattice of double orthoclosed sets is orthomodular. An integrally open concept is closely related to an open set in the Euclidean topology in a finite dimensional ordered vector space. We prove: if V is an ordered Euclidean space, then V is integrally open and directed (and is also Archimedean) if and only if its positive cone, without vertex 0, is an open set in the Euclidean topology (and also the family of all order segments , a < b, is a base for the Euclidean topology). Received January 7, 2005; accepted in final form November 26, 2005.     

Order Dimension of Orthomodular Amalgamations Over Trees

We consider the amalgamation of bounded involution posets over a strictly directed graph as applied to orthomodular lattices, orthomodular posets or orthoalgebras. In the finite setting, we show that the order dimension of the amalgamation does not exceed that of the amalgamated structures by more than one. We also present conditions under which equality obtains.      

Kripke‐style Semantics of Orthomodular Logics

We present here a Kripke‐style semantics for propositional orthomodular logics that is based on the representation theorem for orthomodular lattices by D.J. Foulis ([2]), in which a sort of semigroups is employed. This semantics can characterize the logics above the orthomodular logic by some elementary conditions.     

Quasivarieties of orthomodular lattices determined by conditions on states

In this paper we carry on the research initiated in [13] and [14]. We consider classes of orthomodular lattices which satisfy certain state and polynomial conditions. We show that these classes form quasivarieties. We then exhibit basic examples of these quasivarieties (some of these examples originated in the quantum logic theory). We finally show how the quasivarieties in question can be described in terms of implicative equations. (It should be noted that in some cases we have not been able to clarify whether or not a class shown to be a quasivariety is a variety, see Section 2.) Received May 26, 1998; accepted in final form May 19, 1999.     

Orthomodular lattices from 3-dimensional quadratic spaces

If E is a vector space over a field K, then any regular symmetric bilinear form on E induces a polarity on the lattice of all subspaces of E. In the particular case where E is 3-dimensional, the set of all subspaces M of E such that both M and are not N-subspaces (which, in most cases, is equivalent to saying that M is nonisotropic), ordered by inclusion and endowed with the restriction of the above polarity, is an orthomodular lattice T(E, ). We show that if K is a proper subfield of K, with K F₂, and E a 3-dimensional K -subspace of E such that the restriction of to E × E is, up to multiplicative constant, a bilinear form on the K -space E , then T(E , ) is isomorphic to an irreducible 3-homogeneous proper subalgebra of T(E, ). Our main result is a structure theorem stating that, when K is not of characteristic 3, the converse is true, i.e., any irreducible 3-homogeneous proper subalgebra of T(E, ) is of this form. As a corollary, we construct infinitely many finite orthomodular lattices which are minimal in the sense that all their proper subalgebras are modular. In fact, this last result was our initial aim in this paper.Received June 4, 2003; accepted in final form May 18, 2004.     

Consistent lattices and Steinitz spaces

It is well-known that a finite lattice L is isomorphic to the lattice of flats of a matroid if and only if L is geometric. A result due to Edelman (see [1], Theorem 3.3) states that a lattice is meet-distributive if and only if it is isomorphic to the lattice of all closed sets of a convex geometry. In this note we prove that a finite lattice is the lattice of closed sets of a closure space with the Steinitz exchange property if and only if it is a consistent lattice. Received February 28, 1997; accepted in final form February 2, 1998.     

Embeddings into Orthomodular Lattices with Given Centers,State Spaces and Automorphism Groups

We prove that, given a nontrivial Boolean algebra B, a compact convex set S and a group G, there is an orthomodular lattice L with the center isomorphic to B, the automorphism group isomorphic to G, and the state space affinely homeomorphic to S. Moreover, given an orthomodular lattice J admitting at least one state, L can be chosen such that J is its subalgebra.     

Commutators and decompositions of orthomodular lattices

We show that in any complete OML (orthomodular lattice) there exists a commutatorc such that [0,c ] is a Boolean algebra. This fact allows us to prove that a complete OML satisfying the relative centre property is isomorphic to a direct product [0,a] × [0,a ] wherea is a join of two commutators, [0,a] is an OML without Boolean quotient and [0,a ] is a Boolean algebra. The proof uses a new characterization of the relative centre property in complete OMLs. In a final section, we specify the previous direct decomposition in the more particular case of locally modular OMLs.     

Varieties of semigroups with non-trivial identities on subsemigroup lattices

The present notice is devoted to the characterization up to the group case of varieties of semigroups whose subsemigroup lattices satisfy non-trivial identities. Received November 2, 1999; accepted in final form April 23, 2000.     

On lattices embeddable into subsemigroup lattices. III: Nilpotent semigroups

We prove that the class of the lattices embeddable into subsemigroup lattices of n-nilpotent semigroups is a finitely based variety for all n < ω. Repnitski? showed that each lattice embeds into the subsemigroup lattice of a commutative nilsemigroup of index 2. In this proof he used a result of Bredikhin and Schein which states that each lattice embeds into the suborder lattices of an appropriate order. We give a direct proof of the Repnitski? result not appealing to the Bredikhin-Schein theorem, so answering a question in a book by Shevrin and Ovsyannikov.     

Commutative idempotent residuated lattices

We investigate the variety of residuated lattices with a commutative and idempotent monoid reduct.     

On lattices embeddable into subsemigroup lattices. V. Trees

We prove that the class of finite lattices embeddable into the subsemilattice lattices of semilattices which are (n-ary) trees can be axiomatized by identities within the class of finite lattices, whence it forms a pseudovariety.     

Iterative separation in distributive congruence lattices

In [PLOŠČICA, M.: Separation in distributive congruence lattices, Algebra Universalis 49 (2003), 1–12] we defined separable sets in algebraic lattices and showed a close connection between the types of non-separable sets in congruence lattices of algebras in a finitely generated congruence distributive variety and the structure of subdirectly irreducible algebras in . Now we generalize these results using the concept of separable mappings (defined on some trees) and apply them to some lattice varieties. Supported by VEGA Grants 2/4134/24, 2/7141/27, and INTAS Grant 03-51-4110.     

Complete congruence lattices of join-infinite distributive lattices

Received July 26, 1993; accepted in final form July 16, 1996.     

Q-universal varieties of bounded lattices

A quasivariety K of algebraic systems of finite type is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.? It is known that, for every variety K of (0, 1)-lattices, if K contains a finite nondistributive simple (0, 1)-lattice, then K is Q-universal, see [3]. The opposite implication is obviously true within varieties of modular (0, 1)-lattices. This paper shows that in general the opposite implication is not true. A family (A _i : i < 2ω) of locally finite varieties of (0, 1)-lattices is exhibited each of which contains no simple non-distributive (0, 1)-lattice and each of which is Q-universal. Received July 19, 2001; accepted in final form July 11, 2002.     

On order-polynomial completeness of lattices

In this note we prove that the cardinality of an infinite order-polynomially complete lattice (if such a lattice exists) must be greater than each , where is defined such that and for . This strengthens the result in [4]. Received November 10, 1995; accepted in final form April 6, 1998.     

Some nearly Boolean orthomodular posets

Let be an orthomodular partially ordered set (``a quantum logic"). Let us say that is nearly Boolean if is set-representable and if every state on is subadditive. We first discuss conditions under which a nearly Boolean OMP must be Boolean. Then we show that in general a nearly Boolean OMP does not have to be Boolean. Moreover, we prove that an arbitrary Boolean algebra may serve as the centre of a (non-Boolean) nearly Boolean OMP.

     

Hyperidentities of some generalizations of lattices

In the paper we present bases and hyperbases of hyperidentities of some generalizations of the variety L of all lattices and the variety D of distributive lattices. We describe the form of hyperidentities of some varieties with two binary operations. Received January 22, 1997; accepted in final form January 7, 1998.     

Pappian implies Arguesian for modular lattices

We prove the claim in the title, using Hessemberg's analog theorem for projective planes, and the characterization of non-Arguesian modular lattices provided by Day and Jónsson [5,6]. Received January 22, 1999; accepted in final form May 6, 1999.