Direct decompositions of orthomodular lattices

Carréga, J. C.; Chevalier, Georges; Mayet, R.

doi:10.1007/bf01188994

Cited by 8 publications

(9 citation statements)

References 12 publications

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“…In [4], the authors construct an infinite non-modular OML L such that all finite sub-OMLs of L are modular. (Actually, L is a sub-OML of the OML, C(H), of all the closed subspaces of a separate Hilbertian space H.) Let S be an exclusion system for MOL ⊂ OML; then there exists S ∈ S isomorphic to a sub-OML of L and, as S is not modular, it is infinite.…”

Section: Minimal Omls Related To An Exclusion Problemmentioning

confidence: 99%

Coverings of [MO n ] and minimal orthomodular lattices

Carréga

Greechie

2008

Algebra univers.

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If T is an orthomodular lattice (OML), we denote by [T ] the equational class generated by T . In this paper we characterize the finite OMLs T such that [T ] covers some [MOn ]. These OMLs T are the non-modular OMLs such that all proper sub-OMLs of T are modular. An OML satisfying that last property is called minimal. There exist infinitely many minimal OMLs provided by quadratic spaces over finite fields. We describe them and give a new way to represent their Greechie diagrams in two separate parts. Other methods to obtain finite minimal OMLs are given.

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Section: Minimal Omls Related To An Exclusion Problemmentioning

confidence: 99%

Coverings of [MO n ] and minimal orthomodular lattices

Carréga

Greechie

2008

Algebra univers.

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“…[12]), the finite orthocomplemented modular lattice with 2n atoms, in which any block possesses exactly two atoms. Many minimal OMLs have been obtained and studied ( [3], [4], [7]), and the existence of an infinite minimal OML has been proved ( [5]). We will see that an easy consequence of our main Theorem is that there exist infinitely many finite minimal OMLs.…”

Section: Vol 52 2004mentioning

confidence: 99%

“…(b) It was already proved in [5] that there exists an infinite minimal OML. Observe that the one we have obtained by our Corollary, when K = Q, is of special interest since all its proper subOML are modular.…”

Section: Vol 52 2004mentioning

confidence: 99%

Orthomodular lattices from 3-dimensional quadratic spaces

Carréga

Mayet

2004

Algebra univers.

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If E is a vector space over a field K, then any regular symmetric bilinear form ϕ on E induces a polarity M → M ⊥ 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 M ⊥ 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, ϕ). We show that if K is a proper subfield of K, with K = F 2 , and E a 3-dimensional K -subspace of E such that the restriction of ϕ to E × E is, up to multiplicative constant, a bilinear form ϕ on the K -space E , then T (E , ϕ ) is isomorphic to an irreducible 3-homogeneous proper subalgebra of T (E, ϕ). 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, ϕ) 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. PreliminariesLet us recall some basic notions of the theory of orthomodular lattices (abbreviated OMLs). For more details the reader may consult [12] or [14].An OML is an algebra (L, 0, 1, ∨, ∧, ⊥ ), where (L, 0, 1, ∨, ∧) is a bounded lattice, and ⊥ is an antitone (i.e., such that a ≤ b implies b ⊥ ≤ a ⊥ ) and involutive unary operation so that a ∨ a ⊥ = 1 (which implies a ∧ a ⊥ = 0), and the orthomodular law a∨b = a∨(a ⊥ ∧(a∨b)) holds true. The class of OMLs is a variety which contains the variety of Boolean algebras. Moreover, if H is any Hilbert space, then the lattice L(H) of closed subspaces of H, endowed with the usual operation M → M ⊥ , is an OML. If H is finite-dimensional, L(H) is the orthocomplemented modular lattice of all subspaces of H. A block of an OML L is a maximal Boolean subalgebra of L. When L is finite, this algebra can be represented by its Greechie diagram ([12]), a hypergraph, whose vertices correspond to the atoms and whose edges represent the blocks of L. Here we will use such diagrams only in the simplest case described in [11].

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“…The work of Loomis and Maeda led to a theory of decomposition of OMLs into direct summands of various special types, presented first in a classic paper by A. Ramsay [27, §3], and later in more generality by G. Kalmbach [19, §7]. A complementary development of the Ramsay-Kalmbach direct decomposition theory, more in the spirit of universal algebra, but still in the context of OMLs, was subsequently published by J. Carrega, G. Chevalier, and R. Mayet [2].…”

Section: Introductionmentioning

confidence: 95%

Centrally orthocomplete effect algebras

Foulis

Pulmannová

2010

Algebra Univers.

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Motivated by the theory of Loomis dimension lattices, we generalize the notion of a hull mapping to an arbitrary effect algebra (EA). Using hull mappings, we identify certain special types of elements in an EA, including generalizations of the invariant elements and of the simple elements in a dimension lattice. We introduce and study a new class of effect algebras, called centrally orthocomplete effect algebras (COEAs), satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. We show that COEAs admit a central cover mapping and we develop the basic theory of direct decomposition of COEAs.

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Direct decompositions of orthomodular lattices

Cited by 8 publications

References 12 publications

Coverings of [MO n ] and minimal orthomodular lattices

Coverings of [MO n ] and minimal orthomodular lattices

Orthomodular lattices from 3-dimensional quadratic spaces

Centrally orthocomplete effect algebras

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