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.     

Decomposition of ℓ-group-valued measures

We deal with decomposition theorems for modular measures µ: L → G defined on a D-lattice with values in a Dedekind complete ?-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete ?-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for ?-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.     

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.     

Subalgebras of Orthomodular Lattices

Sachs (Can J Math 14:451–460, 1962) showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as well as by its poset of Boolean subalgebras BSub(L). The domain BSub(L) has recently found use in an approach to the foundations of quantum mechanics initiated by Butterfield and Isham (Int J Theor Phys 37(11):2669–2733, 1998, Int J Theor Phys 38(3):827–859, 1999), at least in the case where L is the orthomodular lattice of projections of a Hilbert space, or von Neumann algebra. The results here may add some additional perspective to this line of work.     

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.     

Counting formulas for glued lattices

A tolerance relation of a lattice L, i.e., a reflexive and symmetric relation of L which is compatible with join and meet, is called glued if covering blocks of have nonempty intersection. For a lattice L with a glued tolerance relation we prove a formula counting the number of elements of L with exactly k lower (upper) covers. Moreover, we prove similar formulas for incidence structures and graphs and we give a new proof of Dilworth's covering theorem.     

On the number of join-irreducibles in a congruence representation of a finite distributive lattice

For a finite lattice L, let $ \trianglelefteq_L $ denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form $ \trianglelefteq_L $, as follows: Theorem. Let $ \trianglelefteq $ be a quasi-ordering on a finite set P. Then the following conditions are equivalent:(i) There exists a finite lattice L such that $ \langle J(L), \trianglelefteq_L $ is isomorphic to the quasi-ordered set $ \langle P, \trianglelefteq \rangle $.(ii) $ |\{x\in P|p \trianglelefteq x\}| \neq 2 $, for any $ p \in P $.For a finite lattice L, let $ \mathrm{je}(L) = |J(L)|-|J(\mathrm{Con} L)| $ where Con L is the congruence lattice of L. It is well-known that the inequality $ \mathrm{je}(L) \geq 0 $ holds. For a finite distributive lattice D, let us define the join- excess function:$ \mathrm{JE}(D) =\mathrm{min(je} (L) | \mathrm{Con} L \cong D). $We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that $ \mathrm{JE}(D) \leq (2/3)| \mathrm{J}(D)|$ , for any finite distributive lattice D; the constant 2/3 is best possible.A special case of this formula gives a characterization of congruence lattices of finite lower bounded lattices.Dedicated to the memory of Gian-Carlo Rota     

Comparability graphs of lattices

A theorem of N. Terai and T. Hibi for finite distributive lattices and a theorem of Hibi for finite modular lattices (suggested by R.P. Stanley) are equivalent to the following: if a finite distributive or modular lattice of rank d contains a complemented rank 3 interval, then the lattice is (d+1)-connected.In this paper, the following generalization is proved: Let L be a (finite or infinite) semimodular lattice of rank d that is not a chain (d∈N₀). Then the comparability graph of L is (d+1)-connected if and only if L has no simplicial elements, where z∈L is simplicial if the elements comparable to z form a chain.     

On the Lebesgue decomposition for non-additive functions

A Lebesgue decomposition theorem for non-additive functions, acting on a \(\sigma \) -complete orthomodular lattice and taking values in Hausdorff uniform spaces, is established. No algebraic structure is required on target spaces. The Boolean case is also investigated.     

Convex partitions of a finite lattice

The purpose of this paper is to introduce the lattice of convex partitions for a lattice L. Then we will show some properties of this lattice. Finally, we will show that if the convex partition lattice of L is finite and modular if and only if L is a finite chain. Presented by R. McKenzie. Received December 16, 2004; accepted in final form March 7, 2006.     

Orthomodular semilattices

We present a simple equational characterization of (meet) semilattices with 0 where for each element p the interval [0,p] is an orthomodular lattice or an ortholattice possibly satisfying the compatibility condition.     

A note on “small representations of finite distributive lattices as congruence lattices”

G. Grätzer, H. Lakser, and E. T. Schmidt proved that every distributive lattice with n join-irreducible elements can be represented as the congruence lattice of a small lattice L, that is, a lattice L with O(n ² ) elements. G. Grätzer, I. Rival, and N. Zaguia proved that, for any <2, O(n ² ) can not be improved to O(n ). In this note we show that the theorem about small representation can be improved further to get a more delicate result.     

The convexity lattice of a poset

The authors investigate the lattice Co(P) of convex subsets of a general partially ordered set P. In particular, they determine the conditions under which Co(P) and Co(Q) are isomorphic; and give necessary and sufficient conditions on a lattice L so that L is isomorphic to Co(P) for some P.     

Generalizing semidistributivity

A latticeL is called congruence normal if it can be generated by doubling of convex sets starting with the one-element lattice. In the special case of intervals, the lattice is called bounded. It has been proven thatL is bounded if and only ifL is congruence normal and semidistributive.In this paper we study the connection between certain classes of convex sets and generalized semidistributive laws. These so-called doubling classes are pseudovarieties which can be described by implications as well as by forbiden substructures. In the end, we examine the structure of the lattice of all doubling classes.     

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeL _n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L _n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.     

Automatic boundedness of quantum measures

LetG be a commutative Hausdorff topological group. Letm be aG-valued, completely additive measure on a complete orthomodular posetL. It is shown, among other results, that when the centre ofL is non-atomic thenm must be strictly bounded. WhenL is specialised to being the lattice of projections in a von Neumann algebra this extends some results known for real valued measures. The first author was partially supported by GNAFA and by the project Analisi Real of MURST.     

Congruence lattices of function lattices

Thefunction lattice L ^P is the lattice of all isotone maps from a posetP into a latticeL.D. Duffus, B. Jónsson, and I. Rival proved in 1978 that for afinite poset P, the congruence lattice ofL ^P is a direct power of the congruence lattice ofL; the exponent is |P|.This result fails for infiniteP. However, utilizing a generalization of theL ^P construction, theL[D] construction (the extension ofL byD, whereD is a bounded distributive lattice), the second author proved in 1979 that ConL[D] is isomorphic to (ConL) [ConD] for afinite lattice L.In this paper we prove that the isomorphism ConL[D](ConL)[ConD] holds for a latticeL and a bounded distributive latticeD iff either ConL orD is finite.The research of the first author was supported by the NSERC of Canada.The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

Reducible classes of finite lattices

In this paper we study a notion of reducibility in finite lattices. An element x of a (finite) lattice L satisfying certain properties is deletable if L-x is a lattice satisfying the same properties. A class of lattices is reducible if each lattice of this class admits (at least) one deletable element (equivalently if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step). First we characterize the deletable elements in a pseudocomplemented lattice what allows to prove that the class of pseudocomplemented lattices is reducible. Then we characterize the deletable elements in semimodular, modular and distributive lattices what allows to prove that the classes of semimodular and locally distributive lattices are reducible. In conclusion the notion of reducibility for a class of lattices is compared with some other notions like the notion of order variety.     

An Algebraic Version of the Cantor-Bernstein-Schröder Theorem

The Cantor-Bernstein-Schröder theorem of the set theory was generalized by Sikorski and Tarski to -complete boolean algebras, and recently by several authors to other algebraic structures. In this paper we expose an abstract version which is applicable to algebras with an underlying lattice structure and such that the central elements of this lattice determine a direct decomposition of the algebra. Necessary and sufficient conditions for the validity of the Cantor-Bernstein-Schröder theorem for these algebras are given. These results are applied to obtain versions of the Cantor-Bernstein-Schröder theorem for -complete orthomodular lattices, Stone algebras, BL-algebras, MV-algebras, pseudo MV-algebras, ukasiewicz and Post algebras of order n.