The completion by cuts of an orthocomplemented modular lattice

Adams, D. H.

doi:10.1017/s0004972700041526

Cited by 19 publications

(10 citation statements)

References 2 publications

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“…(1) Obvious. (2) Every ortholattice with more than one element contains a two element Boolean algebra as a subalgebra, and the two element Boolean algebra generates the variety of Boolean algebras.…”

Section: Proposition 71 (1) the Trivial Variety Is The Smallest Varimentioning

confidence: 99%

Algebraic Aspects of Orthomodular Lattices

Bruns

Harding

2000

Current Research in Operational Quantum Logic

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In this survey article we try to give an up-to-date account of certain aspects of the theory of ortholattices (abbreviated OLs), orthomodular lattices (abbreviated OMLs) and modular ortholattices (abbreviated MOLs), not hiding our own research interests. Since most of the questions we deal with have their origin in Universal Algebra, we start with a section discussing the basic concepts and results of Universal Algebra without proofs. In the next three sections we discuss, mostly with proofs, the basic results and standard techniques of the theory of OMLs. In the remaining five sections we work our way to the border of present day research, with no or only sketchy proofs. Section 5 deals with products and subdirect products, section 6 with free structures and section 7 with classes of OLs defined by equations. In section 8 we discuss embeddings of OLs into complete ones. The last section deals with questions originating in Category Theory, mainly amalgamation, epimorphisms and monomorphisms. The later sections of this paper contain an abundance of open problems. We hope that this will initiate further research.

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Section: Proposition 71 (1) the Trivial Variety Is The Smallest Varimentioning

confidence: 99%

Algebraic Aspects of Orthomodular Lattices

Bruns

Harding

2000

Current Research in Operational Quantum Logic

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“…an orthoposet being a lattice) the orthocomplementation on which extends the orthocomplementation of Ρ (see [4, p. 255-256]). In contrast to this fact the MacNeille completion of even a modular ortholattice need not be orthomodular (see [1]). In quantum probability theory an important question seems to be orthocompleteness of orthomodular posets.…”

Section: The Macneille Completion and Orthocompletionmentioning

confidence: 99%

On Completions of Orthoposets

Riečanová¹

1994

Demonstratio Mathematica

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“…Then for every «i, «2 € H and every a £ A Kl , b £ A KJ we have a ^ b 1 -. Now let us denote F = { x E L | x = 0 or x is a join of a finite set of atoms in 1 } and L ={y £ L \ either y E F or y 1 (in L). Using Theorems 3.1 and 3.3 we obtain a Kn ->0 (in L).…”

Section: Order Topological O M L S In the Language Of Filters; Countementioning

confidence: 99%

“…A well-known fact is that the MacNeille completion of an orthomodular lattice L need not be orthomodular, even if L is a modular ortholattice ( [1,5] and others). Little is known about completions of orthomodular lattices (abbreviated OMLs).…”

Section: Introductionmentioning

confidence: 99%