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.


INTRODUCTION
The motivation of using intervals instead of exact values can also be perceived from the fact that the amount of imprecision can be codified through intervals in terms of its width. Since the operations in Ł∞ are continuous, the resulting interval operations are correct and optimal in the sense of Hickey [10] and Santiago [15], which means that imprecision stored in input intervals are controlled by such operations.
BL-algebras -which were introduced by Hájek [9] -are an algebraic counterpart to Basic Logic (BL) which generalizes the three most commonly used logics in the theory of fuzzy sets; namely: Łukasiewicz logic, product logic and Gödel logic [7,8]. This together with the fact that intervalvalued fuzzy set theory has been revealed as an increasingly promising extension of usual fuzzy sets [4,5,6,14] -namely: the usual membership degrees are replaced by closed intervals in [0, 1] -lead us to consider the investigation on the intervalization of BL algebras.
Although BL-algebras has been widely investigated (c.f. [1,2,3,13,17]) we found no reference on the literature which takes into account its interval counterpart. Following this standpoint, we investigate the extension of BL-algebras to interval structures. Namely, by using the notion of abstract intervals and the notion of best interval representation investigated by Santiago et.al. [15], we show how the notion of correctness on intervals affects the algebraic structure of BLalgebras. For example, we prove that there is no best interval representation, , for a given BL-algebra implication, →, (see Theorem 1). Some other properties are also showed. This paper is organized as follows: Section 1 gives a brief introduction to BL-algebras. Section 2 shows how would be an interval BL-algebra. Section 3 shows the incompatibility between the notions of interval correctness and BL-algebras, provide a way to build an interval-based structure starting from two BL-algebras and give some properties of such a system. Finally, section 4 provides some final remarks.
In this section, we expose a brief introduction of BL-algebras, some of its properties and some examples.
2. (Algebra of Product). This is the algebraic semantics for known product logic, the structure [0, 1], min, max, * , →, 0, 1 , where * it is the usual multiplication of real numbers on the unit interval [0, 1] and for all x, y ∈ X and 0, 1 : X → L are the constant functions associated with 0, 1 ∈ L.
Proposition 1. (see [9]) If L, ∧, ∨, * , →, 0 L , 1 L is a BL-algebra and x, y, z ∈ L then: Some of the above examples have elements which are not finitely representable. That is, they are algebras which contain irrational numbers, like π − 3 ∈ [0, 1]. A question posed is how can we represent the underlying imprecision of such structures? One answer for that is the application of Interval Mathematics, which model the imprecision in numerical calculations ( [11], [12]) and provides algorithms with rigorous control of errors. In the next section, we show that this approach induces algebraic structures which cannot be BL-algebras. Hence, a new algebraic structure will be revealed in order to obtain a suitable interval counterpart of a BL-algebra.

INTERVAL BL-ALGEBRAS
In what follows we introduce some required concepts, like the abstract notion of intervals.
The aim is to provide the ability to use intervals to represent the elements of a BL-algebra L, ∧, ∨, * , →, 0 L , 1 L . We also define a partial order on I(L) called Kulisch-Miranker order: For all X,Y ∈ I(L), The fundamental property of interval mathematics is the notion of interval correctness. It was studied by Santiago et al 1 [15]. Instead of correctness the authors used the term representation. Essentially, correctness or representation means that if F is correct with respect to f , then we can enfold any exact value r in a closed interval [a, b] and then simply operate with such "envelopes" by using F, because the resulting interval F([a, b]) will enfold the desired result f (r), in symbols: In what follows we show this notion for binary operations: a binary interval operation defined on I(L) represents a binary operation ♦ defined on L whenever, Example 4 (Arithmetic operations on real intervals). Let [a, b] and [c, d] be real intervals. The interval operations of sum, difference and product are defined in the following way: 1 In this paper the authors use the term representation instead of correctness because interval expressions could be faced not just as machine representations of an exact calculation, but also as an instance of mathematical representation of real numbers.
Notice that for each interval operation • ∈ {⊕, , ⊗} and their respective elementary real oper- Therefore, in each case, the binary interval operation • defined on I(R) represents a binary operation • defined on R. The following example ratifies the thesis that not all extension of arithmetic operations for intervals is correct. Another desirable property according to Hickey [10] is Optimality: The resulting interval should be the smallest possible which satisfy the correctness criterion. The process of giving the correct and optimal interval version F for a function f is called: "intervalization".
Since BL-algebras are partially ordered systems L, ≤ it is possible to apply Definition 2 to obtain the partial order I(L), . The question is: From this partial order is it possible to define a BL-algebra which represents L? The following propositions will show that the answer is negative.
For now, observe that it is possible to obtain a BL-algebra of intervals from some BL-algebras.
Definition 3. Given a BL-algebra: L, ∧, ∨, * , →, 0 L , 1 L in which L is a complete lattice, we define the following binary operations on I(L): Proof. According to the definition of and operators, just consider for each Thus I(L), , , 0 L , 1 L is a lattice. Now consider the non-empty set X ⊆ I(L). It is obvious that [0 L , 0 L ] is a lower bound of X, then the set:   Proof. Since that the operator * is associative, commutative and has identity 1 L , just check these properties for operator what is straightforward. Proof. Notice that the axioms (BL1) and (BL2) follow, respectively, from propositions 2 and 3. Moreover, given X,Y, Z ∈ I(L), In fact, This means that Z ≤ X → Y and Z ≤ X → Y . From such inequalities we can establish the following Therefore, for all X,Y, Z ∈ I(L) the pair ( , ⇒) is Galois connection, hence (BL3) is satisfied. Now let's check out the Axiom (BL4). Indeed, for all X,Y ∈ I(L), Finally, for the axiom (BL5), we can simplify writing, for all X,Y ∈ I(L) This completes the proof.
Although it is possible to obtain interval BL-algebras from BL-algebras, the next theorem shows that none of them will provide correct implications. This is informally stated in [16]. . Therefore, in both cases the interval binary operator is not correct with respect to binary operator →.

THE BEST INTERVAL REPRESENTATION
The most important property of Moore Interval arithmetic [12] is not just its correctness, i.e, is the tightest interval contaning x + y. Santiago et al [15] call this feature as: "The best interval representation" of "+". More generally: Definition 4. Given a lattice L, ∧, ∨ , an interval operator .
∆ : I(L) × I(L) → I(L) is representable if there exist operators, ∆ 1 , ∆ 2 : L × L → L such that for each X,Y ∈ I(L), with α ∈ X and β ∈ Y , we have that In this case ∆ 1 and ∆ 2 are called representants of .
Similarly to the previous item, since ∆ i (α, β ) = α → i β with i ∈ {1, 2}, we can obtain the following inequalities: Using the fact that → 2 ≤→ 1 we have the chain which allows us to complete the desired result.
In particular, when the operators ∆ 1 and ∆ 2 coincide, we have the following: Corollary 1. If L, ∧, ∨, * , →, 0 L , 1 L is a BL-algebra then the following items provide the best interval representation of their corresponding operators: In the following, we provide some results of intervalization of BL-algebras.
Proof. The properties of operator presented in Section 1 are related with the respective properties of the best interval operator enrolled above.