Logics for Finite UL and IUL-Algebras Are Substructural Fuzzy Logics

: Semilinear substructural logics UL ω and IUL ω are logics for ﬁnite UL and IUL -algebras, respectively. In this paper, the standard completeness of UL ω and IUL ω is proven by the method developed by Jenei, Montagna, Esteva, Gispert, Godo, and Wang. This shows that UL ω and IUL ω are substructural fuzzy logics.

Substructural logics are logics that lack some of the three basic structural rules of contraction, weakening, and exchange. For a survey, see [13]. Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1], i.e., logics that are standard complete [2]. Our result in this paper thus shows that UL ω and IUL ω are substructural fuzzy logics.
As pointed out in [6], our construction in Lemma 5 also presents a method to construct uninorms and involutive uninorms. Then, the standard completeness for UL ω and IUL ω gives a characterization of uninorms and involutive uninorms and their residua constructed by finite UL and IUL-algebras from our constructions. That is, the identity (Fin) holds in all these standard UL and IUL-algebras; see Definition 5. These new classes of uninorms and involutive uninorms may be used in the theory of evaluation as the aggregation operators or combining functions [14,15].
We have proven that HpsUL * is standard complete in [16]. However, we are unable to prove whether HpsUL * ω is standard complete or complete with respect to finite HpsUL * -algebras and left them as open problems. In addition, we have proven that IUL is standard complete in [17], which is a longstanding open problem in the circle of fuzzy logic. Unfortunately, such a great work has not been accepted by our community since 2015, although one referee thought that the central ideas in our proof are reasonable and could find no significant flaws in the reasoning. The referee also said that he would also not be confident that the proof is correct if the proof were his own and he had spent many months laboring over it.

HpsUL
* ω , UL ω , IUL ω and Algebras Involved The Hilbert system HpsUL is the logic of bounded representable residuated lattices, which is based on a countable propositional language with formulas built inductively as usual from a set of propositional variables, binary connectives ⊙, →, ↝, ∧, ∨, and constants e, f , , ⊺, with definable connectives: Definition 1. HpsUL consists of the following axioms and rules [4]: . A logic is a schematic extension (extension for short) of HpsUL if it results from HpsUL by adding axioms in the same language. In particular, Definition 3. New extensions of HpsUL are defined as follows.
Let L ∈ {HpsUL * , UL, IUL, HpsUL * ω , UL ω , IUL ω } in the remainder of this section. A proof in L of a formula ϕ from a set Γ of formulas is defined as usual. We write Γ ⊢ L ϕ if such a proof exists.

Definition 4 ([4]
). An HpsUL-algebra is a bounded residuated lattice A = ⟨A, ∧, ∨, ⋅, →, ↝, e, f , , ⊺⟩ with universe A, binary operations ∧, ∨, ⋅, →, ↝, and constants e, f , , ⊺ such that: (i) ⟨A, ∧, ∨, , ⊺⟩ is a bounded lattice with top element ⊺ and bottom element ; We use the convention that ⋅ binds stronger than other binary operations, and we shall often omit ⋅; we will thus write xy instead of x ⋅ y, for example. Suitable classes of algebras of extensions of HpsUL are defined as follows. In particular: • A is an HpsUL * -algebra if the weak commutativity (Wcm) holds: xy ≤ e iff yx ≤ e for all x, y ∈ A; • A is a UL-algebra if xy = yx for all x, y ∈ A; • A is an IUL-algebra if it is a UL-algebra such that ¬¬x = x for all x ∈ A; • A is an HpsUL * ω -algebra (UL ω or IUL ω -algebra) if it is an HpsUL * -algebra (UL or IUL-algebra) such that the following identity (Fin) holds: x → e = x 2 → e for all x ∈ A. Proof. For the proof of the necessity part, see Lemma 2.4(i) of [1]. For the sufficiency part, assume that xy ⩽ e iff xy 2 ⩽ e for all x, y ∈ A. Suppose that x → e > x 2 → e. Then, x 2 (x → e) > e by Definition 4 (iii), and hence, (x → e)x 2 > e by (Wcm). Therefore, (x → e)x > e by the assumption. Then, x → e > x → e, a contradiction, and thus, x → e ≤ x 2 → e. x 2 → e ≤ x → e is proven by a similar way. Hence, x → e = x 2 → e for all x ∈ A, i.e., A is an HpsUL * ω -algebra.
Proof. Only (1) is proven as follows; for the others, see [1]. If tu ⩽ e, then tut ⩽ e and utu ⩽ e by Proposition 1 and (Wcm). Thus, stut ⩽ s and stutu ⩽ st. Hence, st ⩽ s and s ⩽ st. Therefore, s t = s. The case of tu > e is proven in the same way.

Definition 7 ([3,5]). Let
A be an UL ω -chain. For each s ∈ A, t is the immediate predecessor of s in A if: (i) t ∈ A, t < s; (ii) ∀u ∈ A, u < s implies u ⩽ t. For each s ∈ A, let s − denote the immediate predecessor of s in A if it exists, otherwise take s − = s.
we define: (s, q) ⩽ (t, r) iff either s < S t, or s = t and q ⩽ r and, Now define, for (s, q), (t, r) ∈ X: where by ∧ X and ∨ X are meant min X and max X with respect to ⩽ X , respectively. We will omit the index if it does not cause confusion.
Suppose that s − t ⩾ s, then s − tt ⩾ st > e. Thus, s − t > e by Proposition 1, a contradiction, and hence, This completes the proof of (i).

Lemma 5. Let
A be an HpsUL * ω -chain, X, and the binary operation ○ on X be as in Definition 7. The following conditions hold: (a) X is densely ordered and has a maximum ⊺ X = (⊺, 1) and a minimum X = ( , 1). (b) ⟨X, ○, ⩽ X , e X ⟩ is a linearly-ordered monoid, where e X = (e, 1).
(c) ○ is left-continuous with respect to the order topology on ⟨X, ⩽ X ⟩.
Proof. Claim (e) has been proven by Lemma 2. As pointed out in [3], the associativity of ○ is mainly dependent on Lemma 1(1)∼(2). Other claims are proven in the same way as that of Theorem 4.5 in [3].