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12.3 Zero and Unit Projections in Algebraic Structures

219

Condition xiv gives us the Law of Noncontradiction for weak conjunction: for any formula P, at least one of P and P has the value 0, and so therefore does the weak conjunction of a formula and its negation. It is left as an exercise to prove that FuzzyPL is a product algebra, and hence that xiii is characteristic of FuzzyP’s adjoint bold conjunction and implication operations. Analogously to earlier results, we have

Result 12.4: A formula is a tautology of FuzzyP if and only if it is a P-tautology in every product algebra.

12.3 Zero and Unit Projections in Algebraic Structures

Recall that we introduced fuzzy Bochvarian external negation and assertion, known as 0- and 1-projections in the fuzzy literature, in Section 11.10 of Chapter 11. We will call the corresponding algebraic operations zero projection and unit projection, and we will use the symbols ! and α, respectively, to denote these operations. The following algebraic conditions characterize unit projection in any BL-algebra:6

αi. αx (αx) = unit αii. α(x y) αx αy αiii. αx x

αiv. αx ααx

αv. αx α(x y) αy

αvi. α unit = unit

and we will call a BL-algebra that meets these additional conditions a unit projection algebra.

Although x and αx are not generally equivalent in a unit projection algebra, αx and ααx are:

αx = ααx

Proof:

αx ααx, by αiv, and ααx αx, by αiii, so αx = ααx.

The expressions x and αx are not generally equivalent because the inequality x αx does not hold in all unit projection algebras. This is as it should be, if unit projection is the algebraic counterpart to fuzzy external assertion: if V(P) = .5, for example,P has the value 0.

We can introduce zero projection in a unit projection algebra as the algebraic counterpart to fuzzy external negation with the definition

!x = def (αx) .

6

Hajek´ (1998b) based these algebraic axioms on derivational axioms in Baaz (1996).

 

220

 

 

 

Fuzzy Algebras

so that, for example,

 

 

αx !x = unit

(by αi)

 

and

 

 

 

 

!unit = zero

 

 

Proof:

 

 

 

 

α unit = unit

 

(αvi)

(α unit) = unit

 

(same operation, both sides)

(i) !unit

=

unit

 

(definition)

 

 

and

 

 

 

 

unit (unit zero) = zero

(BL-ii)

unit zero = zero

 

(identity for bold meet)

(ii) unit

= zero

 

(definition)

so

 

 

 

 

!unit = zero

 

(from (i) and (ii))

Adding the external assertion operator to FuzzyL, the MV-algebraic structure of FuzzyL becomes a unit projection algebra. To establish this, we need to show that the following hold for all values x and y in [0. .1], where α stands for the external assertion operation:

α i. max (α x, 1 – α x) = 1

α ii. α (max (x, y)) max (α x, α y) α iii. α x x

α iv. α x α α x

α v. max (0, α x + α (min (1, 1 – x + y)) – 1) α y α vi. α 1 = 1

For (α i), it suffices to point out that α x is always either 1 or 0. For (α iii), we note that if x =1 then α x = 0 and 0 is less than or equal to any value in the unit interval, while if x = 1 then α x = 1 and certainly 1 1. We leave the remainder, as well as the proof that the Godel¨ and product algebraic structures for FuzzyG and FuzzyP augmented with the external assertion operator are unit projection algebras, as exercises.

12.4 Exercises

SECTION 12.1

1Prove that the algebra FuzzyLMV = <[0. .1], L, L, 1, 1, 0>, where L, L, and 1are FuzzyL’s bold disjunction, bold conjunction, and negation operations, is an MV-algebra.

2Prove that it follows from the MV-algebra definition x y =def (x y ) that x y = x (x y).

12.4 Exercises

221

SECTION 12.2

3Which MV-algebra conditions other than Double Negation, if any, fail to hold for the algebra defined by FuzzyG’s operations?

4Which MV-algebra conditions other than Double Negation, if any, fail to hold for the algebra defined by FuzzyP’s operations?

5Give a complete proof that every MV-algebra is a residuated lattice in the sense described in Section 12.2. You are free to cite previous results along the way.

6Prove 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, by showing that this follows from the definition of the residuum operation

m tn n p if and only if m n p, for all m, n, p [0. .1]

for any t-norm tn. (Hint: consider the case where m = 1 and show that in this case either 1 n p or 1 p n. You will need to use t-norm properties to establish this.)

7Prove that every MV-algebra is a BL-algebra.

8Prove that

(BL-ii) x (x y) = y holds in every BL-algebra.

9Complete the proof of BL-iii:

a.Show that (x y) z x (y z) in every BL-algebra with Double Negation.

b.Show that if x y and y x in a BL-algebra, then x = y—so that the converse inequalities in the proof of BL-3 in fact establish an equality.

10Prove that

(BL-vi) x y = x y

holds in every BL-algebra with Double Negation.

11Complete the proof of Result 12.2, that every BL-algebra with Double Negation is an MV-algebra, by showing that the following hold true in every BL-algebra with Double Negation:

a.condition i of MV-algebras

b.condition ii of MV-algebras

c.condition iii of MV-algebras

d.the second half of condition iv of MV-algebras: x zero = zero

e.condition v of MV-algebras

f.condition vi of MV-algebras

g.condition vii of MV-algebras.

12Prove that FuzzyGL is a Godel¨ algebra.

13Prove that FuzzyPL is a product algebra.

14Why did we not include the condition xiv. (Godel¨ BL-algebra) x x = x

in the definition of product algebras?

222

Fuzzy Algebras

SECTION 12.3

15Prove that the following hold in every unit projection algebra:

a.α zero = zero

b.!zero = unit

c.!x = !!!x, for any x

16Complete the proof that the MV-algebraic structure of FuzzyL augmented with the external assertion operator is a unit projection algebra, by showing that α ii, α iv, α v, and α vi all hold.

17Prove that the Godel¨ and product algebraic structures FuzzyGL and FuzzyPL augmented with the external assertion operator are unit projection algebras.