The logic of pseudo-uninorms and their residua

Our method of density elimination is generalized to the non-commutative substructural logic GpsUL*. Then the standard completeness of GpsUL* follows as a lemma by virtue of previous work by Metcalfe and Montagna. This result shows that GpsUL* is the logic of pseudo-uninorms and their residua and answered the question posed by Prof. Metcalfe, Olivetti, Gabbay and Tsinakis.


Introduction
In 2009, Prof. Metcalfe, Olivetti and Gabbay conjectured that the Hilbert system HpsUL is the logic of pseudo-uninorms and their residua [1]. Although HpsUL is the logic of bounded representable residuated lattices, it is not the case, as shown by Prof. Wang and Zhao in [2]. In 2013, we constructed the system HpsUL * by adding the weakly commutativity rule (WCM) ⊢ (A ↝ t) → (A → t) to HpsUL and conjectured that it is the logic of residuated pseudo-uninorms and their residua [3].
In this paper, we prove our conjecture by showing that the density elimination holds for the hypersequent system GpsUL * corresponding to HpsUL * . Then the standard completeness of HpsUL * follows as a lemma by virtue of previous work by Metcalfe and Montagna [4]. This shows that HpsUL * is an axiomatization for the variety of residuated lattices generated by all dense residuated chains. Thus we also answered the question posed by Prof. Metcalfe and Tsinakis in [5] in 2017.
In proving the density elimination for GpsUL * , we have to overcome several difficulties as follows. Firstly, cut-elimination doesn't holds for GpsUL * . Note that (WCM) and the density rule(D) are formulated as Here, the major problem is how to extend (D) such that it is applicable to G 2 Γ 2 , Π ′ 2 , p, Π ′′ 2 , Σ 2 ⇒ p. By replacing p with t, we get G 2 Γ 2 , Π ′ 2 , t, Π ′′ 2 , Σ 2 ⇒ t. But there exists no derivation of which we can't obtain by (WCM). It seems that (WCM) can't be strengthened further in order to solve this difficulty. We overcome this difficulty by introducing a restricted subsystem GpsUL Ω of GpsUL * . GpsUL Ω is a generalization of GIUL Ω , which we introduced in [6] in order to solve a longstanding open problem, i.e., the standard completeness of IUL. Two new manipulations, which we call the derivation-splitting operation and derivation-splicing operation, are introduced in GpsUL Ω .
The third difficulty we encounter is that the conditions of applying the restricted external contraction rule (EC Ω ) become more complex in GpsUL Ω because new derivation-splitting operations make the conclusion of the generalized density rule to be a set of hypersequents rather than one hypersequent. We continue to apply derivation-grafting operations in the separation algorithm of the multiple branches of GIUL Ω in [6] but we have to introduce a new construction method for GpsUL Ω by induction on the height of the complete set of maximal (pEC)-nodes other than on the number of branches.
Hence G is not a tautology in HpsUL. Therefore it is not a theorem in HpsUL by Theorem 9.27 in [1].
is provable in GpsUL * , as shown in Figure 1.

Figure 1 A proof τ of G
Suppose that G has a cut-free proof ρ. Then there exists no occurrence of t in ρ by its subformula property. Thus there exists no application of (WCM) in ρ. Hence G is a theorem of GpsUL, which contradicts Lemma 2.4.
Remark 2.6. Following the construction given in the proof of Theorem 53 in [4], (CUT ) in the figure 1 is eliminated by the following derivation. However, the application of (WCM) in ρ is invalid, which illustrates the reason why the cut-elimination theorem doesn't hold in GpsUL * .
We call it the weakly cut rule and, denote by (WCT ).
Proof. It is proved by a procedure similar to that of Theorem 53 in [4] and omitted.
GpsUL Ω is a restricted subsystem of GpsUL * such that (i) p is designated as the unique eigenvariable by which we means that it does not be used to built up any formula containing logical connectives and only used as a sequent-formula.
(ii) Each occurrence of p in a hypersequent is assigned one unique identification number i in GpsUL Ω and written as p i . Initial sequent p ⇒ p of GpsUL * has the form such that G ′′ can be obtained by applying σ l to antecedents of sequents in G ′ and σ r to succedents of sequents in G ′ .
(v) A hypersequent G G 1 G 2 can be contracted as G G 1 in GpsUL Ω under certain condition given in Construction 3.15, which we called the constraint external contraction rule and denote by (vi) (EW) is forbidden in GpsUL Ω and, (EC) and (CUT ) are replaced with (EC Ω ) and (WCT ), respectively. and (viii) G 1 S 1 and G 2 S 2 are closed and disjoint for each two-premise inference rule Proof. Although (WCT ) makes t's in its premises disappear in its conclusion, it has no effect on identification numbers of the eigenvariable p in a hypersequent because t is a constant in GpsUL Ω and distinguished from propositional variables.

The generalized density rule (D) for GpsUL Ω
In this section, GL cf Ω is G ps ULΩ without (EC Ω ). Generally, A, B, C, ⋯, denote a formula other than an eigenvariable p i .
There are three cases to be considered.
The case of all focus formula(s) of S ′ contained in ∆ k is dealt with by a procedure dual to above and omitted.
(I)) and, ⟨τ * ⟩ + j ⟨H k+1 ⟩ + j is constructed by combining ⟨τ * ⟩ + j ⟨H k ⟩ + j and Case 3 S ′ ∈ ⟨H k ⟩ + j . It is dealt with by a procedure dual to Case 2 and omitted. Definition 3.2. The manipulation described in Construction 3.1 is called the derivation-splitting operation when it is applied to a derivation and, the splitting operation when applied to a hypersequent.
Then there exist two hypersequents G 1 and G 2 such Ω is constructed by induction on n − m as follows.
• For the base step, let n − m = 0. Then and Π ′ , Π ′′ , Π ′′′ = Π and Γ ′′ , Γ ′ = Γ and ∆ ′ , ∆ ′′ = ∆. It follows from Corollary 3.3 that 6 • For the induction step, let n − m > 0. Then it is treated using applications of the induction hypothesis to the premise followed by an application of the relevant rule. For example, By the induction hypothesis we obtain a derivation of G Γ, Π, ∆ ⇒ A: Definition 3.5. The manipulation described in Construction 3.4 is called the derivation-splicing operation when it is applied to a derivation and, the splicing operation when applied to a hypersequent.
, the procedure terminates and n ∶= k, otherwise v l (G k ) v r (G k ) ≠ ∅ and define i k+1 to be an identification num- Proof. By the construction in the proof of Lemma 3.9, i k ∈ v l (G k−1 ) for all 2 ⩽ k ⩽ n. Then This shows that D K (G G ′ ) is constructed by repeatedly applying splicing operations.
Let G ′ be a splitting unit of G G ′ in the form Γ 1 ⇒ p 1 ⋯ Γ n ⇒ p n . Then each node of Ω G ′ has one and only one child node. Thus there exists one cycle in Ω G ′ by V G ′ = n < ∞. Assume that, without loss of generality, (1, 2), (2, 3), ⋯, (i, 1) is the cycle of Ω G ′ . Then p 1 ∈ Γ 2 , This process also shows that there exists only one cycle in Ω G ′ . Then we introduce the following definition.
Definition 3.13. (i) Γ j ⇒ p j is called a splitting sequent of G ′ and p j its corresponding splitting variable for all 1 ⩽ j ⩽ i.
(ii) Let K = {1, 2, ⋯, n} and D 1 (G Γ,   In this section, we adapt the separation algorithm of branches in [6] to GpsUL * and prove the following theorem.