Orthomodular lattice

A lattice with a zero $0$ and a one $1$ in which for any element $a$ there is an orthocomplement $a^\perp$, i.e. an element such that

1) $ a \vee a^\perp = 1 $, $a \wedge a^\perp = 0$, $(a^\perp)^\perp = a$;

2) $ a \le b \Rightarrow a^\perp \ge b^\perp$;

and such that the orthomodular law:

3) $ a \le b \Rightarrow b = a \vee (b \wedge a^\perp)$

is satisfied.

In an orthomodular lattice one studies distributivity, perspectivity, irreducibility, modularity of pairs, properties of the centre and of ideals, the commutator, solvability, and applications in the logic of quantum mechanics (see [1], [2]).

If $\mathfrak{A}$ is an arbitrary von Neumann algebra, then the set $P(\mathfrak{A})$ of all its projections is a complete orthomodular lattice. In these conditions, if $\mathfrak{A}$ is a factor, then a dimension function can be defined on the set $P(\mathfrak{A})$. Dependent on the set of values of this function, the factors are divided into the types $\mathrm{I}_n$, $\mathrm{I}_\infty$, $\mathrm{II}_1$, $\mathrm{II}_\infty$, $\mathrm{III}$ (the Murray–von Neumann classification, see [4]). It has been established that lattices of projections of factors of the type $\mathrm{I}_n$ and $\mathrm{II}_1$ are continuous geometries, i.e. complete complemented modular lattices (cf. Lattice with complements; Modular lattice; Complete lattice) satisfying the following two continuity axioms:

1) $b \wedge (\vee_{\alpha \in D} a_\alpha) = \vee_{\alpha \in D} b \wedge a_\alpha$ for any directed set of indices $D$ and any set of elements $\{ a_\alpha : \alpha \in D \}$ such that $\alpha \prec \alpha'$ implies $a_\alpha \le a_{\alpha'}$;

2) the condition dual to 1).

The problem of constructing an abstract dimension theory within the framework of such a class of lattices, which would also include, apart from modular lattices of projections of factors of the types $\mathrm{I}_n$ and $\mathrm{II}_1$, non-modular lattices of projections of factors of the remaining types, has arisen. The existence of a dimension function of a complete orthomodular lattice with an equivalence relation which satisfies certain supplementary conditions has been proved. This class of lattices includes both lattices of projections of factors and continuous geometries.

Orthomodular lattices, which are a natural generalization of lattices of projections of factors, also constitute an essentially broader class, in that many properties of lattices of projections are not valid for arbitrary orthomodular lattices. In the same way as continuous geometries are coordinatized by regular rings (see [1]), orthomodular lattices can be coordinatized by Baer ${*}$-semi-groups. If a complete orthomodular lattice is modular, it is continuous (see [7]). There exists an orthocomplemented modular lattice whose completion by sections is not orthomodular (whereas the completion by sections of a semi-modular orthocomplemented lattice is semi-modular and the lattice of projections of a von Neumann algebra is semi-modular).

References

[1]	L.A. Skornyakov, "Complemented modular lattices and regular rings" , Oliver & Boyd (1964) (Translated from Russian)
[2]	Itogi Nauki. Algebra. Topol. Geom. 1968 , Moscow (1970) (In Russian)
[3]	T.S. Fofanova, "General theory of lattices" , Ordered sets and lattices , 3 , Saratov (1975) pp. 22–40 (In Russian)
[4]	F. Murray, J. von Neumann, "On rings of operators" Ann. of Math. , 37 : 1 (1936) pp. 116–229
[5]	L.H. Loomis, "The lattice theoretic background of the dimension theory of operator algebras" Mem. Amer. Math. Soc. , 18 (1955) pp. 1–36
[6]	S. Maeda, "Dimension functions on certain general lattices" J. Sci. Hiroshima Univ. Ser. A , 19 : 2 (1955) pp. 211–237
[7]	I. Kaplansky, "Any orthocomplemented complete modular lattice is a continuous geometry" Ann. of Math. , 61 : 3 (1955) pp. 524–541

Comments

For "completion by sections" see Completion, MacNeille (of a partially ordered set).

References

[a1]	T.S. Blyth, M.F. Janowitz, "Residuation theory" , Pergamon (1972)
[a2]	G. Kalmbach, "Orthomodular lattices" , Acad. Press (1983)
[a3]	G. Kalmbach, "Measures and Hilbert lattices" , World Sci. (1986)
[a4]	L. Beran, "Orthomodular lattices" , Reidel (1985) pp. 4ff