On BL-Algebras and its Interval Counterpart

Interval Fuzzy Logic and Interval-valued Fuzzy Sets have been widely investigated. Some Fuzzy Logics were algebraically modeled by Peter Hájek as BL-algebras. What is the algebraic counterpart for the interval setting? It is known from the literature that there is an incompatibility between some algebraic structures and its interval counterpart. This paper shows that such incompatibility is also present in the level of BL-algebras. Here we show both: (1) the impossibility of match imprecision and the correctness of the underlying BL-implication and (2) some facts about the intervalization of BL-algebras.

Keywords:
Fuzzy Logic; BL-Algebras; intervals; correctness principle

Authorship

R. PAIVA ^**Corresponding author: Rui Paiva - E-mail: rui.brasileiro@ifce.edu.br

Group for Logic, Language, Information, Theory and Applications (LoLITA), Federal Institute of Education, Science and Technology of Ceará (IFCE), Maracanaú - CE, 61939-140, Brazil. E-mail: rui.brasileiro@ifce.edu.brFederal Institute of Education, Science and Technology of CearáBrazilMaracanaú, CE, BrazilGroup for Logic, Language, Information, Theory and Applications (LoLITA), Federal Institute of Education, Science and Technology of Ceará (IFCE), Maracanaú - CE, 61939-140, Brazil. E-mail: rui.brasileiro@ifce.edu.br

https://orcid.org/0000-0002-1775-8489

R. SANTIAGO

Group for Logic, Language, Information, Theory and Applications (LoLITA), Informatics and Applied Mathematics Department, Federal University of Rio Grande do Norte (UFRN), Natal - RN, 59078-970, Brazil. E-mail: regivan@dimap.ufrn.br, bedregal@dimap.ufrn.brFederal University of Rio Grande do NorteBrazilNatal, RN, BrazilGroup for Logic, Language, Information, Theory and Applications (LoLITA), Informatics and Applied Mathematics Department, Federal University of Rio Grande do Norte (UFRN), Natal - RN, 59078-970, Brazil. E-mail: regivan@dimap.ufrn.br, bedregal@dimap.ufrn.br

B. BEDREGAL

*Corresponding author: Rui Paiva - E-mail: rui.brasileiro@ifce.edu.br

SCIMAGO INSTITUTIONS RANKINGS

Formulas

Formulas (44)

(2.1)

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

x \to y = 1 i f f x \leq y and y otherwise .

x \to y = 1 i f f x \leq y and y / x otherwise .

x \to y = 1 i f f x \leq y and \min \{1, 1 - x + y\} otherwise .

A * B = A \cap B a n d A \to B = A^{C} \cup B

(f \land g) (x) = f (x) \land g (x) (f \lor g) (x) = f (x) \lor g (x) (f * g) (x) = f (x) * g (x) (f \to g) (x) = f (x) \to g (x)

X ⊑ Y \Leftrightarrow X \leq Y and X \leq Y .

(x, y) \in [a, b] \times [c, d] implies x ◊ y \in [a, b] ♦ [c, d] .

X ⊖ Y = [\min \{X - Y, X - Y\}, \max \{X - Y, X - Y\}]

X^{l} = {J \in I (L) | J is lower bound of X}

v = \underset{A \in X^{l}}{V} (A) and w = \underset{A \in X^{l}}{V} (A) .

X^{u} = \{T \in I (L) | T is upper bound of X\} \neq 0 .

m = \underset{B \in X^{u}}{\land} (B) and n = \underset{B \in X^{u}}{\land} (B) .

(2.1)

Z ⊑ (X \Rightarrow Y) if and only if [Z, Z] ⊑ [X \to Y, X \to Y] .

Z ⊑ (X \Rightarrow Y) iff Z ⊑ ⊔ \{W | X ⊛ W ⊑ Y\} iff Z \leq ⊔ \{W | X ⊛ W ⊑ Y\} and Z \leq ⊔ \{W | X ⊛ W ⊑ Y\} iff Z \leq \lor \{W | X ⊛ W ⊑ Y, for some W\} and Z \leq \lor \{W | X ⊛ W ⊑ Y, f o r s o m e W\} .

\lor \{W | X ⊛ W ⊑ Y, for some W\} = \lor \{W | X ⊛ W \leq Y and X ⊛ W \leq Y, for some W\} = \lor \{W | X * W \leq Y and X * W \leq Y, for some W\} \leq \lor {\{W | X * W \leq Y\}}^{2} .

\lor \{W | X ⊛ W ⊑ Y, for some W\} = \lor \{W | X ⊛ W \leq Y and X ⊛ W \subseteq Y, for some W\} = \lor \{W | X * W \leq Y and X * W \leq Y, for some W\} \leq \lor \{W | X * W \leq Y\} .

Z ⊑ (X \Rightarrow Y) i f f Z \leq X \to Y a n d Z \leq X \to Y i f f X * Z \leq Y a n d X * Z \leq Y i f f [X * Z, X * Z] ⊑ [Y, Y] i f f X ⊛ Z ⊑ Y .

X ⊛ (X \Rightarrow Y) = [X, X] ⊛ ([X, X] \Rightarrow [Y, Y]) = [X, X] ⊛ (⊔ \{[Z, Z] \in I (L) | [X, X] ⊛ [Z, Z] ⊑ [Y, Y]\}) \overset{E q . (2.1)}{=} [X, X] ⊛ [X \to Y, X \to Y] = [X * (X \to Y), X * (X \to Y)] = [X \land Y, X \land Y] = [X, X] ⊓ [Y, Y] = X ⊓ Y .

(X \Rightarrow Y) ⊔ (Y \Rightarrow X) = (⊔ \{Z \in I (L) | X ⊛ Z ⊑ Y\}) ⊔ (⊔ \{Z \in I (L) | Y ⊛ Z \leq X\}) = [X \to Y, X \to Y] ⊔ [Y \to X, Y \to X] = [(X \to Y) \lor (Y \to X), (X \to Y) \lor (Y \to X)] = [1_{L}, 1_{L}] = 1 .

∆ (X, Y) = [\inf ∆_{1} (α, β) \cup ∆_{2} (α, β), \sup ∆_{1} (α, β) \cup ∆_{2} (α, β)]

a *_{1} x \leq α *_{1} x \leq b *_{1} x and c *_{2} y \leq y *_{2} β \leq y *_{2} d .

a *_{1} c \leq α *_{1} c \leq b *_{1} c and c *_{2} b \leq b *_{2} β \leq b *_{2} d .

a *_{1} c \leq α *_{1} β \leq α *_{2} β \leq b *_{2} d .

b \to_{2} c \leq α \to_{2} c \leq a \to_{2} c and a \to_{1} c \leq a \to_{1} β \leq a \to_{1} d .

b \to_{2} c \leq α \to_{2} β \leq a \to_{2} c \leq a \to_{1} c \leq α \to_{1} β \leq a \to_{1} d,

X \hat{*} 0 = [X, X] \hat{*} [0_{L}, 0_{L}] = [X *_{1} 0_{L}, X *_{2} 0_{L}] \overset{(B L 8)}{=} [0_{L}, 0_{L}] = 0 .

X \hat{\to_{1, 2}} (Y \hat{\to_{1, 2}} Z) = X \hat{\to_{1, 2}} ([Y \to_{2} Z, Y \to_{1} Z]) = [X \to_{2} (Y \to_{2} Z), X \to_{1} (Y \to_{1} Z)] = [Y \to_{2} (X \to_{2} Z), Y \to_{1} (X \to_{1} Z)] = Y \hat{\to_{1, 2}} ([X \to_{1} Z, X \to_{2} Z]) = Y \hat{\to_{1, 2}} (X \hat{\to_{1, 2}} Z) .

(X \hat{\lor} Y) \hat{*_{1, 2}} Z = [X \lor Y, X \lor Y] \hat{*_{1, 2}} [Z, Z] = [(X \lor Y) *_{1} Z, (X \lor Y) *_{2} Z] \overset{(B L 13)}{=} [(X *_{1} Z) \lor (Y *_{1} Z), (X *_{2} Z) \lor (Y *_{2} Z)] = [X *_{1} Z, X *_{2} Z] \hat{\lor} [Y *_{1} Z, Y *_{2} Z] = (X \hat{*_{1, 2}} Z) \hat{\lor} (Y \hat{*_{1, 2}} Z) .

(X \hat{*_{1, 2}} Z) \hat{\to_{1, 2}} Y = ([X, X] \hat{*_{1, 2}} [Z, Z]) \hat{\to_{1, 2}} [Y, Y] = ([X *_{1} Z, X *_{2} Z] \hat{\to_{1, 2}} [Y, Y] = [(X *_{2} Z) \to_{2} Y, (X *_{1} Z) \to_{1} Y] \overset{((B L 18))}{=} [Z \to_{2} (X \to_{2} Y), Z \to_{1} (X \to_{1} Y)] .

Z \hat{\to_{1, 2}} (X \hat{\to_{1, 2}} Y) = [Z, Z] \hat{\to_{1, 2}} ([X, X] \hat{\to_{1, 2}} [Y, Y]) = [Z, Z] \hat{\to_{1, 2}} ([X \to_{2} Y, X \to_{1} Y] = [Z \to_{2} (X \to_{2} Y), Z \to_{1} (X \to_{1} Y)] .

(X \hat{*_{1, 2}} Z) \hat{\to_{1, 2}} (X \hat{*_{1, 2}} Y) = [0_{L}, 1_{L}] .

X \hat{*_{1, 2}} (X \hat{\to_{1, 2}} Y) = [X, X] \hat{*_{1, 2}} ([X, X] \hat{\to_{1, 2}} [Y, Y]) = [X *_{1} (X \to_{2} Y), X *_{2} (X \to_{1} Y)] .

X *_{1} (X \to_{2} Y) \leq X *_{1} (X \to_{1} Y) \overset{(B L 6)}{\leq} Y .

X \leq X \leq Y \leq Y

(3.1)

(X \to Y) * (Y \to Z) = 1 * (Y \to Z) \leq X \to Z .

(X \to Y) * (Y \to Z) \overset{(B L 7)}{\leq} (X \to Y) * (Y \to Z) \overset{(B L 14)}{\leq} X \to Z,

(3.2)

(X \to Y) * (Y \to Z) \leq X \to Z .

[(X \to_{2} Y) *_{1} (Y \to_{2} Z), (X \to_{1} Y) *_{2} (Y \to_{1} Z)] ⊑ [X \to_{2} Z, X \to_{1} Z] .

(3.3)

X *_{2} (X \to_{2} Y) \leq (X \land Y) \overset{(B L 3)}{\Leftrightarrow} X \to_{2} Y \leq X \to_{2} (X \land Y) .

(3.4)

X \to_{2} (X \land Y) \leq X \to_{2} Y .

(3.5)

X \to_{2} Y = X \to_{2} (X \land Y) .

(3.6)

X *_{1} (X \to_{1} Y) \leq (X \land Y) \overset{(B L 3)}{\Leftrightarrow} X \to_{1} Y \leq X \to_{1} (X \land Y) .

(3.7)

X \to_{1} (X \land Y) \leq X \to_{1} Y .

(3.8)

X \to_{1} Y = X \to_{1} (X \land Y) .

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On BL-Algebras and its Interval Counterpart

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[1] *Corresponding author: Rui Paiva - E-mail: rui.brasileiro@ifce.edu.br