12Fuzzy Algebras

which, when spelled out, gives

x y = x (x y)

(proof is left as an exercise).

We noted in Chapter 11 that the Laws of Excluded Middle and Noncontradiction hold in FuzzyL when they are expressed using bold disjunction and conjunction but fail when expressed with the weak connectives. The bold version of the Law of Excluded Middle appears in MV-algebras as the first complementation condition x x = unit, while the bold version of the Law of Noncontradiction appears as the equation (x x ) = zero, which is derivable by complementing both sides of the second complementation condition and then replacing zero with unit by virtue of condition viii, the duality of zero and unit.

In fact, we have a result relating FuzzyL and MV-algebras that is analogous to Result 4.3 of Chapter 4 relating classical propositional logic and Boolean algebras. First, some definitions. When we interpret formulas of propositional logic in an MV-algebra by assigning a member of the algebra’s domain to each atomic formula and using the algebra’s bold join, bold meet, and complement operations to define the respective values of bold disjunctions, bold conjunctions, and negations, we call this an algebraic interpretation based on that MV-algebra. We will say that a formula of propositional logic is a tautology of an MV-algebra (or an MV-tautology) if the formula evaluates to unit under every algebraic interpretation based on that algebra. Then, defining the other FuzzyL connectives in terms of bold disjunction, bold conjunction, and negation (as we’ve seen we can do), we have

Result 12.1: A formula is a tautology of FuzzyL if and only if it is an MV-tautology in every MV-algebra.

Proof of 12.1 is beyond the scope of this text; interested readers may consult Gottwald (2001) (also for proofs of Results 12.3 and 12.4 later in this chapter).

It is important to note that this result does not say that the set of FuzzyL tautologies coincides with the set of MV-tautologies for any MV-algebra. For example, in Chapter 9 we examined the MV-algebraic structure of L3. We know that the set of L3 tautologies is distinct from the set of FuzzyL tautologies, so it follows that the set of MV-tautologies for any three-valued MV-algebra (which, by Result 9.3, coincides with the set of L3 tautologies) is different from the set of MV-tautologies for FuzzyLMV. (Moreover, in Section 11.3 of Chapter 11 we introduced Lukasiewicz logics for all finite sets of truth-values taken from the unit interval. Each one of these has an MV-algebraic structure but no two have the same set of tautologies, and each of the tautology sets differs from the set of FuzzyL tautologies.) A formula must be an MV-tautology of every MV-algebra if it is to be a FuzzyL tautology, and vice versa.

12.2 Residuated Lattices and BL-Algebras

The algebraic structures characterizing FuzzyG and FuzzyP (based on their respective bold disjunction, bold conjunction, and negation operations) are not MValgebras.1 Most notably, neither of those structures satisfies the Double Negation condition (vii) for MV-algebras. This was to be expected, given the nonvalidity of the inference

P

P

in both systems. It is left as an exercise to determine which other conditions on MV-algebras fail for one or both of these systems.

We’ll present algebras for FuzzyG and FuzzyP that are special cases of residuated lattices (MV-algebras are also special cases of residuated lattices). A residuated lattice is an algebra <L, , ∩, , , unit, zero> that meets the following conditions:2

Recall that conditions i–v define a lattice with zero and unit elements.

Conditions vi–viii define the bold meet as a commutative, associative operation with unit as its identity element. Condition ix states that the bold meet operation is isotonic, or nondecreasing in both arguments, and condition x states that and form an adjoint pair. The connection with t-norms and their residuation operations should be obvious. Conditions vi–ix are the conditions for t-norm operations, and condition x is the adjointness condition defining the residuum operation for a t-norm.

1This section provides just a glimpse of the relations among MV-algebras, residuated lattices, and BL-algebras. An excellent summary of literature exploring the relations among these algebras (along with Boolean algebras) appears in Novak,´ Perfilieva, and Mockoˇˇr (1999, pp. 23–33).

2Residuated lattices were first studied in Dilworth and Ward (1939).

Negation is defined in a residuated lattice as

x =def x zero

This should also look familiar from Chapter 11; it is the way that negation is standardly defined in fuzzy logical systems. Given this operation, a dual operation forin residuated lattices is definable as

x y =def (x y ).

If MV = <M, , , , unit, zero> is an MV-algebra, and we define

x ∩ y =def x (x y) x y =def x (x y) x y =def x y,

as we did for MV-algebras in Section 12.1, then R(MV) = <M, , ∩, , , unit, zero> is a residuated lattice; it is in this sense that we say that MV-algebras are special cases of residuated lattices.3 For example, we know that x ≤ y if and only if x ∩ y = x in an MV-algebra, so the definition of inequality required for a residuated lattice holds in R(MV). We can establish the first part of bold meet isotonicity as follows:

First part of Isotonicity of MV-Algebra Meet: If x ≤ y then x z ≤ y z.

Proof:

Assume that x ≤ y in an MV-algebra. Then:

A complete proof that every MV-algebra is a residuated lattice, many pieces of which we have already seen by now, is left as an exercise.

The converse does not generally hold; some residuated lattices are not MValgebras. One reason is that the conditions defining residuated lattices do not entail Double Negation, so in some residuated lattices it is not true that x = x for all x in L. This should come as no surprise since we have stated that the algebraic structures for FuzzyG and FuzzyP are residuated lattices, and we know that Double Negation fails in those logical systems.

3Because Boolean algebras are MV-algebras (this was proved in Chapter 9), it follows that Boolean algebras are also special cases of residuated lattices.

The algebraic structure corresponding to FuzzyG is a residuated lattice FuzzyGL = <[0. .1], G, ∩G, G, G, 1, 0>, where G and G are FuzzyG’s bold conjunction and conditional operations (we saw in Section 11.7 of Chapter 11 that G and G form an adjoint pair), and G and ∩G are FuzzyG’s weak disjunction and conjunction operations (which are identical to FuzzyG’s bold operations). The algebraic structure FuzzyPL = <[0. .1], P, ∩P, P, P, 1, 0> corresponding to FuzzyP is also a residuated lattice. Both of these algebraic structures, along with that of FuzzyL, are special types of residuated lattices called BL-algebras.4 A BL-algebra is a residuated lattice that satisfies the additional conditions

xi.x ∩ y = x (x y)

xii.(x y) (y x) = unit

Condition xi should be familiar from the general definition of weak conjunction in fuzzy logics introduced in Chapter 11:

P Q =def P & (P → Q)

Condition xii captures the fact that for any formulas P and Q in a t-norm-based fuzzy logical system, at least one of P → Q or Q → P has the value 1 (to be proved in the exercises).

Every MV-algebra is a BL-algebra (also to be proved in the exercises). Conversely, we have

Result 12.2: Every BL-algebra that satisfies Double Negation (x = x) is an MValgebra.

Proof: We’ll establish that each of the MV-algebra conditions holds in any BLalgebra that satisfies Double Negation. We’ll use the following properties, which hold for all BL-algebras with Double Negation (BL-i through BL-iii hold for BL-algebras generally, not just those with Double Negation):

(BL-i) if unit ≤ x, then unit = x Proof: Assume unit ≤ x. Then:

(BL-ii) x (x y) = y Proof: Left as an exercise.

(BL-iii) x (y z) = (x y) z

4BL-algebras were introduced in Hajek´ (1998a, 1998b) to capture the commonalities of systems of fuzzy logic based on continuous t-norms (thus including Lukasiewicz, Godel,¨ and product logics). BL stands for basic logic. The definitions later of Godel¨ and product algebras as special types of BL-algebras are from Hajek´ (1998b, pp. 91 and 100).

218 Fuzzy Algebras

Condition viii:

Condition ix:

The rest of this proof reverses the steps leading to (x y) y = (y ∩ x ) .

The lattice FuzzyGL exemplifies another special case of BL-algebras: the Godel¨- algebras.5 A Godel¨ algebra is a BL-algebra that satisfies the additional condition

xiii. (Godel¨ BL-algebra) x x = x.

This condition says that Godel¨ bold conjunction (identical to Godel¨ weak conjunction) is idempotent. Proof that FuzzyGL is a Godel¨ algebra is left as an exercise. In addition we have

Result 12.3: A formula is a tautology of FuzzyG if and only if it is a G-tautology in every Godel¨ algebra,

where G-tautologies of Godel¨ algebras are defined in the by now obvious way. The algebraic structure FuzzyPL = <[0. .1], P, ∩P, P, P, 1,0>, where P

and P are FuzzyP’s bold conjunction and conditional operations, and P and ∩P are FuzzyP’s weak disjunction and conjunction operations, is an instance of another special case of BL-algebras, the product algebras. A product algebra is a BL-algebra that also satisfies the additional conditions

xiii.(Product BL-algebra) z ≤ ((x z) (y z)) (x y)

xiv.(Product BL-algebra) x ∩ x = zero

where

x =def x zero.

5Godel¨ algebras are identical to so-called Heyting algebras that satisfy the “prelinearity” condition: (x y) (y x) = unit.