Weakly Orthomodular and Dually Weakly Orthomodular Lattices

We introduce so-called weakly orthomodular and dually weakly orthomodular lattices which are lattices with a unary operation satisfying formally the orthomodular law or its dual although neither boundedness nor complementation is assumed. It turns out that lattices being both weakly orthomodular and dually weakly orthomodular are in fact complemented but the complementation need not be neither antitone nor an involution. Moreover, every modular lattice with complementation is both weakly orthomodular and dually weakly orthomodular. The class of weakly orthomodular lattices and the class of dually weakly orthomodular lattices form varieties which are arithmetical and congruence regular. Connections to left residuated lattices are presented and commuting elements are introduced. Using commuting elements, we define a center of such a (dually) weakly orthomodular lattice and we provide conditions under which such lattices can be represented as a non-trivial direct product.


Introduction
Orthomodular lattices play an important role both in algebra and quantum mechanics. They were introduced independently by Birkhoff and von Neumann [3] and Husimi [9] as an algebraic semantic of the logic of quantum mechanics. Let us recall that an algebra (L, ∨, ∧, , 0, 1) of type (2, 2, 1, 0, 0) is called an ortholattice if (L, ∨, ∧, 0, 1) is a bounded lattice and satisfies the following conditions: • for all x, y ∈ L, x ≤ y implies y ≤ x ( is antitone), • (x ) ≈ x ( is an involution).
Since these conditions can be rewritten in the form of identities as follows the class of orthomodular lattices forms a variety. This variety is arithmetical, congruence regular and congruence uniform (cf. [1,6] and [10]). Since ortholattices do not have these important properties except congruence distributivity, it is apparent that the strongness of orthomodular lattices is due to the orthomodular law (1), respectively (2). This motivated us to investigate algebras (L, ∨, ∧, ) such that (L, ∨, ∧) is a lattice and is a unary operation satisfying (1) or (2) or both. It is our goal to show that the conditions (1) and/or (2) are so strong that we can prove a lot of results valid for orthomodular lattices (see e.g. [5,10]) also in our more general setting. So one aim of the present paper is to demonstrate the strongness of the orthomodular law independently from the fact if the corresponding unary operation is antitone or an involution.

Basic Properties
Let L = (L, ∨, ∧, 0, 1) be a bounded lattice and a unary operation on L. We say that (L, ∨, ∧, , 0, 1) is a lattice with complementation if x ∨ x ≈ 1 and x ∧ x ≈ 0. In this case is called a complementation. In general, we do not ask that the complementation is either antitone or an involution. Hence, we cannot use the De Morgan laws in general. Now, we show that conditions (1) and (2) are independent even in the case that the underlying lattice is a lattice with complementation.
Since in the lattice with the following Hasse diagram (see e.g. [4]) We obtain the following Corollary 2.5 Every lattice L = (L, ∨, ∧, ) being both weakly orthomodular and dually weakly orthomodular is bounded, is a complementation, 0 ≈ 1 and 1 ≈ 0 where 0 and 1 denote the smallest respectively greatest element of L.
Moreover, conditions (1) and (2) remain valid in arbitrary intervals provided the complementation is defined suitably.  It should be remarked that if L is modular or orthomodular then * and + coincide. Contrary to the fact that every interval of a weakly orthomodular lattice is weakly orthomodular, that does not mean that certain properties of the corresponding complementation remain valid. For example, the complementation of the first weakly orthomodular lattice from Example 2.3 is an involution whereas the corresponding complementation in the interval [d, 1] is not an involution as the following calculation shows: This can be seen as follows: If L satisfies (1) and (2) then both sides of (3) are equal to x. If, conversely, L satisfies (3) then, because of we obtain (1) and (2). Hence, the class of lattices being both weakly orthomodular and dually weakly orthomodular forms a variety. It is clear that such a lattice is orthomodular if and only if the De Morgan laws In what follows we describe a construction of lattices being both weakly orthomodular and dually weakly orthomodular which need not be modular. For this, we recall the concept of a horizontal sum. ∨ and ∧ and a unary operation on L as follows: It is easy to see that the horizontal sum of lattices being both weakly orthomodular and dually weakly orthomodular is again a lattice of this type. The horizontal sum of the lattices from Example 2.3 is a non-modular lattice being both weakly orthomodular and dually weakly orthomodular whose complementation is neither antitone nor an involution.
Let P = (P , ≤, 0) be a poset with smallest element 0. Recall that the neighbors of 0 are called atoms, P is called atomic if every element of P \ {0} lies over some atom and P is called atomistic if every element of P is a join of atoms.
For weakly orthomodular lattices with complementation we can prove the following important result whose proof is adopted from that for orthomodular lattices.

Theorem 2.9 Every atomic weakly orthomodular lattice with complementation is atomistic.
Proof Let L = (L, ∨, ∧, , 0, 1) be an atomic lattice with complementation satisfying (1), a, b ∈ L and A denote the set of all atoms x of L with x ≤ a and assume x ≤ b for all x ∈ A. Suppose, a ∧b < a. Because of a = (a ∧b)∨(a ∧(a ∧b) ) we have a ∧(a ∧b) = 0. Since L is atomic there exists an atom c of L with c ≤ a ∧ (a ∧ b) . This shows c ≤ a, i.e.
This shows a = A.

Commuting Elements
It is well known that commuting elements play an important role in orthomodular lattices. A similar concept can be introduced also for weakly orthomodular lattices but with a slight modification caused by the fact that the De Morgan laws cannot be used here. Hence, we define In this case we say that a commutes with b. It should be noticed that if L is orthomodular then (4) and (5)  In every lattice being both weakly orthomodular and dually weakly orthomodular we have for arbitrary elements x and y: we have ∨ (a ∧ b ) and One can easily see that if L 1 = (L 1 , ∨, ∧, 0, 1), L 2 = (L 2 , ∨, ∧, 0, 1) are lattices being both weakly orthomodular and dually weakly orthomodular, L := L 1 × L 2 and a ∈ {(0, 1), (1, 0)} then a ∈ C(L) and (x ∧a) = x ∨a and (x ∧a ) = x ∨a for all x ∈ L. Hence, these elements can be used for directly decomposing L as shown in the following theorem. Apparently, they satisfy rather strong properties.  (c, d). Hence f and g are mutually inverse bijections between L and [0, a] × [0, a ]. Since they are obviously order-preserving, they are order isomorphisms and therefore lattice isomorphisms. Finally, we have a) a , (b ∧ a )  The lattice from Example 3.4 coincides with that of Example 2.3, but the complementation is defined in a different way. One can easily check that the element e in Example 2.3 does not satisfy the assumptions of Theorem 3.3 and hence this theorem cannot be applied to the dually weakly orthomodular lattice of Example 2.3.
In the following, for a subset M of a lattice, M denotes the sublattice generated by M.
The well-known Foulis-Holland-Theorem for orthomodular lattices says that if an element of an orthomodular lattice commutes with two other elements then the lattice generated by all three elements is distributive. Using Theorem 3.3 we obtain a similar result for dually weakly orthomodular lattices.  , 0), (b ∧ a, b ∧ a ), (c ∧ a, c ∧ a )} which is a sublattice of ∧ a, c ∧ a} × {0, b ∧ a , c ∧ a } = {a, b ∧ a, c ∧ a, b ∧ c ∧ a, (b ∧ a) ∨ (c ∧ a)}  ×{0, b ∧ a , c ∧ a , b ∧ c ∧ a , (b ∧ a ) ∨ (c ∧ a )}.
Since the last two lattices have at most one pair of incomparable elements, they cannot contain M 3 nor N 5 as a sublattice and hence they are distributive. This shows that also {a, b, c} is distributive.

Example 3.6
Since the element e of Example 3.4 satisfies the assumption of Theorem 3.5 we have that {e, x, y} is distributive for all elements x, y of this lattice.
The following theorem generalizes the corresponding theorem for orthomodular lattices. The proof follows the same lines as that of the corresponding theorem for orthomodular lattices.

Left Residuated Lattices
Orthomodular lattices form an algebraic semantic for the logic of quantum mechanics (see e.g. [3] and [9]). In particular, MV-algebras are residuated lattices which form an algebraic semantic for many-valued Łukasiewicz logics; moreover, divisible and prelinear residuated lattices form a semantic for Hajek's BL-logics etc. All of these logics are so-called substructural logics. We have already developed several attempts to connect various semantics, see e.g. [7] and [8]. Hence, the question arises if a similar approach is possible for lattices being both weakly orthomodular and dually weakly orthomodular. At first, we define the main concept of this section.
The algebra R is called idempotent if the identity x x ≈ x holds.
A connection between lattices being both weakly orthomodular and dually weakly orthomodular with left residuated lattices can be established if the operations and → are defined as shown in the following theorem:

) is a lattice being both weakly orthomodular and dually weakly orthomodular and satisfying the identity (x ) ≈ x and one defines
x y := (x ∨ y ) ∧ y and for all x, y ∈ L then (L, ∨, ∧, , →, 0, 1) is an idempotent left residuated lattice satisfying the identity x (x ∨ y) ≈ x.
Proof Let a, b, c ∈ L. Then we have If the left residuated lattice is modular, we can prove the converse.
Proof If a ∈ R then 1 ≤ a → 1 = a ∨ (a ∧ 1) = a ∨ a since 1 a ≤ 1 and a ∧ a = (0 ∨ a ) ∧ a = 0 a ≤ 0 since 0 ≤ a → 0. This shows that is a complementation. According to Lemma 2.2 we are done.
If modularity is not assumed we must ask for stronger assumptions.

Congruence Properties
Congruence properties form an important tool for revealing essential features of algebras. In particular, this is true for varieties of algebras. It is well-known (see e.g. [1] and [10]) that the variety of orthomodular lattices is arithmetical and congruence regular. In what follows we show that the same is true also for the variety of weakly orthomodular lattices. For the reader's convenience, we firstly recall these properties.
• arithmetical if it is both permutable and distributive and • congruence regular if for all A = (A, F ) ∈ V, a ∈ A and , ∈ Con A, [a] = [a] implies = .
For the following proposition, see e.g. the monograph [6].
and t 1 (x, y, z) = t 2 (x.y, z) = z implies u = u ∧ (u ∨ z) = u ∧ (u ∧ z) = 0 where u := (x ∨ y) ∧ (x ∧ y) and hence According to Theorem 5.3, every weakly orthomdular lattice with complementation is congruence regular and thus each of its congruences is determined by any single class. This dependence is described in the following Proposition 5.4 Let L = (L, ∨, ∧, , 0, 1) be a weakly orthomodular lattice with complementation, a, b, c ∈ L and ∈ Con L. Then the following are equivalent: We have (ii) ⇒ (i): That (iii) and (iv) are equivalent to (i) follows from the equivalence of (i) and (ii) in case c = 0 respectively in case c = 1.