Densification of FL chains via residuated frames

We introduce a systematic method for densification, i.e., embedding a given chain into a dense one preserving certain identities, in the framework of FL algebras (pointed residuated lattices). Our method, based on residuated frames, offers a uniform proof for many of the known densification and standard completeness results in the literature. We propose a syntactic criterion for densification, called semianchoredness. We then prove that the semilinear varieties of integral FL algebras defined by semi-anchored equations admit densification, so that the corresponding fuzzy logics are standard complete. Our method also applies to (possibly non-integral) commutative FL chains. We prove that the semilinear varieties of commutative FL algebras defined by knotted axioms xm≤xn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x^{m} \leq x^{n}}$$\end{document} (with m,n>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m, n > 1}$$\end{document}) admit densification. This provides a purely algebraic proof to the standard completeness of uninorm logic as well as its extensions by knotted axioms.


Introduction
Given a class K of ordered algebras and a chain A (i.e., a totally ordered algebra) in it, we would like to embed A into a dense chain in the same class K. This construction, referred to as densification, is important both on its own and as a key step towards standard completeness of various fuzzy logics, i.e., a completeness theorem with respect to the valuations of propositional variables into the real unit interval [0, 1]. Many fuzzy logics fall into the class of substructural logics, whose algebraic semantics are given by FL algebras (or pointed residuated lattices) [19]. We thus consider densification of FL chains and standard completeness of associated fuzzy logics. Some classes of FL chains, such as Gödel chains, are well known to be densifiable, while others, such as Boolean algebras, are clearly not densifiable; see, e.g., [21] for background. However, there is no general criteria for classifying densifiable and non-densifiable FL chains.
Our aim in this paper is not to prove a new result, but rather to provide a uniform, algebraic account of densification. This is important since standard completeness is often proved by a proof theoretic argument, known as density elimination in the hypersequent calculus [1,26,11,12,4]. While density elimination is no doubt an interesting application of proof theory, it obscures the algebraic essence of densification. Even when densification is proved algebraically (as in [23,24,21]), it is not clear to what extent the employed technique generalizes. To fulfill our goal, we employ residuated frames [18], that are effective devices to construct (complete) FL algebras with various properties. They are also a key to connect proof theory with algebraic studies; for instance, one can naturally define a residuated frame W from the sequent calculus FL, so that validity in the dual algebra W + directly implies cut-free derivability in FL. This strong connection allowed us to prove that for a certain class of substructural logics, a strong form of cut-admissibility (proof theory) is equivalent to closure under completions (algebra), thus promoting a new approach to substructural logics, dubbed algebraic proof theory for substructural logics [9,10].
In this paper, we use residuated frames to densify a given FL chain, preserving certain identities. Although our argument was originally inspired by density elimination following the spirit of algebraic proof theory, the resulting construction can be understood without any reference to proof theory: it provides a purely algebraic account. Further steps in a purely algebraic analysis of densification can be found in [17].
The rest of this paper is organized as follows. Section 2 discusses densification in a general setting, and Section 3 specializes it to FL algebras. Section 4 reviews residuated frames, and Section 5 applies them to densify integral FL chains. This encompasses the standard completeness of monoidal t-norm logic and its noncommutative variant [23,24]. Section 6 then addresses a more involved case: (possibly non-integral) commutative FL chains. It provides a purely algebraic proof of the standard completeness of uninorm logic, for which only proof theoretic arguments were previously known [26]. Section 7 recalls the concept of a substructural hierarchy [7,8,9,10], that is useful to classify equations in the language of FL. Based on the hierarchical classification, we introduce the class of semi-anchored P 3 equations in Section 8, and prove a general result that every nontrivial semilinear variety of integral FL algebras defined by semi-anchored P 3 equations admits densification and standard completeness. We then turn our attention to varieties of commutative FL algebras in Section 9. We prove that every semilinear variety of commutative FL algebras defined by knotted axioms x m ≤ x n with m, n > 1 admits densification and standard completeness. These results are again inspired by the proof theoretic arguments in [4,2]. We conclude the paper with some remarks and open problems.

Densifiability
We begin with a general consideration of densification and standard completeness. In the sequel, we assume that every algebra A comes equipped with Proof. For simplicity, let us assume κ, |L| ≤ ℵ 0 ; the argument below clearly works also for an arbitrary κ > 1 and L, provided that one uses the axiom of choice. Let X be a countable set of variables and T = T (X) the set of terms in the language L over X. Let (t 0 , u 0 ), (t 1 , u 1 ), . . . be a countable sequence of elements of T 2 such that each (t, u) ∈ T 2 occurs infinitely many times in it.
For each n ∈ N, we define a chain B n in V as well as a partial function f n : X B n . Let B 0 := A and f 0 be any surjective partial function onto A such that X\ dom(f 0 ) is infinite.
For n ≥ 0, if one of f n (t n ), f n (u n ) is undefined or f n (t n ) < f n (u n ), then let B n+1 := B n and f n+1 := f n .
Otherwise, let x be a variable taken from X\ dom(f n ). If there is p ∈ B n such that f n (t n ) < p < f n (u n ), then let B n+1 := B n . If not, let B n+1 be a chain in V that fills the gap (f n (t n ), f n (u n )) by p ∈ B n+1 . We assume B n ⊆ B n+1 and define f n+1 : X B n+1 by extending f n with f n+1 (x) := p. Let B := B n , f := f n , and C be the subalgebra of B generated by . Moreover, C is dense since for every pair (g, h) ∈ C 2 with g < h, there is n ∈ N such that g = f n (t n ) and h = f n (u n ), so that we have g < f n+1 (x) < h.
Given a class K of algebras of the same type, the semantic consequence relation |= K is defined as usual. Namely, given a set E ∪ {s = t} of equations in the language of K, E |= K s = t holds if E entails s = t in every algebra where V C consists of all chains in V. This is equivalent to saying that every subdirectly irreducible algebra in V is a chain (cf. [21]). We say 1] consists of all standard chains in V, namely those over the real unit interval ([0, 1], ≤). This conforms to the terminology in fuzzy logics under the identification of an (algebraizable) logic with the corresponding variety. There are actually two weaker notion of standard completeness. The finite strong standard completeness defined as 1] for E a finite set of equations and the weak standard completeness defined as In the rest of this paper, we mean by standard completeness the strongest version.
We need a few concepts concerning completions. Given an algebra A, a completion of A consists of a complete algebra B together with an embedding e : A join-dense and meet-dense completion is called a MacNeille completion. It is known that the lattice reduct of a MacNeille completion is uniquely determined (up to isomorphism that fixes A) by join and meet density [5,29]. For instance, the MacNeille completion of the rational unit interval Proposition 2.3. Let L be a finite or countable language of algebras and V a variety of type L. If V is semilinear, densifiable and every chain in V has a MacNeille completion in it, then V is standard complete.
Proof. Let E ∪ {s = t} be a set of equations, and suppose that E |= V s = t. By semilinearity, there is a finite or countable chain A such that E |= A s = t. By Proposition 2.2, A is embeddable into a countable dense chain B in V. It is well known that (B, ≤ B ) is isomorphic to one of (0, 1) Q , (0, 1] Q , [0, 1) Q , and [0, 1] Q , whose MacNeille completion is [0, 1]. Hence, B is embeddable into a standard chain C in V [0,1] , and we have E |= V [0,1] s = t.
Hence, there are two key factors for standard completeness: densifiability and closure under MacNeille completions.

FL algebras
In this section, we recall the concept of an FL algebra. A standard reference on this topic is [19]. Definition 3.1. A residuated lattice is an algebra A = (A, ∧, ∨, ·, \, /, 1) such that (A, ∧, ∨) is a lattice, (A, ·, 1) is a monoid, and for all x, y, z ∈ A, An FL algebra is a residuated lattice A with a distinguished element 0 ∈ A. The constant 0 is used to define negations: ∼x := x\0, Densification of FL chains via residuated frames 5 We remark that A is a chain if and only if the following communication property holds for all x, y, z, w ∈ A: x ≤ z and y ≤ w =⇒ x ≤ w or y ≤ z. (com) Also important is the fact that if (g, h) is a gap in a FL chain, then we have h\g < 1. Indeed, 1 ≤ h\g would imply h ≤ g. The two divisions \ and / coincide in any commutative FL algebra. So we write x → y := x\y = y/x in that case.
We write FL for the variety of FL algebras and use subscripts e, c, i, o to indicate the properties (e), (c), (i), (o) above. For instance, FL ei denotes the variety of commutative integral FL algebras. It is known (cf. [19,21]) that a variety of FL algebras is semilinear if and only if it satisfies the four-variable equation λ a (x∨y\y)∨ρ b (x∨y\x) = 1, where λ a and ρ b are conjugate operators defined by: λ a (x) := (a\xa) ∧ 1 and ρ b (x) := (bx/b) ∧ 1.
Given a variety V of FL algebras, we denote by V the subvariety obtained by imposing the above equation. Notice that it is equivalent to the familiar prelinearity axiom (x → y) ∨ (y → x) = 1 in FL ei algebras.
Unfortunately, FL is not standard complete (cf. [31,21]). On the other hand, it is not hard to see that every subvariety of FL i defined by a combination of (e), (c), (o) is densifiable and closed under MacNeille completions, so is standard complete [23,24].
A short proof of the densifiability of FL i is as follows. Let A be an integral FL chain with a gap (g, h). We insert a new element p between g and h: take A p := A ∪ {p}, with g < p < h. The meet and join operations are naturally extended to A p . To extend multiplication · and divisions \, /, note that for every a ∈ A, either ah = h or ah ≤ g holds. For every a ∈ A, we define: The other cases p · a, p/p, a/p, and p/a are defined analogously.
This gives rises to a new algebra A p in FL i that fills the gap (g, h) of A. While it is possible to check the correctness manually, our approach is rather to derive A p by a general residuated frame construction (Section 5). Our approach will explain the rationale behind A p , and provide a generic recipe for proving further standard completeness results. It also leads to an algebraic proof of the standard completeness of uninorm logic [26] in Section 6.

Residuated frames and MacNeille completions
Just as Kripke frames are useful devices to build various Heyting and modal algebras, residuated frames are useful devices to build various FL algebras. In this section, we introduce residuated frames and recall some relevant facts from [18,9].
We often omit • and write xy for x • y.
Given a frame W = (W, W , N, •, ε, ), there is a canonical way to make it residuated: letW := W × W × W and defineÑ ⊆ W ×W by ThenW := (W,W ,Ñ, •, ε, (ε, , ε)) is a residuated frame, since As we have said at the beginning, the primary purpose of residuated frames is to build residuated lattices. We now describe the construction. Let W = (W, W , N, •, ε, ) be a residuated frame. Given X, Y ⊆ W and Z ⊆ W , let where X N z iff x N z for every x ∈ X, and x N Z iff x N z for every z ∈ Z.
We write x and z instead of {x} and {z} . The pair ( , ) forms a Galois connection: X ⊆ Z ⇐⇒ X ⊇ Z, so that γ(X) := X defines a closure operator on P(W ) (the powerset of W ): Furthermore, γ is a nucleus, namely it satisfies It is for this property that a frame has to be residuated.
Let P(W ) be the powerset of W and γ[P(W )] ⊆ P(W ) be its image under γ.
is a complete FL algebra. Vol. 00, XX Densification of FL chains via residuated frames 7 As an example, let A = (A, ∧, ∨, ·, \, /, 1, 0) be an FL algebra. Then we may define a frame by W A := (A, A, N, ·, 1, 0), where N is the lattice ordering ≤ of A. W A is residuated precisely because A is residuated: Hence, by the previous proposition, W + A is a complete FL algebra. We want W + A to be commutative (resp. contractive, integral, 0-bounded, totally ordered) whenever A is. The following rules ensure that.
, whenever A is commutative (resp. contractive, integral, 0-bounded, totally ordered). These properties are in turn propagated to the dual algebra W + A . This holds for any residuated frame.
Proof. Let us only prove that (com N ) implies W + being totally ordered. Suppose that there are X, Y ∈ γ[P(W )] for which X ⊆ Y and Y ⊆ X. The former means that there are x ∈ X and w ∈ Y such that x N w does not hold (since Y = Y ). Similarly, the latter means that there are y ∈ Y and z ∈ X such that y N z does not hold. On the other hand, we have x N z and y N w by definition of X , Y . Hence, the rule (com N ) implies that at least one of x N w and y N z should hold, a contradiction.
Note also that W + is complete, hence it is always a bounded FL algebra. Finally, we would like to have an embedding of A into W + A . More generally, let A be an FL algebra and W = (W, W , N, •, ε, ) a residuated frame. Suppose that there are injections i : A −→ W and i : A −→ W by means of which we identify A with a subset of W and of W . In such a situation, the rules in Figure 1, called Gentzen rules, ensure the existence of a homomorphism. embedding.
When A is bounded, the homomorphism e preserves both the least and greatest elements. It should be remarked that Lemma 4.4 actually holds for much more general situations. For instance, A can be an arbitrary, even 176 P. Baldi and K. Terui Algebra Univers. 8 P. Baldi and K. Terui Algebra univers.
x N a a N z holds for every Galois-closed set X. The last equality holds because e(a) = a = a and X N a iff X ⊆ a .
The varieties FL x with x ⊆ {e, c, i, o} are just a few examples. We will see in Section 8 that the same holds for many more subvarieties of FL.

Densification of integral FL chains
Residuated frames are useful not just for completion, but also for densification. In this section, we prove the densifiability of FL x with {i} ⊆ x ⊆ {e, c, i, o} by using residuated frames. Our proof gives a rationale behind the concrete definition of A p in Section 3, and moreover serves as a warm-up before the more involved case of (non-integral) commutative FL chains in the next section. Vol. 00, XX Densification of FL chains via residuated frames 9 Let us fix an integral FL chain A, a gap (g, h) in it, and a new element p. Our purpose is to define a residuated frame whose dual algebra fills the gap (g, h) by p.
We define a frame W p A = (W, W , N, •, ε, ) as follows.
Thus, each element x ∈ W is a finite sequence of elements from A ∪ {p}. We denote by A * the subset of W that consists of finite sequences of elements from A (without any occurrence of p). Also, given Let us now define N . Under the intuition that g < p < h should hold and N should be an extension of ≤ A , it is natural to require that a N p iff a ≤ g, and p N a iff h ≤ a for every a ∈ A. We also require that p N p. The definition below embodies these requirements. For every x ∈ W and a ∈ A: x N p always holds (otherwise).
As explained in Section 4, the frame W p A induces a residuated frameW p A . Notice that since A is an integral chain, W p A satisfies uv N z uxv N z for every u, x, v ∈ W and z ∈ W . Hence,W p A satisfies the rule (i N ), so the dual algebraW p+ A is integral by Lemma 4.3. To have a closer look at the residuated frameW p A , it is convenient to partition the setW = W × W × W into three, in accordance with the case distinctions in the definition of N : Just as we associated an element x ∈ A to each x ∈ W , we associate an element z ∈ A to each z ∈W as follows: Finally, we define The following lemma explains why we have defined the sets A * , A • , and the concept of stability.
When z ∈W 2 and x ∈ A * , xÑ z always holds.
(2): x ∈ A * means that the sequence x contains an occurrence of p, which is interpreted by h. Hence, the claim holds by integrality.
(3): This is proved in a similar way.
In case at least one of the conclusions is not stable, (com N ) immediately holds by Lemma 5.1 (1). Notice that this is always the case when both premises xÑ z and yÑ w are not stable. Hence, we only need to consider the cases when both conclusions are stable and either both or only one of the premises is stable.
(i) If both premises xÑ z and yÑ w are stable, (com N ) boils down to x ≤ z and y ≤ w =⇒ x ≤ w or y ≤ z, that holds by the communication property in A.
(ii) Assume only one premise is stable. For instance, let yÑ w be stable and xÑ z not stable, so that x ∈ A * and z ∈ A • . We have either y ≤ g or h ≤ y since (g, h) is a gap. If y ≤ g, then y ≤ g ≤ z by Lemma 5.1(3), so the right conclusion holds. If h ≤ y, Lemma 5.1(2) and the right premise implies x ≤ h ≤ y ≤ w, so the left conclusion holds. The case where yÑ w is not stable and xÑ z is stable is symmetrical.
For the next lemma, we consider injections i, i from A to W and W given by i(a) := a ∈ W and i (a) := (ε, a, ε) ∈W , and identify a with i(a) and i (a).  a ≤ A b for every a, b ∈ A. Hence, e(a) := γ(a) is an embedding of A intoW p+ A . Proof. Observe that all Gentzen rules (Figure 1, where N is replaced byÑ ) have stable premises. If the conclusion is also stable, then the claim follows from the premises by Lemma 5.1 (1). Otherwise, (as may happen for the rule (Cut)), the conclusion holds automatically. Vol. 00, XX Densification of FL chains via residuated frames 11 Lemma 5.4. Let e be the embedding of A intoW p+ A in Lemma 5.3. The following hold.

2) e(g) e(p) e(h).
Proof. (1): Suppose that x = a ∈ A. We have a ∈ a by (Id). Hence, a ⊆ a . To show the other inclusion, let y ∈ a and z ∈ a . Then yÑ a and aÑ z, so yÑ z by (Cut). This shows that a ⊆ a .
For x = p, the above reasoning shows that it is sufficient to verify (Id) and (Cut) for p, too: Here, we identify p on the right-hand side with (ε, p, ε) ∈W . (Id) is obvious. For (Cut), if the conclusion is unstable, it holds automatically. Otherwise, we distinguish three cases. If x ∈ A * and z ∈ A • , Lemma 5.1(3) and the left premise (which is stable) imply x ≤ g ≤ z. If x ∈ A * and z ∈ A • , Lemma 5.1(2) and the right premise (which is stable) imply Thus, we have given a uniform proof for the standard completeness of Gödel logic (FL ecio ), monoidal t-norm logic (FL eio ), and its noncommutative counterpart (FL io ) [23,24].
The structure ofW p+ A . We have obtained a chainW p+ A filling a gap of A, but we have not yet seen what kind of chain it is. By looking into its structure, it turns out that it is just a MacNeille completion of the chain A p presented in Section 3.
We will show that the restriction ofW p+ A to e[A]∪{γ(p)} forms a subalgebra by giving a concrete description. To simplify the notation, we writê   Proof. Notice thatx ·ŷ = γ(γ(x) • γ(y)) = (xy) . Hence, to see the equivalence betweenx ·ŷ andû, it is sufficient to check (xy) = u , which holds exactly when xyÑ z iff uÑ z for every z ∈W .
If z ∈ A • , stability implies If z ∈ A • , both sides of the above four hold by Lemma 5.1(1) and (3).
To prove the equalities for \, notice thatŵ = w andx\ẑ = x \z = {x}\z for every w, x, z ∈ A ∪ {p}. Hence, to seex\ẑ =ŵ, it is sufficient to check that xyÑ z iff yÑ w for every y ∈ W .
For the equality forâ\p, we distinguish two cases. If y ∈ A * , stability implies • ayÑ p iff ay ≤ g iff y ≤ g iff yÑ p (when ah = h). To see the second equivalence, y ≤ g obviously implies ay ≤ g. Conversely, suppose that y ≤ g does not hold. Then h ≤ y, so h = ah ≤ ay. Hence, ay ≤ g does not hold. • ayÑ p iff ay ≤ g iff y ≤ a\g iff yÑ a\g (when ah ≤ g).
If y ∈ A * , both sides of the above two hold. In particular, yÑ a\g holds since ay ≤ ah ≤ g by Lemma 5.1(2), so y ≤ a\g.
and any Galois-closed set X is both a join of elements from the second set and a meet of elements from the third set (cf. ( * ) in Section 4).
One might call into question the significance of our construction based on residuated frames, since the resulting algebra admits a much simpler presentation as given in Section 3. Our justifications are as follows.
• Our method has a heuristic value, as it provides a general recipe how to Vol. 00, XX Densification of FL chains via residuated frames 13 find a chain that fills a gap. In essence, it amounts to a combinatorial task of finding a residuated frame satisfying (com N ) and Gentzen rules. Once such a frame has been found, we are done. See the next section for another application of our method.
• To prove standard completeness, we usually need to show both densifiability and closure under MacNeille completions (Proposition 2.3). Since residuated frames unify the two tasks to a large extent, the method may in fact be considered quite economical.
• Residuated frames are intimately connected to the sequent calculus in proof theory, as one can see in the Gentzen rules of Figure 1. This allows us to translate various proof theoretic arguments into algebraic ones. Indeed, our construction was inspired by a proof theoretic argument for standard completeness: first introduce the density rule in the hypersequent calculus, which enforces the intended algebraic models to be dense chains, and then show that it can be eliminated from a given proof, thus relating dense chains with nondense ones [1,26,11,12,4]. Further references on the subject can be found in [27,25]. Our frame construction precisely mirrors the way that the density rule is eliminated in [12,4]. It is amazing that such a proof theoretic argument, devised independently of algebraic considerations, translates into an algebraic one quite smoothly. It suggests a deep connection between proof theory and algebra, perhaps much deeper than usually believed. Finding such a connection is the main goal of our long-term project: algebraic proof theory for substructural logics [9,10].

Densification of commutative FL chains
We now turn to another class of chains: commutative FL chains. The class of bounded commutative FL chains provides a general algebraic semantics for so-called uninorm logic, which is known to be standard complete [26]. However, all the known proofs of this fact are proof theoretic, based on elimination of the density rule in a hypersequent calculus. In this section, we translate the proof theoretic argument into an algebraic one based on residuated frames. This gives rise to a first algebraic proof of standard completeness for uninorm logic.
Let A be a commutative FL chain with a gap (g, h) and p a new element. We again build a residuated frame whose dual algebra fills the gap (g, h). Although we could define W as before, we can exploit commutativity to simplify the construction.
We define a frame W  Algebra Univers. 14 P. Baldi and K. Terui Algebra univers.
• There are three types of elements in W ×W : (ap n , b), (a, p), and (ap n+1 , p) with a, b ∈ A and n ∈ N. N is defined accordingly: Notice that this is compatible with the previous definition. In particular, ap n+1 N p always holds if A is integral. As before, the frame W p A induces a residuated frameW p A := (W,W ,Ñ, •, ε, (ε, )). Because of commutativity, the definitions ofW andÑ are slightly simplified: As in the integral case, the setW can be partitioned into three: Elements of W,W are interpreted by elements of A. For x = ap n ∈ W , let x := ah n ∈ A. For z ∈W , we define ah n → 1, if z = (ap n+1 , p) ∈W 3 .
As before, Similarly to Lemma 5.1(1), we have the following result.
Since a, b ∈ A, this reduces to the case (i).
(iii): Suppose that w ∈ A • and z ∈ A • . We write w = (w 1 , w 2 ) and z = (a, p). There are three subcases.
First, suppose that x, y ∈ A. Then we may write x = x p and y = y p, so that (com N ) becomes x Ñ (pa, p) and y Ñ (pw 1 , w 2 ) x Ñ (pw 1 , w 2 ) or y Ñ (pa, p) .
Second, suppose that x ∈ A and y ∈ A, so that we may write y = y p. Notice that xÑ z iff xa N p iff xa ≤ g. Also, yÑ z iff y pa N p iff y a ≤ 1. Thus, what we have to check is xa ≤ g and y ≤ w =⇒ x ≤ w or y a ≤ 1.
By the communication property, the premises imply either x ≤ w or ya ≤ g. If x ≤ w, we are done. Otherwise, we have y ha = ya ≤ g, so y a ≤ h → g < 1 (since g < h). So, we are done.
Finally, suppose that x ∈ A and y ∈ A, so that we may write x = x p. Note that xÑ z iff x pa N p iff x a ≤ 1. Also, yÑ z iff ya N p iff ya ≤ g. Thus, what we have to check is x a ≤ 1 and y ≤ w =⇒ x ≤ w or ya ≤ g.
If ya ≤ g, we are done. Otherwise, h ≤ ya. Hence, together with the premises, we obtain x = x h ≤ x ya ≤ y ≤ w.  , and moreover a N b  implies a ≤ A b for every a, b ∈ A. Hence, e(a) := γ(a) is an embedding of A intoW p+ A .
Proof. As in the proof of Lemma 5.3, notice that all Gentzen rules except (Cut) have stable premises and conclusion. Hence, we only have to check the (Cut) rule xÑ a aÑ z xÑ z (Cut), where (x, z) is unstable. We may write x = x p and z = (b, p). By noting that xÑ z iff x pb N p iff x b ≤ 1, this amounts to x h ≤ a and ab ≤ g =⇒ x b ≤ 1.
Lemma 6.4. Let e be the embedding of A intoW p+ A in Lemma 6.3. The following hold.

(2) e(g) e(p) e(h).
Proof. In view of the proof of Lemma 5.4, it is sufficient to show that (Id) and (Cut) hold for p. For (Cut), xÑ p pÑ z xÑ z (Cut).
If x ∈ A and z ∈ A • , then all of (x, z), (x, p), and (p, z) are stable. Hence, the premises imply x ≤ g < h ≤ z. If x ∈ A and z ∈ A • , we may write x = x p. The premises amount to x ≤ 1 and h ≤ z, so we obtain x = x h ≤ z.
If x ∈ A and z ∈ A • , we may write x = x p and z = (a, p). The premises amount to x ≤ 1 and a ≤ 1, so we obtain x a ≤ 1.
Finally, if x ∈ A and z ∈ A • , we may write z = (a, p). The premises amount to x ≤ g and a ≤ 1, so we obtain xa ≤ g.
In any case, we obtain the conclusion xÑ z.
We have proved that the chainW p+ A fills the gap (g, h) of A. An alternative construction of this algebra and a finer analysis of its structure are provided in [17]. It is easy to see thatW p A satisfies (e N ), (i N ), (o N ) whenever A satisfies (e), (i), (o). Thus, we have the following. Theorem 6.5. Every variety FL x with {e} ⊆ x ⊆ {e, i, o} is densifiable, hence is standard complete.
As we mentioned at the beginning of the section, uninorm logic is complete with respect to the class of bounded commutative FL chains. Since boundedness is clearly preserved by our construction, we have obtained a purely algebraic proof of the standard completeness of uninorm logic [26].

Substructural hierarchy and MacNeille completions
The rest of this paper is devoted to densification of subvarieties of FL i and FL e . The concept of substructural hierarchy [7,8,9,10] is useful to deal with those subvarieties systematically.
Definition 7.1. For each n ≥ 0, the sets P n and N n are defined as follows.
By residuation, any equation u = v can be written as 1 ≤ t. We say that u = v belongs to P n (N n , resp.) if t does.
The classes (P n , N n ) constitute the substructural hierarchy ( Figure 2). Among those classes, relevant to subsequent arguments, are N 2 and P 3 . The former includes the following: The classes N 2 and P 3 are intimately related to the classes of structural quasiequations and structural clauses defined below. Definition 7.2. By a clause, we mean a classical first-order formula of the form t 1 ≤ u 1 and · · · and t m ≤ u m =⇒ t m+1 ≤ u m+1 or · · · or t n ≤ u n , (q) where t i , u i are terms of FL and all variables are assumed to be universally quantified. Each t i ≤ u i (1 ≤ i ≤ m) is called a premise, while each t j ≤ u j (m + 1 ≤ j ≤ n) is a conclusion. We say (q) is a quasiequation if n = m + 1. It is structural if t 1 , . . . , t n are products of variables (including the empty product 1) and u 1 , . . . , u n are either a variable or 0. Given a structural clause (q), let L(q) be the set of variables occurring in t m+1 , . . . , t n , and R(q) the set of variables occurring in u m+1 , . . . , u n . We say (q) is analytic if the following conditions are satisfied: Separation: L(q) and R(q) are disjoint. Linearity: Each variable in L(q) ∪ R(q) occurs exactly once in the conclusions t m+1 ≤ u m+1 , . . . , t n ≤ u n . Inclusion: Each of t 1 , . . . , t m is a product of variables in L(q), while each of u 1 , . . . , u m is either a variable in R(q) or 0.  Proof.
(1) is proved in [9]. For (2), we have A |= t · u = 1 ⇐⇒ A |= (t = 1 and u = 1), for every integral FL chain A. Thus, each P 3 equation is equivalent to a set of disjunctions of the form (t 1 = 1 or · · · or t n = 1). The rest of the proof proceeds as in [8,10].
Example 7.4. Our running example is the weak nilpotent minimum axiom 1 ≤ ∼(xy) ∨ (x ∧ y\xy) that belongs to P 3 . It is equivalent in integral FL chains to xy ≤ z and xv ≤ z and vy ≤ z and vv ≤ z =⇒ xy ≤ 0 or v ≤ z. (wnm) Structural clauses are useful because they can be expressed as rules for residuated frames. Moreover, analytic ones are preserved under the dual algebra construction. To make it more precise, consider a structural clause of the form (q) above, and let W = (W, W , N, •, ε, ) be a residuated frame. We can naturally translate each t i into a term over (•, ε), and each u i into either Vol. 00, XX Densification of FL chains via residuated frames 19 a variable or . The resulting terms are still denoted by t i , u i . Corresponding to the clause (q), we have t 1 N u 1 and · · · and t m N u m =⇒ t m+1 N u m+1 or · · · or t n N u n . (q N ) Example 7.5. The clause (wnm) corresponds to the following rule for residuated frames: xy N z and xv N z and vy N z and vv N z =⇒ xy N or v N z.
By definition, if an FL algebra A satisfies (q), then the residuated frame W A satisfies (q N ). Moreover, Lemma 4.3 generalizes to all analytic clauses.
Theorem 7.6. Let (q) be an analytic clause. If a residuated frame W satisfies (q N ), then the dual algebra W + satisfies (q).
The correctness of the above theorem should be clear from the example below as well as the case of (com) handled by Lemma 4.3. The case of quasiequations is detailed in [9] and the case of clauses is implicit in [8]; [10] contains a more general result.
Example 7.7. Suppose that W satisfies (wnm N ). Our goal is to show that W + satisfies (wnm), namely holds for all Galois-closed sets X, Y, V, Z. Suppose that neither of the conclusions holds. Then there are x ∈ X, y ∈ Y , v ∈ V , and z ∈ Z such that neither xy N nor v N z holds (since Z = Z ). On the other hand, the premises yield xy N z, xv N z, vy N z, and vv N z, which contradict the assumption that W satisfies the rule (wnm N ).

(1) If V is defined by equations equivalent to analytic quasiequations and
A ∈ V, then its MacNeille completion belongs to V. (2) If V is defined by equations equivalent to analytic clauses (over chains) and A is a chain in V, then its MacNeille completion belongs to V.

Densification of subvarieties of FL i
We now focus on subvarieties of FL i defined by P 3 equations. By Theorems 7. 3  whose only nontrivial chain is the two element one. Notice that BA is defined by excluded middle x ∨ ¬x = 1 in P 2 , which is equivalent to We should rule out such a clause by introducing some criteria. The criteria below are inspired by the proof-theoretical approach in [4], which extends [12]. Before we proceed further, let us make it precise what it means that the specific residuated frameW p A defined in Section 5 satisfies (q N ). Recall that an analytic clause (q) is of the form t 1 ≤ z 1 and · · · and t m ≤ z m =⇒ t m+1 ≤ z m+1 or · · · or t n ≤ z n .
For the purpose of this section, it is convenient to write (q) as P =⇒ C, where Recall that each equation in P and C consists of variables L(q) and R(q). To each x ∈ L(q), we associate an element x • ∈ W = (A ∪{p}) * so that each term t is interpreted by t • ∈ W . Likewise, to each z ∈ R(q) we associate a triple z • ∈W = W ×W ×W , where W = A∪{p}. The interpretations of constants 1, 0 are already fixed: 1 • := ε ∈ W and 0 • := (ε, , ε) = (ε, 0, ε) ∈W . It is now clear whenW p A satisfies (q N ). It is true just in case the following holds for each such interpretation • : Suppose that t is a product of variables: t = x 1 · · · x n . Then t • ∈ A * iff x • i ∈ A * for every 1 ≤ i ≤ n. This implies: Let us now come back to criteria for densifiability. That is, DP (q) is the set of pairs of variables "connected" by one of the premises, and similarly for DC(q). We say that (q) is anchored if DP (q) ⊆ DC(q).
Clearly, the clause (em) is not anchored since DP (em) = {(x, z), (y, z)} and DC(em) = {(y, z)}. On the other hand, any analytic quasiequation is anchored due to the inclusion condition. Proof. Assume that A satisfies an anchored clause (q). Our goal is to verify ( * ) above for any interpretation •. If there is a conclusion t ≤ z ∈ C such that (t • , z • ) is not stable, then we have t •Ñ z • by Lemma 5.1(1), so ( * ) holds. Vol. 00, XX Densification of FL chains via residuated frames 21 Otherwise, (t • , z • ) is stable for every t ≤ z ∈ C, and the same holds for every t ≤ z ∈ P by ($) and DP (q) ⊆ DC(q). Hence, by Lemma 5.1(1), ( * ) amounts to {t holds since A satisfies (q).
The previous lemma does not apply to many clauses. For instance, it does not apply to (wnm): since (x, z), (y, z) ∈ DP (wnm)\DC(wnm). To deal with this and more involved clauses, we need to extend the definition of anchored clause.
In the sequel, we write t = t(x 1 , . . . , x n ) to indicate variable occurrences x 1 , . . . , x n in term t. Then t(y 1 , . . . , y n ) denotes the result of substituting y i for x i . Definition 8.3. Let (q) : P =⇒ C be an analytic clause. We say that (q) is semi-anchored if for every premise t ≤ z in P , if t can be written as t(x 1 , . . . , x n ), with (x i , z) ∈ DC(q) (1 ≤ i ≤ n), one of the following holds: (t may contain a variable x 0 ∈ {x 1 , . . . , x n } with (x 0 , z) ∈ DC(q).) Notice that checking semi-anchoredness amounts to checking finitely many conditions on the premises and the conclusions of a clause; hence, it is decidable. The PROLOG-system AxiomCalc automates the conversion of equations into analytic clauses in Section 7 and checks whether an analytic clause satisfies a condition similar to (1) in Definition 8.3; see [4]. Condition (2) in Definition 8.3 does not present any particular challenge for further automation. (AxiomCalc is available online at http://www.logic.at/people/lara/ axiomcalc.html.) For the first premise xy ≤ z, the term xy can be written as t 1 (x) = t 2 (y) = t 3 (x, y) = xy, so that t 1 (v) = vy, t 2 (v) = xv, and t 3 (v, v) = vv. In any case, we have premises t 1 (v) ≤ z, t 2 (v) ≤ z, and t 3 (v, v) ≤ z with (v, z) ∈ DC(wnm), so the case (1) applies. Similarly for the other premises.
Likewise, we can show that (wnm n ) is equivalent to a semi-anchored clause for every n. This conforms to the standard completeness of monoidal t-norm logic with (wnm n ) proved in [4].
Example 8.5. The equations (Ω n ) are equivalent in integral FL chains to the conjunction of x n−1 ≤ x n and (x n−1 \y) ∨ (y\x · y). FL ew chains satisfying these equations are called Ω(S n MT L) chains and are shown to be densifiable Vol. 00, XX Densification of FL chains via residuated frames 25 Otherwise k = 1 and e 1 = 1. Since m > 1, we have m − k ≥ 1. Hence, ( * ) and ( * * ) with l := m − k implies a n 1 · · · a n m b m ≤ 1, which leads to the conclusion.
Finally, we obtain the main theorem of this section.
Theorem 9.4. Let V be the subvariety of FL e defined by x m ≤ x n with distinct m, n > 1. Then V is densifiable, and so is standard complete.
Proof. Let A ∈ V be a chain with a gap (g, h). By Lemmas 9.2 and 9.3,W p A satisfies (knot nN m ). Hence,W p+ A , filling the gap (g, h) of A, satisfies (knot n m ) by Theorem 7.6, i.e.,W p+ A ∈ V. Notice that x m ≤ x 0 with m > 0 is equivalent to the integrality x ≤ 1, that has already been dealt with; indeed, the former implies x ≤ (1 ∨ x) m ≤ (1 ∨ x) 0 = 1. Likewise, x 0 ≤ x n with n > 0 is equivalent to 1 ≤ x, which defines the trivial variety.
The only remaining cases are x m ≤ x 1 and x 1 ≤ x n with m, n > 1, which are respectively equivalent to x 2 ≤ x and x ≤ x 2 in FL . Unfortunately, our result in this section, as well as its proof theoretic origin [2], does not cover these cases. [26] shows proof-theoretically only that the subvariety of FL axiomatized by both x 2 ≤ x and x ≤ x 2 is densifiable. In a recently submitted work [3], all these cases are addressed by general proof-theoretic means. Translating them into our algebraic framework would require the construction of a residuated frame different from the one given in Section 6.
Unfortunately, N 3 -subvarieties (i.e., subvarieties defined by N 3 equations) of FL , or even FL eio , cannot be dealt with by our method. This is the case, for instance, for the varieties corresponding to basic logic, Lukasiewicz logic, product logic, WCMTL and ΠMTL (see, e.g., [13,14,15,28,20]). It is certainly a limitation of our approach, but notice that these varieties only enjoy the finite strong form of standard completeness (E |= V s = t ⇔ E |= V [0,1] s = t for finite E), and it has actually been proved that none of them admits the strong form studied in this paper.
It is an open problem to what extent the N 2 -subvarieties of FL e admit densification and standard completeness. Section 9 only gives a partial solution (for knotted axioms x m ≤ x n with m, n > 1). In this paper, we have not considered involutive subvarieties of FL, i.e., those defined by −∼x = ∼−x = x. For involutive FL ei , which corresponds to involutive monoidal t-norm logic, strong standard completeness has been proved algebraically in [15] and prooftheoretically in [26]. We believe that this result can be reproved by employing involutive residuated frames of [18]. On the other hand, an important open problem in this direction is the standard completeness of involutive FL e , which correspond to involutive uninorm logic, for which we are not sure whether our method applies or not.