12.3 Zero and Unit Projections in Algebraic Structures

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 1− are 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).

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?