Exploring Unsharp Logic: Operators on Complemented Lattices Without Unique Complements

Okay, buckle up, because we’re diving into a pretty fascinating corner of mathematics: complemented lattices. Now, you might have encountered lattices before – they’re structures with operations like “meet” (think intersection or AND) and “join” (think union or OR), along with a bottom element (0) and a top element (1). A complemented lattice is one where every element has at least one “complement”.

But here’s where it gets interesting, and where this particular exploration really shines: we’re *not* assuming that complement is unique. Nope. In many standard setups, there’s a specific operation, often denoted by a prime (‘), that gives you *the* complement. But in the world we’re looking at right now, an element can totally have *multiple* complements. Think of it like having several ways to “cancel out” an element to get back to the extremes (0 and 1).

The Star of the Show: The Set of Complements (a⁺)

Since we don’t have a single complement, the natural thing to do is consider the set of *all* complements for a given element ‘a’. We call this set a⁺. This is our primary focus – treating a⁺ as an operator that takes an element and gives us back a whole bunch of its complements. Because our lattice is complemented, we know this set a⁺ is never empty, which is a relief!

We can even extend this idea. If you have a whole subset of elements, say ‘A’, we can define A⁺ as the set of elements that are complements to *every* element in A. This A⁺ might sometimes be empty, like L⁺ (the set of complements for the whole lattice L) is empty, but the set of complements for the empty set (∅⁺) is the whole lattice L.

We looked at some examples to get a feel for this. In the lattice N₅ (a non-modular one), the sets of complements can be a bit complex. But in modular lattices like M₃, things can behave a bit more nicely. For instance, in M₃, we noticed that for any element ‘x’, the set of complements of its set of complements (x⁺⁺) is just ‘x’ itself (if we identify singletons with the element). This isn’t true in N₅, where x⁺⁺ might be a set containing ‘x’ and other elements.

Properties of the a⁺ Operator

So, what kind of properties does this a⁺ operator have? We found some neat ones:

• For any element ‘a’, ‘a’ is always in a⁺⁺, and applying the operator three times (a⁺⁺⁺) gets you back to a⁺.
• The set of complements (a⁺), when ordered by the lattice’s relation (≤), forms what’s called an antichain. This means no two distinct elements in a⁺ are comparable (neither is less than or equal to the other). Interestingly, this property holds if and only if the lattice doesn’t contain a specific non-modular structure called N₅ (that also includes 0 and 1). Pretty cool how the structure of the complement sets tells you something about the overall lattice shape!
• The set a⁺ is also convex. If you have two complements ‘b’ and ‘c’ for ‘a’, and some element ‘d’ is between them (b ≤ d ≤ c), then ‘d’ must also be a complement of ‘a’.
• If the mapping that sends an element ‘x’ to its double complement set x⁺⁺ isn’t injective (meaning different elements can have the same x⁺⁺), then the lattice definitely doesn’t satisfy that nice x⁺⁺ ≈ x identity we saw in M₃.

We also saw how a⁺ relates to the lattice order. For elements ‘x’ and ‘y’, if x ≤ y, then the set of complements of y (y⁺) is “less than or equal to” the set of complements of x (x⁺) in a specific way (denoted ≤₁). This is kind of like an antitone property, but for sets of complements.

When x⁺⁺ = x Holds

Characterizing when x⁺⁺ ≈ x holds is a key question. We know it doesn’t always happen in non-modular lattices like N₅. So, we focused on complemented modular lattices. In this setting, we found that the identity x⁺⁺ ≈ x holds if and only if for every element ‘x’ and every element ‘y’ in x⁺⁺, there’s a complement ‘z’ of ‘y’ such that either (x ∨ y) ∧ z = 0 or (x ∧ y) ∨ z = 1. It’s a bit technical, but it gives us a handle on this property using the complement sets.

Introducing Logical Operators: → and ⊙

Inspired by work on orthomodular lattices and their “Sasaki operations” (which use the unique complement ‘x”), we decided to define two new operators using our set of complements, a⁺. These are meant to be like implication (→) and conjunction (⊙) in some kind of propositional logic.

For elements ‘a’ and ‘b’, we define:

• a → b as the join (∨) of the set a⁺ with the meet (∧) of ‘a’ and ‘b’ (a⁺ ∨ (a ∧ b)). Remember, a⁺ is a *set*, so this join is taken element-wise or in some aggregated sense (the text defines A ∨ B and A ∧ B for sets A, B).
• a ⊙ b as the meet (∧) of ‘b’ with the join (∨) of ‘a’ and the set b⁺ (b ∧ (a ∨ b⁺)).

Crucially, these operators are “unsharp”. Unlike standard logical connectives that give you a single truth value (True/False) or a single element in an algebraic structure, these operators assign a *non-empty subset* of the lattice to each pair of elements.

Properties of the → Operator (Implication)

Even though it’s “unsharp”, the → operator shares quite a few properties with implication in logics like intuitionistic logic or quantum logic. For a complemented lattice:

• a → 0 gives you the set of complements a⁺.
• 1 → a is just ‘a’.
• If a ≤ b, then a → b equals 1 (the top element).
• a → b equals 1 if and only if a ∧ b is in the set a⁺⁺.
• If ‘b’ is in the set of complements a⁺, then a → b is exactly a⁺.
• If b ≤ c, then a → b is “less than or equal to” a → c (in the ≤₁ or ≤₂ sense for sets).

And in complemented modular lattices, we see even more logic-like behavior:

• Modus Ponens: a ∧ (a → b) ≤ b. This is a fundamental rule of logic!
• Modus Tollens: If a⁺ ≤ b⁺, then (a → b) ∧ b⁺ = a⁺.
• a → (a → b) = a → b. This is like saying “if A implies B, then A implies (A implies B)” is equivalent to “A implies B”.
• If a⁺ is less than or equal to b (in the ≤₁ sense), then a → b = b.

We also found that the equivalence “a → x = 1 if and only if a ≤ x” holds if and only if ‘a’ is a minimal element in its double complement set a⁺⁺.

Properties of the ⊙ Operator (Conjunction)

The ⊙ operator also has some expected and some unique properties:

• 0 ⊙ a = a ⊙ 0 = 0.
• 1 ⊙ a = a ⊙ 1 = a.
• a ∧ b ≤ a ⊙ b ≤ b. This tells us the result is somewhere between the standard meet and ‘b’.
• It’s idempotent: x ⊙ x = x.
• If a ≤ b, then a ⊙ c is “less than or equal to” b ⊙ c.

In complemented modular lattices, we get a nice connection to the order: a ≤ b if and only if a ⊙ b = a. Also, (a ⊙ b) ⊙ b = a ⊙ b.

The Adjoint Pair

Here’s a really significant finding: in complemented modular lattices, the operators ⊙ and → form an adjoint pair. This is a big deal in algebraic logic! What it means is that for any elements a, b, and c, the following is equivalent:
a ⊙ b ≤ c if and only if a ≤ b → c
This is a fundamental relationship that links these two “unsharp” operators together in a very structured way.

Deductive Systems

Since our → operator behaves a bit like implication, we introduced the concept of deductive systems for complemented lattices. These are subsets ‘D’ of the lattice that capture a basic logical derivation rule. A subset D is a deductive system if:

• The top element 1 is in D.
• If an element ‘a’ is in D, and the set a → b is a subset of D, then ‘b’ must also be in D. This is like a set-based version of Modus Ponens: if you have ‘a’, and ‘a’ “implies” ‘b’ (in this set sense), then you have ‘b’.

The collection of all deductive systems of a lattice itself forms a complete lattice under inclusion. We explored the deductive systems of the M_n lattices and found they form a Boolean algebra.

Deductive Systems, Filters, and Equivalence Relations

We looked at how these deductive systems relate to other standard lattice concepts.

• Every deductive system is an order filter (if x is in D and x ≤ y, then y is in D).
• If, additionally, for any x, y in D, the set x → y is a subset of D, then D is a standard filter (closed under meet).
• In complemented modular lattices, things simplify: every filter is automatically a deductive system.

Finally, we touched upon the relationship between deductive systems and equivalence relations (like congruences). For a deductive system D, we defined an equivalence relation Θ(D) where (a, b) is in Θ(D) if both a → b and b → a are subsets of D. We showed that congruences on complemented modular lattices induce deductive systems. However, not all deductive systems come from congruences in this simple way.

We introduced the idea of an equivalence relation having the “Substitution Property” with respect to a⁺ and →. This property means that if two elements are related by the equivalence, their sets of complements are also related, and applying the → operator with related elements results in related sets. We found that if an equivalence relation has the Substitution Property with respect to →, it also has it for a⁺, and its kernel (the set of elements related to 1) is a deductive system.

This led us to define “compatible deductive systems” – those satisfying two extra conditions related to the → operator. We showed that these compatible deductive systems are precisely the ones that induce an equivalence relation (Θ(D)) that *does* have the Substitution Property with respect to →, and for which the kernel of Θ(D) is exactly D itself.

Wrapping Up

So, there you have it! By stepping away from the assumption of a unique complement and focusing on the *set* of complements, we uncovered a rich structure. We saw how this set operator a⁺ behaves, how it relates to the lattice’s properties (like modularity or the absence of N₅), and how it allows us to define “unsharp” logical operators → and ⊙. These operators, particularly in the modular setting, share many properties with classical and non-classical logic connectives and even form an adjoint pair. This framework also gives rise to a notion of deductive systems, which connect back to filters and equivalence relations, albeit with some nuances due to the set-valued nature of the operators. It’s a fascinating glimpse into how algebraic structures can model different forms of logic, even when things aren’t perfectly “sharp” and elements can have multiple “opposites”.

Source: Springer