An orthomodular lattice

An orthomodular lattice is constructed by taking a homomorphic image of the ortholattice obtained from a certain orthogonality relation on an infinite set.Presented by B. Jónsson.This research was supported by N.S.E.R.C. grant 0041702.     

Lattice uniformities and modular functions on orthomodular lattices

It is shown that the lattice of all exhaustive lattice uniformities on an orthomodular latticeL is isomorphic to the centre of a natural completion (of a quotient) ofL, and is thus a complete Boolean algebra. This is applied to prove a decomposition theorem for exhaustive modular functions on orthomodular lattices, which generalizes Traynor's decomposition theorem [14].     

Greechie diagrams of orthomodular partial algebras

Greechie diagrams are well known graphical representations of orthomodular partial algebras, orthomodular posets and orthomodular lattices. For each hypergraph D a partial algebra ⟦D⟧ = (A; ⊕, ′, 0) of type (2,1,0) can be defined. A Greechie diagram can be seen as a special hypergraph: different points of the hypergraph have different interpretations in the corresponding partial algebra ⟦D⟧, and each line in the hypergraph has a maximal Boolean subalgebra as interpretation, in which the points are the atoms. This paper gives some generalisations of the characterisations in [K83] and [D84] of diagrams which represent orthomodular partial algebras (= OMAs), and we give an algorithm how to check whether a given hypergraph D is an OMA-diagram whose maximal Boolean subalgebras are induced by the lines of the hypergraph. Received July 22, 2004; accepted in final form February 1, 2007.     

Irreducible orthomodular lattices which are simple

It is well known that for a chain finite orthomodular lattice, all congruences are factor congruences, so any directly irreducible chain finite orthomodular lattice is simple. In this paper it is shown that the notions of directly irreducible and simple coincide in any variety generated by a set of orthomodular lattices that has a uniform finite upper bound on the lengths of their chains. The prototypical example of such a variety is any variety generated by a set ofn dimensional orthocomplemented projective geometries.Presented by B. Jónsson.Supported by a grant from NSERC.     

Completions of orthomodular lattices

If K is a variety of orthomodular lattices generated by a finite orthomodular lattice the MacNeille completion of every algebra in K again belongs to K.     

Different orthomodular orthocomplementations on a lattice

A new method for constructing separable Hilbert lattices is described. Examples of lattices are given that admit infinitely many different orthomodular orthocomplementations.     

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. Received June 28, 1995; accepted in final form June 27, 1996.     

A characterization of state spaces of orthomodular lattices

It is shown that every rational polytope is affinely equivalent to the set of all states of a finite orthomodular lattice, and that every compact convex subset of a locally convex topological vector space is affinely homeomorphic to the set of all states of an orthomodular lattice.     

Completeness of the bounded Boolean powers of orthomodular lattices

Completeness of orthomodular lattices is frequently assumed in axiomatic treatments of the foundations of quantum mechanics. We show that the bounded Boolean power of an orthomodular lattice by a Boolean algebra is complete if and only if one of these is complete and the other is finite.Dedicated to the memory of Alan DayPresented by J. Sichler.     

Orthmodular lattices whose MacNeille completions are not orthomodular

The only known example of an orthomodular lattice (abbreviated: OML) whose MacNeille completion is not an OML has been noted independently by several authors, see Adams [1], and is based on a theorem of Ameniya and Araki [2]. This theorem states that for an inner product space V, if we consider the ortholattice ?(V,⊥) = {A \( \subseteq \) V: A = A ^⊥⊥} where A ^⊥ is the set of elements orthogonal to A, then ?(V,⊥) is an OML if and only if V is complete. Taking the orthomodular lattice L of finite or confinite dimensional subspaces of an incomplete inner product space V, the ortholattice ?(V,⊥) is a MacNeille completion of L which is not orthomodular. This does not answer the longstanding question Can every OML be embedded into a complete OML? as L can be embedded into the complete OML ?(V,⊥), where V is the completion of the inner product space V. Although the power of the Ameniya-Araki theorem makes the preceding example elegant to present, the ability to picture the situation is lost. In this paper, I present a simpler method to construct OMLs whose Macneille completions are not orthomodular. No use is made of the Ameniya-Araki theorem. Instead, this method is based on a construction introduced by Kalmbach [7] in which the Boolean algebras generated by the chains of a lattice are glued together to form an OML. A simple method to complete these OMLs is also given. The final section of this paper briefly covers some elementary properties of the Kalmbach construction. I have included this section because I feel that this construction may be quite useful for many purposes and virtually no literature has been written on it.     

The free Banach lattice generated by a Banach space

The free Banach lattice over a Banach space is introduced and analyzed. This generalizes the concept of free Banach lattice over a set of generators, and allows us to study the Nakano property and the density character of non-degenerate intervals on these spaces, answering some recent questions of B. de Pagter and A.W. Wickstead. Moreover, an example of a Banach lattice which is weakly compactly generated as a lattice but not as a Banach space is exhibited, thus answering a question of J. Diestel.     

The lattice structure of pseudo-random vectors generated by matrix generators

In many areas of science the problems treated by Monte-Carlo simulations become more and more complex and more extensive. Because of that generators like linear congruential matrix generators are needed which produce enormously many pseudo-random numbers. In order to assess stochastical properties of the generated pseudo-random vectors the lattice structure of these matrix generators is studied here.     

On a strong property of the weak subalgebra lattice

In the present paper, we apply results from [Pió1] to prove that for an arbitrary total and locally finite unary algebra A of finite unary type K, its weak subalgebra lattice uniquely determines its strong subalgebra lattice (recall that in the case of total algebras the strong subalgebra lattice is the well-known lattice of all (total) subalgebras). More precisely, we prove that for every unary partial algebra B of the same unary type K, if weak subalgebra lattices of A and B are isomorphic (with A as above), then the strong subalgebra lattices of A and B are isomorphic, and moreover B is also total and locally finite. At the end of this paper we also show the necessity of all the three conditions for A. Received September 5, 1997; accepted in final form October 7, 1998.     

On lattice profile of the elliptic curve linear congruential generators

Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter and Winterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions. We also use a result of Brandstätter and Winterhof on the linear complexity profile related to the correlation measure of order $k$ to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG.     

Congruence kernels of orthomodular implication algebras

Abstracting from certain properties of the implication operation in Boolean algebras leads to so-called orthomodular implication algebras. These are in a natural one-to-one correspondence with families of orthomodular lattices. It is proved that congruence kernels of orthomodular implication algebras are in a natural one-to-one correspondence with families of compatible p-filters on the corresponding orthomodular lattices.     

An efficient lattice reduction method for F2-linear pseudorandom number generators using Mulders and Storjohann algorithm

Recent simulations often use highly parallel machines with many processors, and they need many pseudorandom number generators with distinct parameter sets, and hence we need an effective fast assessment of the generator with a given parameter set. Linear generators over the two-element field are good candidates, because of the powerful assessment via their dimensions of equidistribution. Some efficient algorithms to compute these dimensions use reduced bases of lattices associated with the generator. In this article, we use a fast lattice reduction algorithm by Mulders and Storjohann instead of Schmidt’s algorithm, and show that the order of computational complexity is lessened. Experiments show an improvement in the speed by a factor of three. We also report that just using a sparsest initial state (i.e., consisting of all 0 bits except one) significantly accelerates the lattice computation, in the case of Mersenne Twister generators.     

Modal‐type orthomodular logic

In this paper we enrich the orthomodular structure by adding a modal operator, following a physical motivation. A logical system is developed, obtaining algebraic completeness and completeness with respect to a Kripkestyle semantic founded on Baer*‐semigroups as in [22] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Boolean topological orthomodular poset

Wilce introduced the notion of a topological orthomodular poset and proved any compact topological orthomodular poset whose underlying orthomodular poset is a Boolean algebra is a topological Boolean algebra in the usual sense. Wilce asked whether the compactness assumption was necessary for this result. We provide an example to show the compactness assumption is necessary.     

Projections in a Synaptic Algebra

A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice, and give several sufficient conditions for modularity of the projection lattice.