Modal MTL-algebras

Available online at www.sciencedirect.com

Fuzzy Sets and Systems 222 (2013) 58–77www.elsevier.com/locate/fss

Modal MTL-algebras

W. Mortona,∗ C.J. van Altenb

a Department of Mathematics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africab School of Computer Science, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa

Received 28 April 2012; received in revised form 8 October 2012; accepted 7 November 2012Available online 21 November 2012

Abstract

A modal MTL-algebra is an algebra in the variety generated by the modal MTL-chains—linearly ordered commutative, bounded,integral, residuated lattices equipped with a unary order-preserving operation. Reverse modal MTL-algebras can be defined similarlyby equipping a unary order-reversing operation instead. We axiomatize the variety of (reverse) modal MTL-algebras. Two construc-tions are considered on (reverse) modal MTL-chains: the MacNeille completion of the underlying order and a finite embeddabilityconstruction. In both cases we define a suitable extension of the unary order-preserving (-reversing) operation. Properties preservedvia these constructions are investigated using approximations. In particular, a large class of identities preserved by each of theconstructions is described syntactically.© 2012 Elsevier B.V. All rights reserved.

Keywords: Non-classical logics; Algebra; MTL-algebras; Modal operator; Prelinear; MacNeille completion; Finite embeddability property

1. Introduction

The addition of modal operators to various non-classical logics has been studied increasingly in recent years. Forexample, the exponentials ! and ? of linear logic [1] can be viewed as modal operators, since they have some similaritieswith the modalities � and �. Another example is the Baaz Delta �, intended to mean complete (classical) truth, added tofuzzy logic [2]. In [3,4] storage operators and truth stresser modalities are added to many-valued logics and, in particular,to monoidal t-norm logics, respectively. For more examples of the addition of modalities to various (substructural) logicsthe reader can refer to, for example, [5,6] or [7].

In this paper we consider the algebraic semantics of the monoidal t-norm logic (MTL) with unary operations(modalities). In [8], MTL was introduced as the logic of left-continuous t-norms and an algebraic semantics for thelogic, namely the variety of ‘MTL-algebras’, was defined. MTL-algebras are bounded, commutative, integral residuatedlattices (as defined, e.g., in [9]) satisfying the prelinearity identity: (x → y) ∨ (y → x) = 1. A main result concerningthe variety of MTL-algebras is that it is generated by the class of linearly ordered MTL-algebras, or ‘MTL-chains’.Motivated by this, we first define a ‘modal MTL-chain’ to be an MTL-chain equipped with an additional order-preserving (unary) operation f, and a ‘modal MTL-algebra’ as any algebra in the variety generated by the class ofmodal MTL-chains. As we show, this class is strictly smaller than the class of all MTL-algebras that have an additional

∗ Corresponding author. Tel.: +27 82 634 4263; fax: +27 11 559 2874.E-mail addresses: [email protected] (W. Morton), [email protected] (C.J. van Alten).

0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.fss.2012.11.008

http://www.elsevier.com/locate/fss

http://dx.doi.org/10.1016/j.fss.2012.11.008

http://www.elsevier.com/locate/fss

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.fss.2012.11.008

W. Morton, C.J. van Alten / Fuzzy Sets and Systems 222 (2013) 58 –77 59

order-preserving operation. In particular, we show that modal MTL-algebras are axiomatized by the axioms of MTL-algebras together with f (x ∨ y) = f (x) ∨ f (y) and ( f (y) → f ((x → z) ◦ y)) ∨ (z → x) = 1.

We note that the less general notion (in the sense that there are additional constraints) of a ‘modal residuatedlattice’ has been consider by Ono in [6], where such algebras are defined to be residuated lattices equipped with anoperation f that satisfies f (x) ≤ x , f (x) ≤ f ( f (x)), 1 ≤ f (1) and f (x) ◦ f (y) ≤ f (x ◦ y) in addition to beingorder-preserving.

We also consider ‘reverse modal MTL-chains’ and algebras in which the modality is order-reversing ratherthan order-preserving. A natural motivating example is the operation g(x) = 1 − x on any standard MTL-algebra,that is, an MTL-algebra whose universe is the real interval [0, 1]. An axiomatization for the class of reverse modalMTL-algebras consists of the axioms for MTL together with g(x ∨ y) = g(x) ∧ g(y) and (g((x → z) ◦ y) → g(y))∨(z → x) = 1.

Our research is motivated by considerations in modal algebras. An important problem in modal algebras isthat of canonicity—the preservation of identities by the canonical extension. A classical result there is that the classof Sahlqvist identities [10], a syntactically defined class, is preserved by the canonical extension [11]. We prove aSahlqvist-like result by giving a syntactic description of a class of identities preserved by the MacNeille completion.Our algebraic approach is motivated by the algebraic approach for modal algebras in [11,12]. We extend the methodsused in [13,14]. As shown, the MacNeille completion can be adapted to obtain ‘finite embeddability properties’ formany of the classes considered here. The finite embeddability property, or FEP, for a finitely axiomatized class ofalgebras implies decidability of the universal theory. Here we prove the FEP for the classes of modal MTL-chains andmodal MTL-algebras and, as in the case of the MacNeille completion, give a syntactic description of a class of identitiespreserved by the FEP construction. Thus, we obtain the FEP for any class of modal MTL-algebras axiomatized by afinite subset of these identities. We also obtain similar results for the MacNeille completion and FEP for reverse modalMTL-algebras.

2. Preliminaries

For more details on residuated lattices and MTL-algebras we refer the reader to [8,9].Recall that an (integral, bounded, commutative) residuated lattice is an algebra of the form 〈A, ◦, →, ∧, ∨, 0, 1〉,

where 〈A, ∧, ∨, 0, 1〉 is a bounded lattice (with 1 and 0 as greatest and least elements, respectively) and ◦ is a binaryoperation that is associative, commutative, has identity 1 and is residuated with residual →, i.e., for all a, b, c ∈ A

a ◦ c ≤ b ⇔ c ≤ a → b.

An MTL-algebra is a residuated lattice that satisfies the prelinearity identity:

(x → y) ∨ (y → x) = 1.

Definition 2.1.

(i) A modal residuated lattice is an algebra 〈A, ◦, →, ∧, ∨, f, 0, 1〉, where 〈A, ◦, →, ∧, ∨, 0, 1〉 is a residuated latticeand f is a unary operation that is order-preserving, i.e., x ≤ y ⇒ f (x) ≤ f (y).

(ii) A reverse modal residuated lattice is an algebra 〈A, ◦, →, ∧, ∨, g, 0, 1〉, where 〈A, ◦, →, ∧, ∨, 0, 1〉 is a residuatedlattice and g is a unary operation that is order-reversing, i.e., x ≤ y ⇒ g(y) ≤ g(x).

We shall use the following properties of residuated lattices. The operation ◦ is order-preserving in each argument,while → is order-preserving in its second argument and order-reversing in its first argument. Furthermore, the followinghold:

1 → x = x, x → 1 = 1, x → x = 1,

x ◦ (x → y) ≤ y,

(x ◦ y) → z = x → (y → z),

x ◦ y ≤ x ∧ y,

x ≤ y ⇔ x → y = 1.

60 W. Morton, C.J. van Alten / Fuzzy Sets and Systems 222 (2013) 58 –77

The operation ◦ distributes over all existing joins, i.e., if∨

i∈I xi exists, then y ◦∨i∈I xi = ∨

i∈I y ◦ xi . The residualoperation → satisfies:

x → y =∨

{z : x ◦ z ≤ y} = max{z : x ◦ z ≤ y}.The following hold in all MTL-algebras:

x → (y ∨ z) = (x → y) ∨ (x → z),

x → (y ∧ z) = (x → y) ∧ (x → z),

(x ∨ y) → z = (x → z) ∧ (y → z),

(x ∧ y) → z = (x → z) ∨ (y → z).

We use the abbreviation ¬x := x → 0, which defines a negation operation. From the properties listed above it followsthat the negation is order-reversing, ¬0 = 1 and ¬1 = 0. Inductively define the terms xn , for n ∈ N, as follows: x0 = 1and xn+1 = x ◦ xn .

3. Axiomatization of (reverse) modal MTL-algebras

Let A = 〈A, ◦, →, ∧, ∨, f, 0, 1〉 be a fixed modal residuated lattice.

Definition 3.1. A subset F of A is a filter 1 of A if: 1 ∈ F , F is upward closed, closed under ◦, and f (c) → f (c◦a) ∈ Fwhenever a ∈ F and c ∈ A.

The set of all filters of A, ordered by inclusion, forms a complete lattice, denoted by Fil A. If � is a congruence onA and a ∈ A, we use [a]� to denote the congruence class of a with respect to �.

Proposition 3.2. The congruence lattice of A, Con A, is isomorphic to the filter lattice, Fil A; the isomorphisms aregiven by: ��[1]� and F��F = {(a, b) : a → b, b → a ∈ F}.

Proof. Since the result is known for residuated lattices, we need only to check that the result extends to the operationf. Let F be a filter of A. To see that �F is compatible with f, suppose (a, b) ∈ �F , i.e., a → b, b → a ∈ F . Sincea ◦ (a → b) ≤ b and f is order preserving it follows that f (a ◦ (a → b)) ≤ f (b). By the definition of a filter,f (a) → f (a ◦ (a → b)) ∈ F , and f (a) → f (a ◦ (a → b)) ≤ f (a) → f (b), so f (a) → f (b) ∈ F . Similarly, we canshow that f (b) → f (a) ∈ F , hence ( f (a), f (b)) ∈ �F . Next, let � be a congruence on A. To see that [1]� is a filter,suppose a ∈ [1]� and c ∈ A. Then (a, 1) ∈ �, so ( f (c) → f (c◦a), f (c) → f (c◦1)) ∈ �, i.e., ( f (c) → f (c◦a), 1) ∈ �.Thus, f (c) → f (c ◦ a) ∈ [1]�. �

It is easy to see that a residuated lattice whose underlying lattice order is linear is an MTL-algebra; we call suchalgebras MTL-chains. Every MTL-algebra is a subdirect product of MTL-chains (see, e.g., [8]); thus, the MTL-chainsgenerate the variety of MTL-algebras. To extend this result to the modal case, we make the following definitions.

Definition 3.3.

(i) A modal MTL-chain is a modal residuated lattice whose underlying lattice order is linear.(ii) A modal MTL-algebra is any algebra in the variety generated by modal MTL-chains.

Since f is order-preserving, the following identity holds in all modal MTL-chains, and hence also modalMTL-algebras:

f (x ∨ y) = f (x) ∨ f (y). (1)

The identity f (x ∧ y) = f (x) ∧ f (y) also holds. Note that the order-preserving property of f can be inferred from (1).

1 Sometimes called an ‘ideal’, as it satisfies the notion of ideal from [15].


Observe that the variety of modal MTL-algebras is congruence distributive since its algebras contain a lattice reduct,so Jonsson’s theorem (see, e.g., [16]) says that every subdirectly irreducible modal MTL-algebra is a homomorphicimage of a subalgebra of an ultraproduct of modal MTL-chains. But the class of modal MTL-chains is closed underultraproducts, subalgebras and homomorphic images. Thus, the variety generated by modal MTL-chains may beobtained by taking subdirect products only. In particular, this means that the variety coincides with the quasivarietygenerated by modal MTL-chains. In order to axiomatize the variety of modal MTL-algebras it is sufficient, therefore,to determine identities that a modal residuated lattice must satisfy to be embeddable into a product of modal MTL-chains, and hence a subdirect product of modal MTL-chains. From the theory of universal algebra we know that if theintersection of a set of congruences of an algebra is the trivial congruence, then the algebra is a subdirect product of theassociated factor algebras. Thus, we shall characterize the congruences of a modal MTL-algebra for which the factoralgebra is a modal MTL-chain. Since the congruence lattice of a modal MTL-algebra is isomorphic to the filter lattice,we characterize the filters F for which A/�F is a modal MTL-chain (see, e.g., [8]).

A filter of A is called prime if, for all a, b ∈ A, a ∨ b ∈ F implies that at least one of a ∈ F or b ∈ F .The following result can be obtained for modal MTL-algebras in the same way that it is obtained for

MTL-algebras.

Lemma 3.4. If A satisfies the prelinearity identity, then a filter F of A is prime if, and only if, A/�F is linearlyordered.

Lemma 3.5. The variety of modal MTL-algebras satisfies

x ∨ z = 1 ⇒ ( f (y) → f (x ◦ y)) ∨ z = 1. (2)

Proof. If a ∨ c = 1 in a chain, then either a = 1, in which case f (b) → f (a ◦ b) = 1, or c = 1, in which case( f (b) → f (a ◦ b)) ∨ c = 1. Thus, by Jonsson’s theorem every modal MTL-algebra satisfies (2). �

Recall that a nonempty subset K ⊆ A is a lattice-ideal of A if K is downward- and join-closed.

Definition 3.6. For every lattice-ideal K of A define:

FK = {a ∈ A : (∃c ∈ K )(a ∨ c = 1)}.

Lemma 3.7. Suppose A satisfies the quasi-identity (2).

(i) If K is a lattice-ideal of A, then FK is a filter of A.(ii) If K is a maximal (proper) lattice-ideal of A, then FK is prime.

Proof. (i) It is clear that 1 ∈ FK . Let a, b ∈ A. If a, b ∈ FK , then there exist c1, c2 ∈ K such that a ∨ c1 = 1 andb ∨ c2 = 1. To see that a ◦b ∈ FK , observe that c1 ∨ c2 ∈ K and (a ∨ c1)◦ (b ∨ c2) = 1. After distributing the left-handside, we obtain (a ◦b)∨ (c1 ∨c2) = 1, so a ◦b ∈ FK . If a ∈ FK and a ≤ b, then there exists c ∈ K such that a ∨c = 1,hence also b ∨ c = 1, so b ∈ FK . If a ∈ FK , say a ∨ c = 1 for some c ∈ K , then ( f (b) → f (a ◦ b)) ∨ c = 1, by (2),so f (b) → f (a ◦ b) ∈ FK .

(ii) Suppose K is a maximal lattice-ideal of A and a ∨ b ∈ FK . Then there exists c ∈ K such that (a ∨ b) ∨ c = 1.Suppose that a /∈ FK . Then a ∨ c � 1 for every c ∈ K so the lattice-ideal of A generated by K ∪ {a} is a proper lattice-ideal containing K. Since K is maximal, we must have a ∈ K . Thus, a ∨ c ∈ K hence b ∈ FK since b ∨ (a ∨ c) = 1.A similar argument shows that if b /∈ FK , then a ∈ FK . Therefore, at least one of a ∈ FK or b ∈ FK . �

Theorem 3.8. The variety of modal MTL-algebras is axiomatized by the axioms of MTL-algebras together with (1)and (2).

Proof. Let A be a modal residuated lattice that satisfies the prelinearity identity, (1) and (2). Note that the identity (1)replaces the quasi-identity for the order-preserving property of f. For each a ∈ A\{1} there exists, by Zorn’s Lemma,a maximal lattice-ideal Ka of A with a ∈ Ka . Note that a /∈ FKa else there exists c ∈ Ka such that a ∨ c = 1, which


implies that 1 ∈ Ka , but Ka is proper. It follows that⋂{FKa : a ∈ A\{1}} = {1}.

By the isomorphism between the congruence lattice and filter lattice of A, it follows that A is a subdirect productof {A/�FKa

: a ∈ A\{1}}. Furthermore, from Lemmas 3.7 and 3.4 it follows that A is a subdirect product of modalMTL-chains. �

As we shall show, the quasi-identity (2) in the above results may be replaced by the following identity:

( f (y) → f ((x → z) ◦ y)) ∨ (z → x) = 1. (3)

Corollary 3.9. The variety of modal MTL-algebras is axiomatized by the axioms of MTL-algebras together with (1)and (3).

Proof. Since every chain satisfies: x ≤ z or z ≤ x , it follows easily that every modal MTL-chain, hence every modalMTL-algebra, satisfies (3). Suppose A is a modal residuated lattice that satisfies the prelinearity condition and (3);we show that it also satisfies (2), from which the result follows. If a, b, c ∈ A and a ∨ c = 1, then c = 1 → c =(a ∨ c) → c = (a → c) ∧ (c → c) = a → c. Similarly, a = c → a. By (3), ( f (b) → f (a ◦ b)) ∨ c = 1. �

We show, by example, that the quasi-identity (2) cannot be dropped from our axiomatization of modal MTL-algebras.Let A be the modal residuated lattice described as follows. The universe A = {0, a, b, 1}, the lattice order is given by0 ≤ a, b ≤ 1 with a and b incomparable. For x, y ∈ A, let x ◦A y = x ∧ y and x →A y = max{z : x ∧ z ≤ y}.Let f A be the operation defined by f A(0) = 0, f A(a) = b, f A(b) = a and f A(1) = 1. Then the f A-free reduct ofA is an MTL-algebra and hence a subdirect product of MTL-chains. In addition f distributes over joins, i.e., (1) holds,however, A does not satisfy (2). To see this, observe that a ∨ b = 1 but f A(1) →A f A(1 ◦A a) = 1 →A b = b. Ahas only two filters, namely {1} and {0, a, b, 1}, and therefore only two congruences. Thus, A is subdirectly irreducibleand hence cannot be represented as a subdirect product of modal MTL-chains.

The above results can be adapted to reverse modal MTL-algebras as follows. We omit the proofs as they are similarto those for modal MTL-algebras. Let B = 〈B, ◦, →, ∧, ∨, g, 0, 1〉 be a fixed reverse modal residuated lattice.

Definition 3.10. A subset F of B is a filter of B if: 1 ∈ F , F is upward closed, closed under ◦, and g(c ◦a) → g(c) ∈ Fwhenever a ∈ F and c ∈ B.

Corollary 3.11. The congruence lattice, Con B, is isomorphic to the filter lattice, Fil B; the isomorphisms are givenby: ��[1]� and F��F .

Definition 3.12.

(i) A reverse modal MTL-chain is a reverse modal residuated lattice whose underlying lattice order is linear.(ii) A reverse modal MTL-algebra is any algebra in the variety generated by reverse modal MTL-chains.

Since g is order-reversing, the identities

g(x ∨ y) = g(x) ∧ g(y) (4)

and g(x ∧ y) = g(x) ∨ g(y) hold in all reverse modal MTL-chains and MTL-algebras; either identity implies that g isorder-reversing. Consider the following quasi-identity and identity:

x ∨ z = 1 ⇒ (g(x ◦ y) → g(y)) ∨ z = 1, (5)

(g((x → z) ◦ y) → g(y)) ∨ (z → x) = 1. (6)

Theorem 3.13. The variety of reverse modal MTL-algebras is axiomatized by the axioms for MTL-algebras togetherwith (4), and (5) or (6).


We show, by example, that the quasi-identity (5) cannot be dropped from our axiomatization of reverse modal MTL-algebras. Let B be the reverse modal residuated lattice defined as follows. The g-free reduct of B is the same as thef-free-reduct in the previous example, so it is an MTL-algebra. Let gB be defined by gB(0) = 1, gB(a) = a, gB(b) = band gB(1) = 0. Then B satisfies (4) but not (5) since a ∨B b = 1 but

(gB(1 ◦B a) →B gB(1)) ∨B b = b.

Again, B has only two filters, namely {1} and {0, a, b, 1}, and therefore only two congruences. Thus, B is subdirectlyirreducible and hence cannot be represented as a subdirect product of reverse modal MTL-chains.

4. The MacNeille completion

Throughout this section A = 〈A, ◦, →, ∧, ∨, f, 0, 1〉 will be a fixed modal MTL-chain.We shall use the MacNeille completion to construct a complete lattice 〈F, ∧F, ∨F, 0F, 1F〉 into which the underlying

ordering on A embeds. We then extend the operations ◦, → and f on A to operations ◦F, →F and f F on F so thatthe resulting algebra F is a modal MTL-chain into which A embeds, called the ‘completion’ of A. The definitions of◦F and →F are described in [13] (and given below); here we define the modality f F. Thereafter, various preservationproperties of the completion of A into F are considered, that is, properties of A that are also satisfied by F. For example,if A is ‘involutive’, i.e., satisfies ¬¬x = x , then so is F. If A is ‘strict’, i.e., satisfies x ∧ (¬x) = 0, then so is F. Thefollowing identities are also preserved [13]: ¬(x ◦ y) ∨ ((x ∧ y) → (x ◦ y)) = 1 (weak nilpotent minimum), x ∨ ¬xn

(weak excluded middle), xn+1 = xn (n-contraction) and ¬xn+1 = ¬xn (weak n-contraction). We describe a generalscheme of inequalities s � t whose preservation by the completion is determined by the form of terms s and t. Theresults obtained here build on the results obtained in [13]; in particular, the scheme given there is extended to includeterms built up with an additional modal operator f.

4.1. The construction

We recall some definitions and results for the MacNeille completion (see [17], for example) as well as definitions of◦F and →F.

For K , L ⊆ A, let

K � = {a ∈ A : (∀b ∈ K ) a ≤ b},K u = {a ∈ A : (∀b ∈ K ) b ≤ a}.

We call X ⊆ A stable if X �u = X . Let F denote the set of all stable sets.

Lemma 4.1. For K , L ⊆ A and a, b ∈ A

(i) if a ≤ b, then {b}u ⊆ {a}u and {a}� ⊆ {b}�,(ii) if K ⊆ L , then Lu ⊆ K u and L� ⊆ K �,

(iii) K ⊆ K u�, K ⊆ K �u ,(iv) K u�u = K u , i.e., K u is stable,(v) K �u� = K �.

Lemma 4.2. For Ki ⊆ A, i ∈ I , and K , L ⊆ A

(i)⋂

i∈I K ui = (

⋃i∈I Ki )u ,

(ii)⋂

i∈I K �i = (

⋃i∈I Ki )�.

Thus, ⊇ is a complete lattice order on F; in fact, it is the MacNeille completion of A. We use∧F and

∨F for thelattice operations on F associated with ⊇: for Xi ∈ F , i ∈ I

F∨i∈I

Xi =⋂i∈I

Xi ,


F∧i∈I

=⋂

{Y ∈ F : (∀i ∈ I )Xi ⊆ Y }.

In the case that A is a chain, the completion 〈F, ⊇ 〉 is also a chain and, for X, Y ∈ F , X ∧F Y = X ∪ Y .

Lemma 4.3. Let X, Y ∈ F . If A is an MTL-chain, and K , L ⊆ A such that X = K u and Y = Lu , then

X ∧F Y = K u ∧F Lu = (K ∧ L)u,

where K ∧ L = {a ∧ b : a ∈ K and b ∈ L}.

Proof. Observe that for each a ∈ Y �, either Y = [a) or there exists b ∈ L such that a � b. To see this, suppose thatb < a for all b ∈ L; so a ∈ Lu = Y . Then a ∈ Y ∩ Y �, which is only possible if Y = [a).

Since F is a chain we may assume, without loss of generality, that X ⊇ Y , so X ∧F Y = X . Note that K ⊆ X � henceK ∧ L ⊆ X �, so X = X �u ⊆ (K ∧ L)u .

For the reverse inclusion, if K ⊆ K ∧ L , then (K ∧ L)u ⊆ K u = X . If K�K ∧ L , then there exists a ∈ K such thatfor all b ∈ L , a � b, i.e., b < a. But then a ∈ Lu = Y ⊆ X . Moreover, since a ∈ K ⊆ X �, it follows that a ∈ X ∩ X �,so X = [a). Now Y = X = [a) since a ∈ Y . Since a /∈ L and Lu = [a), there must be an increasing sequence (cn) inL such that

∨cn = a. But a ∈ K this implies that K ∧ L contains (cn), hence (K ∧ L)u = [a) = X . �

For K , L ⊆ A, define K ◦ L = {a ◦ b : a ∈ K and b ∈ L} and for X, Y ∈ F define X ◦F Y = (X � ◦ Y �)u .

Theorem 4.4 (van Alten [13]).

(i) Let X, Y ∈ F . If K , L ⊆ A is such that X = K u and Y = Lu , then X ◦F Y = K u ◦F Lu = (K ◦ L)u .(ii) The operation ◦F on F is residuated with respect to ⊇ and the residual of X, Y ∈ F is X →F Y = {a ∈ A :

(X � ◦ {a})u ⊇ Y }u .(iii) The algebra F = 〈F, ◦F, →F, ∧F, ∨F, 0F, 1F〉, where 0F = A and 1F = {1}, is a complete MTL-chain and the

map a�{a}u is an embedding of A into F that preserves all existing meets and joins in A.

Define a unary operation (the modal operator) on F as follows: for X ∈ F

f F(X ) = { f (a) : a ∈ X �}u .

Lemma 4.5. The operation f F is order-preserving on F.

Proof. Let X, Y ∈ F such that X ⊇ Y . Then X � ⊆ Y �, so { f (a) : a ∈ X �} ⊆ { f (a) : a ∈ Y �}, hence { f (a) :a ∈ X �}u ⊇ { f (a) : a ∈ Y �}u , i.e., f F(X ) ⊇ f F(Y ). �

Lemma 4.6. The embedding a�{a}u of A into F preserves f, i.e., for a ∈ A, we have f F({a}u) = { f (a)}u .

Proof. We have f F({a}u) = { f (b) : b ∈ {a}u�}u = { f (b) : b � a}u , which is equal to { f (a)}u since f is order-preserving. �

Theorem 4.7. The algebra F = 〈F, ◦F, →F, ∧F, ∨F, f F, 0F, 1F〉 is a complete modal MTL-chain and the mapa�{a}u an embedding of A into F that preserves all existing meets and joins in A.

4.2. Preservation theorems

We say that a modal MTL-identity is preserved by the completion if, whenever a modal MTL-chain A satisfies theidentity, then the completion F, obtained from A as in the previous section, also satisfies the identity. We similarlydefine what it means for a quasi-identity, inequality or any other property to be preserved by the completion. Notethat for MTL-chains, an inequality s � t is equivalent to an identity s = s ∧ t or t = s ∨ t or s → t = 1. Following


van Alten [13], we use ‘approximation terms’ to obtain preservation results. This is related to the methods used in[11,12] to obtain preservation results in completions of modal algebras. (See also [18] for similar results on orderedalgebras.)

Let t be a (modal MTL-) term. If the variables occurring in t are in the sequence �x = x1, . . . , xn , then we denote thisby t(x1, . . . , xn) or t(�x). If �a = a1, . . . , an is a sequence of elements of A, then we write t(�a) to denote the evaluationof the term t in A under the assignment xi�ai . If �X = X1, . . . , Xn is a sequence of elements of F, then we write tF( �X )to denote the evaluation of the term t in F under the assignment xi�Xi . Where a term t(�x) and either �a ∈ A or �X ∈ Fare given, it is assumed that �x and �a or �X are sequences of the same length.

Given a term t(�x) and �X ∈ F , the evaluation of tF( �X ) can be approximated by the set of t(�a)’s where each ai ∈ X �i ,

which we write as {t(�a) : �a ∈ �X �}. Since this set is not stable it is necessary to close it in F, which can be done in twoways, namely

t∃( �X ) = {t(�a) : �a ∈ �X �}u,

t∀( �X ) = {t(�a) : �a ∈ �X �}�u,

which are then our approximations to tF( �X ). We say that t(�x) is

∃ -stable if tF( �X ) = t∃( �X ),

∃ -expanding if tF( �X ) ⊆ t∃( �X ),

∃ -contracting if tF( �X ) ⊇ t∃( �X ),

∀ -stable if tF( �X ) = t∀( �X ),

∀ -expanding if tF( �X ) ⊆ t∀( �X ),

∀ -contracting if tF( �X ) ⊇ t∀( �X ) for all �X ∈ F.

If A satisfies the inequality s(�x) ≤ t(�x), then s∃( �X ) ⊇ t∃( �X ) and s∀( �X ) ⊇ t∀( �X ) for all �X ∈ F . Thus, for example, if Asatisfies s(�x) � t(�x) and s is ∃-contracting and t is ∃-expanding, then for any �X ∈ F , sF( �X ) ⊇ s∃( �X ) ⊇ t∃( �X ) ⊇ tF( �X ),i.e., sF( �X ) � FtF( �X ). More generally we have the following result.

Theorem 4.8 (van Alten [14]).

(i) If both s and t are ∃-stable terms, or both s and t are ∀-stable terms, then s = t is preserved by the completion.(ii) If s is an ∃-contracting term and t is an ∃-expanding term, or s is a ∀-contracting term and t is a ∀-expanding term,

then s � t is preserved by the completion.

In the remainder of this section we describe some classes of (∃-, ∀-) stable, expanding and contracting terms, whichmay be used in conjunction with the above theorem to obtain a class of identities preserved by the MacNeille completion.We will need the following notions.

Definition 4.9. The sets of positive and negative terms are the smallest sets of terms closed under the following rules:

(i) 0 and 1 are both positive and negative;(ii) the term t(x) = x is positive for each variable x;

(iii) if s is negative and t is positive, then s → t is positive and t → s is negative;(iv) if s(x1, . . . , xn) is a {◦, ∧, ∨, f }-term and each ti is positive (resp. negative) terms, then s(t1, . . . , tn) is positive

(resp. negative).

We show that every positive term is ∃-expanding, while every negative term is ∃-contracting. However, in orderto classify positive and negative terms in this way, it is useful to show the stronger result that every negative termis ∀-contracting, which implies that it is ∃-contracting. Note also that a (∃-, ∀-) stable term is both contracting andexpanding. We recall some results regarding MTL-algebras.

Lemma 4.10 (van Alten [13]).

(i) The constants 1 and 0, every variable x, every {◦, ∨}-term and every {∧, ∨}-term is ∃-stable.


(ii) If s is ∃-stable, then ¬s is ∀-stable.(iii) If s1 and s2 are ∃-stable (resp., ∃-contracting, ∃-expanding) terms, then s1 ∨ s2 is ∃-stable (resp., ∃-contracting,

∃-expanding).(iv) If s1 and s2 are ∃-stable (resp., ∃-contracting) terms that have no variables in common, then s1 ◦ s2 is ∃-stable

(resp., ∃-contracting).(v) If t(�x) is an ∃-expanding term and y is a variable not in �x , then t(�x) → y is ∃-contracting.

It was shown in [14] that every positive term in the language {◦, →, ∧, ∨, 1, 0} is ∃-expanding and every suchnegative term is ∀-contracting (hence also ∃-contracting). We shall extend these results to include the modal operator.

Lemma 4.11. If t is an ∃-expanding term, then f (t) is ∃-expanding.

Proof. Let s(�x) = f (t(�x)) and �X ∈ F . Then

tF( �X ) ⊆ t∃( �X ) ⇒ (tF( �X ))� ⊇ (t∃( �X ))�

⇒ { f (�a) : �a ∈ tF( �X )�} ⊇ { f (�a) : �a ∈ t∃( �X )�}⇒ { f (�a) : �a ∈ tF( �X )�}u ⊆ { f (�a) : �a ∈ t∃( �X )�}u

⇒ f F(tF( �X )) ⊆ { f (�a) : a ∈ {t(�b) : b ∈ �X �}u�}u .

Since {t(�b) : b ∈ �X �} ⊆ {t(�b) : b ∈ �X �}u� we have

{ f (�a) : a ∈ {t(�b) : b ∈ �X �}}u ⊇ { f (�a) : a ∈ {t(�b) : b ∈ �X �}u�}u

and hence

sF( �X ) = f F(tF( �X )) ⊆ { f (�a) : �a ∈ t∃( �X )�}u ⊆ { f (t(�a)) : a ∈ �X �}u = s∃( �X ). �

Lemma 4.12. If t is a ∀-contracting term, then f (t) is ∀-contracting.

Proof. Let s(�x) = f (t(�x)) and �X ∈ F . Then

sF( �X ) = f F(tF( �X )) ⊇ f F(t∀( �X ))

= { f (b) : b ∈ {t(�a) : �a ∈ �X �}�u�}u

= { f (b) : b ∈ {t(�a) : �a ∈ �X �}�}u,

which we must show to include s∀( �X ) = { f (t(�a)) : �a ∈ �X �}�u . Let b ∈ {t(�a) : �a ∈ �X �}�, so b � t(�a) for all �a ∈ �X �,hence also f (b) � f (t(�a)) for all �a ∈ �X �. Thus, f (b) ∈ { f (t(�a)) : �a ∈ �X �}�, from which we deduce s∀( �X ) ⊆ sF( �X )after taking upper bounds. �

Combining Proposition 13.27 in [14] with Lemmas 4.11 and 4.12 gives the following result.

Proposition 4.13. Every positive term is ∃-expanding and every negative term is ∀-contracting, hence also∃-contracting.

Consequently, any inequality s � t in which s is negative and t is positive is preserved by the completion. Observethat such an inequality is equivalent to 1 = s → t , and s → t is positive. We next consider ∃-stable terms; since suchterms are both ∃-contracting and ∃-expanding, they may appear on either side of the inequality.

Lemma 4.14. If s1 and s2 are ∃-stable (resp., ∃-contracting) terms that have no variables in common, then s1 ∧ s2 is∃-stable (resp., ∃-contracting).

Proof. Let t(�x, �y) = s1(�x) ∧ s2(�y), where �x and �y have no variables in common, and �X , �Y ∈ F . Then, usingLemma 4.3

tF( �X , �Y ) = sF1 ( �X ) ∧F sF

2 ( �Y ) = (resp., ⊇)s∃1 ( �X ) ∧F s∃

2 ( �Y )


= {s1(�a) : �a ∈ �X �}u ∧F {s2(�b) : �b ∈ �Y �}u

= {s1(�a) ∧ s2(�b) : �a, �b ∈ �X �, �Y �}u = t∃( �X , �Y ). �

Lemma 4.15. Let t1 and t2 be terms that are order-preserving in each co-ordinate and have exactly one variable incommon, say x, and let �y1 and �y2 be the remainder of the variables occurring in t1 and t2, respectively. If s(x, �y1, �y2) =t1(x, �y1)◦ t2(x, �y2) or t1(x, �y1)∧ t2(x, �y2) and both t1 and t2 are ∃-stable (resp., ∃-contracting), then s is ∃-stable (resp.,∃-contracting).

Proof. For X, �Y1, �Y2 ∈ F and s = t1 ◦ t2, we have

sF(X, �Y1, �Y2) = tF1 (X, �Y1) ◦F tF

2 (X, �Y2) = (resp., ⊇) t∃1 (X, �Y1) ◦F t∃2 (X, �Y2).

Using Theorem 4.4(i)

t∃1 (X, �Y1) ◦F t∃2 (X, �Y2) = {t1(a, �b) : a ∈ X �, �b ∈ �Y1�}u ◦F {t2(c, �d) : c ∈ X �, �d ∈ �Y2

�}u

= {t1(a, �b) ◦ t2(c, �d) : a, c ∈ X �, �b ∈ �Y1�, �d ∈ �Y2

�}u

⊆ {t1(a, �b) ◦ t2(a, �d) : a ∈ X �, �b ∈ �Y1�, �d ∈ �Y2

�}u

= s∃(X, �Y1, �Y2).

For the inclusion in the other direction, let e ∈ s∃(X, �Y1, �Y2), i.e., t1(a, �b) ◦ t2(a, �d) ≤ e for all a ∈ X �, �b ∈ �Y1�

and�d ∈ �Y2

�and let a, c ∈ X �. Since A is a chain either a ≤ c or c ≤ a. Suppose c ≤ a; then t2(c, �d) ≤ t2(a, �d) for all

�b ∈ �Y1�

and �d ∈ �Y2�

since t2 is order-preserving in each co-ordinate. Then t1(a, �b) ◦ t2(c, �d) ≤ t1(a, �b) ◦ t2(a, �d) ≤ e,so e ∈ {t1(a, �b) ◦ t2(c, �d) : a, c ∈ X �, �b ∈ �Y1, �d ∈ �Y2}u , as required. If a ≤ c the proof is similar. The proof fors = t1 ∧ t2 follows similarly, using Lemma 4.3. �

By the definition of f F it is immediate that f (x) is ∃-stable. Thus, a particular consequence of the above lemma isthat ( f (x))n is ∃-stable for each n � 1, as are ( f (x1))n1 ◦ · · · ◦ ( f (xk))nk and ( f (x1))n1 ∧ · · · ∧ ( f (xk))nk , where eachxi is a variable, 0 or 1 and each ni � 1. More generally, we have the following:

Lemma 4.16. The following terms are ∃-stable: any term built up inductively from 0, 1, x, f (x), for any variable x, bytaking ◦ or ∧ of terms that share at most one variable, or any ∨.

Combining the above results we obtain the theorem below.

Theorem 4.17. An inequality s � t is preserved by the completion if t is any positive term and s is

(i) a negative term,(ii) a ∃-stable term (as in Lemma 4.16),

(iii) a term t(�x) → y, where t(�x) is ∃-expanding term and y is a variable not in �x ,(iv) any term built up inductively from terms in (i-iii) by taking ◦’s or ∧’s of any terms that have no variables in

common or any ∨’s.

4.3. Special subclasses of modal MTL-chains

In [4], Ciabattoni et al. study the addition of truth stresser modalities to MTL and its extensions. When consideringthe semantics of these logics a number of classes of algebras are studied, all of which are subclasses of modal MTL-algebras as considered here. The various logics studied in [4] are the monoidal t-norm logic (MTL), the involutivemonoidal t-norm logic (IMTL) and the strict monoidal t-norm logic (SMTL). The last two of the aforementioned logicsaxiomatize t-norm logics whose negations are, respectively, involutive and strict. Furthermore, t-norm logics satisfyingan n-contraction property were also studied—with involutive negations (CnIMTL) and without (CnMTL).


Let Logics = {MTL, IMTL, SMTL} ∪ {CnMTL : n ≥ 2}∪ {CnIMTL : n ≥ 3}. For L ∈ Logics, an L-algebra is an MTL-algebra such that all the MTL-axioms as well as theadditional axioms of the logic L are all valid. Then, for L ∈ Logics, an

• LKr -algebra is a modal MTL-algebra satisfying:

(i) f (x → y) ≤ f (x) → f (y),(ii) f (x ∨ y) = f (x) ∨ f (y),

(iii) f (1) = 1.

• LKTr -algebra is an LKr -algebra additionally satisfying: f (x) ≤ x .• LS4r -algebra is an LKTr -algebra additionally satisfying: f ( f (x)) � f (x) (hence also f ( f (x)) = f (x)).• L!r -algebra is an LS4r -algebra additionally satisfying: f (x) ◦ f (x) = f (x).• Lr

�-algebra is an LS4r -algebra additionally satisfying: f (x) ∨ ( f (x) → 0) = 1.

It is shown in [4] that an LKr -algebra is a subdirect product of linearly ordered algebras, hence the quasi-identity (2)holds in such algebras.

Observe that in a modal MTL-algebra, the identity

f (x) ◦ f (y) � f (x ◦ y) (7)

is equivalent to the identity

f (x → y) ≤ f (x) → f (y). (8)

To see that (7) implies (8) we recall that x ◦ (x → y) ≤ y. Since f is order-preserving and by (7), f (x) ◦ f (x →y) � f (x ◦ (x → y)) ≤ f (y), hence f (x → y) ≤ f (x) → f (y). Conversely, to see that (8) implies (7), recallthat x � y → (x ◦ y). Since f is order-preserving and by (8), f (x) ≤ f (y → (x ◦ y)) � f (y) → f (x ◦ y), hencef (x) ◦ f (y) � f (x ◦ y).

Corollary 4.18. If A is a linearly ordered LKr -, LKTr -, LS4r -, L!r - or Lr�-algebra, then so is F.

The proof is a straightforward consequence of Theorem 4.17.

4.4. Left-continuous modalities

If, in addition to being order-preserving, we assume that the operation f is left-continuous, a wider class of propertiesis preserved by the completion. Firstly, the left-continuity of an operator is preserved by the completion.

Lemma 4.19. Let K ⊆ A. If d ∈ K u� such that d � e for any e ∈ K , i.e., e < d for all e ∈ K , then d = ∨K .

Proof. Suppose d ∈ K u� such that e ≤ d for every e ∈ K . Then d is an upper bound for K. Let a ∈ K u , then d ≤ asince d ∈ K u�. Thus, d is the least upper bound for K. �

Lemma 4.20. If f is left-continuous, then for all K ⊆ A

(i) f F(K u) = ( f (K ))u , where f (K ) = { f (a) : a ∈ K },(ii) if X = K u is stable, then f F(X ) = ( f (K ))u .

Proof. We shall prove part (i); part (ii) then follows directly. Since K ⊆ K u� we have f (K ) ⊆ f (K u�) and also( f (K u�))u ⊆ ( f (K ))u , i.e., f F(K u) ⊆ ( f (K ))u . Conversely, let a ∈ ( f (K ))u , i.e., a ≥ f (b) for every b ∈ K , and letc ∈ K u�. If c ≤ d for some d ∈ K , then f (c) ≤ f (d) ≤ a since a ∈ ( f (K ))u . If not, then d < c for every d ∈ K ,so c = ∨

K , by Lemma 4.19. Thus, f (c) = f (∨

K ) = ∨f (K ) by the left-continuity. But a is an upper bound for

f (K ), so∨

f (K ) ≤ a and, in particular, f (c) ≤ a. In either case we find that f (c) ≤ a and therefore a ∈ ( f (K u�))u .We conclude that ( f (K ))u ⊆ ( f (K u�))u . �


Lemma 4.21. If f is left-continuous, i.e., f (∨

i ai ) = ∨i ( f (ai )) whenever

∨ai exists, then f F is also left-continuous.

Proof. By Lemma 4.2(i), f F(∨F

i Xi ) = f F(⋂

i Xi ) = f F(⋂

i X �ui ) = f F((

⋃i X �

i )u). Then, by Lemma 4.20,

f F(∨F

i Xi ) = f (⋃

i X �i )u . Finally, by Lemma 4.2(i), f F(

∨Fi Xi ) = (

⋃i f (X �

i ))u = ⋂i f (X �

i )u = ∨Fi f F(Xi ). �

It is well known that residuated unary operations distribute over all existing joins. Examples of left-continuousoperators therefore include all residuated operators. However, only a partial converse holds: if the underlying lattice ofan algebra A is complete, then a unary operation f on A that distributes over all joins is residuated. Hence, a consequenceof the operator f being left-continuous is that f F is residuated.

Lemma 4.22. If f is left-continuous and s is an ∃-contracting term, then f (s) is also ∃-contracting.

Proof. Let t = f (s(�x)) and X ∈ F . Then

tF( �X ) = f F(sF( �X )) ⊇ f F(s∃( �X )) = f F({s(�a) : �a ∈ �X �}u)

= { f (s(�a)) : �a ∈ �X �}u by Lemma 4.20

= t∃( �X ). �

Corollary 4.23. If f is left-continuous and s is an ∃-stable term, then f (s) is also ∃-stable.

Proof. An ∃-stable term is both ∃-contracting and ∃-expanding. The result follows from Lemmas 4.11 and 4.22. �

Lemma 4.24. If f is left-continuous, then the following terms are ∃-stable: any term built up inductively from 0, 1, x,f (x), for any variable x, by taking ◦ or ∧ of terms that share at most one variable, taking f of any term or ∨ of any twoterms.

Combining the above results we obtain the theorem below.

Theorem 4.25. If f is left continuous, then an inequality s � t is preserved by the completion if t is any positive termand s is

(i) a negative term,(ii) a ∃-stable term (as in Lemma 4.24),

(iii) a term t(�x) → y, where t(�x) is ∃-expanding term and y is a variable not in �x ,(iv) a term built up inductively from terms in (i–iii) by taking ◦ or ∧ of any terms have no variables in common, any f

or any ∨.

4.5. Order-reversing modalities

If A = 〈A, ∧, ∨, ◦, →, g, 0, 1〉 is a reverse modal MTL-algebra, then we define the unary operation gF on F by

gF(X ) = {g(a) : a ∈ X �}�u

and let F = 〈F, ◦F, ∧F, ∨F, gF, 0F, 1F〉, where the other operations are as described in Section 4.1.

Lemma 4.26. The embedding a�{a}u of A into F preserves g, i.e., for a ∈ A, gF({a}u) = {g(a)}u .

Lemma 4.27. The operation gF is order-reversing.

The proofs of Lemmas 4.26 and 4.27 are similar to the proofs of Lemmas 4.6 and 4.5, respectively.

Proposition 4.28. The algebra F = 〈F, ◦F, →F, ∧F, ∨F, gF, 0F, 1F〉 is a complete reverse modal MTL-algebra andthe map a�{a}u is an embedding of A into F that preserves all existing meets and joins in A.


We now turn our attention to properties preserved by this construction. Clearly, g gives the impression of a negativeterm and, indeed, we show that this is the case.

Lemma 4.29. If s is an ∃-expanding term, then g(s) is ∀-contracting.

Proof. Let t = g(s(�x)) and �X ∈ F . Since s is ∃-expanding and gF is order-reversing

tF( �X ) = gF(sF( �X )) ⊇ gF(s∃( �X )) = {g(b) : b ∈ {s(�a) : �a ∈ �X �}u�}�u,

which we must show to be a superset of t∀( �X ) = {g(s(�a)) : �a ∈ X �}�u . Since {s(�a) : �a ∈ �X �} ⊆ {s(�a) : �a ∈ �X �}u�,{g(s(�a)) : �a ∈ �X �} ⊆ {g(b) : b ∈ {s(�a) : �a ∈ �X �}u�} and the result follows after taking the �u-closures. �

Lemma 4.30. If s is a ∀-contracting term, then g(s) is ∃-expanding.

Proof. Let t = g(s(�x)) and �X ∈ F . Since s is ∀-contracting and gF is order-reversing

tF( �X ) = gF(sF( �X )) ⊆ gF(s∀( �X )) = {g(b) : b ∈ {s(�a) : �a ∈ �X �}�u�}�u

= {g(b) : b ∈ {s(�a) : �a ∈ �X �}�}�u,

which we must show to be a subset of t∃( �X ) = {g(s(�a)) : �a ∈ X �}u . Let �a ∈ �X � and b ∈ {s(�a) : �a ∈ �X �}�. Thenb � s(�a) so g(s(�a)) � g(b), hence {g(s(�a)) : �a ∈ X �}� ⊆ {g(b) : b ∈ {s(�a) : �a ∈ �X �}�}� and the result follows aftertaking u’s. �

Extend the notion of positive and negative terms to the language of reverse modal MTL-algebras by defining g(s)negative whenever s is positive and g(s) positive whenever s is negative. The above two results give the following.

Proposition 4.31. Every positive term is ∃-expanding and every negative term is ∀-contracting, hence also∃-contracting.

Theorem 4.32. If g is a reverse modality, an inequality s � t is preserved by the completion if t is any positive termand s is

(i) a negative term,(ii) a ∃-stable term on the language {◦, ∧, ∨, 0, 1} (as in Lemma 4.16),

(iii) a term t(�x) → y, where t(�x) is ∃-expanding term and y is a variable not in �x ,(iv) a term built up inductively from terms in (i-iii) by taking ◦ or ∧ of any terms have no variables in common,

or any ∨.

In addition, we have the following preservation result.

Proposition 4.33. If A satisfies x � g(g(x)) or g(g(x)) � x then F satisfies the same. Thus, if g is involutive, then sois gF.

Proof. The inequality x ≤ g(g(x)) is preserved by Theorem 4.32 since x is ∃-stable and g(g(x)) is positive.For X ∈ F

gF(gF(X )) = {g(b) : b ∈ {g(a) : a ∈ X �}�}�u,

where we have used the fact that K �u� = K � for any K ⊆ A. Suppose A satisfies g(g(x)) � x and let X ∈ F . Letc ∈ {g(b) : b ∈ {g(a) : a ∈ X �}�}�, i.e., c � g(b) for all b ∈ {g(a) : a ∈ X �}�. If d ∈ X , then a � d for alla ∈ X � hence g(d) � g(a) so g(d) ∈ {g(a) : a ∈ X �}�. Thus, c � g(g(d)) � d , so c ∈ X �, hence {g(b) : b ∈ {g(a) :a ∈ X �}�u�}� ⊆ X � and the result follows after taking the superscript u’s. �


5. The finite embeddability property (FEP)

Let A = 〈A, {tAi : i ∈ I }〉 be (any type of) algebra and let B be any subset of A. The partial subalgebra B of A with

domain B is the partial algebra 〈B, {tBi : i ∈ I }〉, where for i ∈ I , ti k-ary, b1, . . . , bk ∈ B

tBi (ba, . . . , bk) =

{tAi (b1, . . . , bk), tA

i (b1, . . . , bk) ∈ B,

undefined tAi (b1, . . . , bk) /∈ B.

By an embedding of a partial subalgebra B into an algebra C we mean a 1−1 map � : B → C that preserves allexisting operations; i.e., if ti is some k-ary operation such that tB

i (b1, . . . , bk) is defined for b1, . . . , bk ∈ B, then�(tB

i (b1, . . . , bk)) = tCi (�(b1), . . . , �(bk)).

A class K of algebras has the finite embeddability property (FEP, for short) if every finite partial subalgebra of somemember of K can be embedded into some finite member of K.

If a class K of algebras has the FEP, then every universal sentence that fails in K will fail in some finite memberof the class K. Therefore, the universal theory of K is decidable if K is finitely axiomatized. On the other hand, for a(quasi-)variety K of algebras, the FEP implies that K is generated as a quasivariety by its finite members. For a moreextensive background on the FEP the reader can refer to [19] or [20].

We first describe the construction for obtaining the FEP for MTL-chains in [13] and then extend this to modalMTL-chains. Thereafter, as for the MacNeille completion, we consider the preservation of various identities underthe FEP construction, and thereby obtain the FEP for various classes of modal MTL-chains. Any variety V of modalMTL-algebras is generated by its subclass of modal MTL-chains and, in fact, the FEP for V follows from the FEP forits subclass of modal MTL-chains [13]. Thus, we need only to consider modal MTL-chains.

5.1. The FEP construction

Throughout the following sections let A be a fixed modal MTL-chain, B a finite subset of A containing 1 and 0, andB the partial subalgebra of A with domain B.

Let M and M∗ be two sets satisfying:

(M1) B ⊆ M ⊆ A and B ⊆ M∗ ⊆ A,(M2) M is closed under ◦,(M3) if a ∈ M and b ∈ M∗, then a → b ∈ M∗,(M4) 〈M, ≤ 〉 is reverse well-ordered and 〈M∗, ≤ 〉 is well-ordered.

The first choice of M and M∗ we shall consider is M the {◦}-closure of B in A and M∗ = {a → b : a ∈ M, b ∈ B}.By definition, B ⊆ M , and B ⊆ M∗ since b ∈ B can be written as 1 → b. Note that if c = a → b ∈ M∗ for a ∈ Mand b ∈ B, then for any d ∈ M , d → c = (a ◦ d) → b ∈ M∗. 〈M, ≤ 〉 is reverse well-ordered and 〈M∗, ≤ 〉 arewell-ordered is proved in [13].

For M and M∗ satisfying (M1–M4) define, for each a ∈ A,

a� = max{b ∈ M : b ≤ a}, au = min{c ∈ M∗ : a ≤ c}.

The well-ordering and reverse well-ordering assumptions ensure that the relevant maxima and minima of the abovesets exist. The maps u and � are both order-preserving. In addition, the following properties are easily derived:

Lemma 5.1. For any a ∈ M , c, d ∈ M∗ and e ∈ A

(i) a ≤ au� and c�u ≤ c,(ii) a ≤ c iff au ≤ c,

(iii) a ≤ c iff a ≤ c�,(iv) au�u = au , c�u� = c�,(v) e�u�u = e�u .


An element c ∈ M∗ is said to be stable if c = c�u . Let C denote the set of stable elements. By Lemma 5.1(iv) and(v), au is stable for a ∈ M and e�u is stable for e ∈ A. By the well-ordering and reverse well-ordering assumptions, itfollows that C is a finite set [13].

We define an MTL-chain with universe C. Since C ⊆ A, the order on A, restricted to C is linear and defines latticeoperations ∧C and ∨C which coincide with the corresponding operations on B. The product operation is defined forc, d ∈ C by:

c ◦C d = (c� ◦ d�)u .

The following property holds: If c, d ∈ C and a, b ∈ M for which c = au and d = bu , then c ◦C d = (a ◦ b)u . Usingthis property one can show that ◦C is associative, commutative, has identity 1 and is residuated with respect to � ; forc, d ∈ C , the residual is

c →C d = (c� → d)�u .

The algebra C = 〈C, ◦C, →C, ∧C, ∨C, 0, 1〉 is therefore a finite MTL-chain and the identity map is an embedding ofB into C; that is, if a ◦B b is defined in B, then a ◦B b = a ◦C b, and, similarly, for →B.

In order to extend the above construction to modal MTL-chains, we define the operation f C on C by:

f C(c) = f (c�)�u .

For ease of notation, we assume that f binds more strongly than u and �.

Lemma 5.2. The identity embedding of B into C preserves the operation f, i.e., if f (b) ∈ B for some b ∈ B, thenf C(b) = f B(b) = f (b).

Proof. Note that, by the definitions of u and � and the fact that B ⊆ M ∩ M∗, if b ∈ B, then bu = b = b�. Thus, iff (b) ∈ B as well, we have f C(b) = f (b�)�u = f (b)�u = f (b). �

Lemma 5.3. f C is order-preserving, hence f C distributes over ∧C and ∨C.

Proof. If a, b ∈ C such that a ≤ b, then a� ≤ b� and also f (a�) ≤ f (b�). Thus, f C(a) = f (a�)�u ≤ f (b�)�u = f C(b).�

Theorem 5.4. For M, M∗ satisfying (M1–M4) the algebra C = 〈C, ◦C, →C, ∧C, ∨C, f C, 0, 1〉 is a finite modalMTL-chain and the identity map is an embedding of B into C.

Since a choice of M and M∗ exists that satisfies (M1–M4), namely, M the {◦}-closure of B in A and M∗ = {a → b :a ∈ M, b ∈ B}, we have the following result.

Theorem 5.5. The class of modal MTL-chains has the FEP, hence the variety of modal MTL-algebras has the FEP.

5.2. Preservation theorems for the FEP construction

As for completions, we identify a class of inequalities that are preserved by the FEP construction of the previoussubsection, meaning that if A satisfies s � t , then so does C. Clearly this implies that the subclass of modal MTL-chainscharacterized by s � t has the FEP, and hence the corresponding subvariety of modal MTL-algebras has the FEP. Weshall again make use of an approximation term to establish preservation of properties. We obtain different preservationresults depending on the choice of M and M∗. Generally, the larger the M is, the stronger the results are, but we mustalways ensure that 〈M, � 〉 is reverse well-ordered and 〈M∗, � 〉 is well-ordered.

Let t(�x) = t(x1, . . . , xn) be any term. If �c = c1, . . . , cn is a sequence of elements of C, then tC(�c) denotes theevaluation of t in C under the assignment xi�ci . Where the term t(�x) and �c ∈ C are given, �x and �c are assumed to besequences of the same length. If �c = c1, . . . , cn is a sequence of elements in C, then �c� denotes the sequence c�

1, . . . , c�n

of elements of M.


For each term s(�x) and �c ∈ C , define

s∗(�c) = s(�c�)�u .

Note that s∗(�c) ∈ C by Lemma 5.1. A term s(�x) is called:

∗-stable if sC(�c) = s∗(�c),

∗-expanding if sC(�c) ≥ s∗(�c),

∗-contracting if sC(�c) ≤ s∗(�c) for all �c ∈ C.

If A satisfies an inequality s � t , then s∗(�c) � t∗(�c) for all �c ∈ C . Thus, if s is ∗-contracting and t is ∗-expanding,then C satisfies s � t and the inequality is preserved. Observe that ∗-stable implies both ∗-contracting and ∗-expanding.This gives the following result.

Theorem 5.6 (van Alten [13]). The following hold for all terms s and t:

(i) If s and t are both ∗-stable, then s = t is preserved by the FEP construction.(ii) If s is ∗-contracting and t is ∗-expanding, then s ≤ t is preserved by the FEP construction.

The following proposition summarizes the results regarding MTL-terms.

Proposition 5.7 (van Alten [13]).

(i) If s(�x) is a {◦, ∨, ∧, 0, 1}-term and �c ∈ C , then sC(�c) = s(�c�)u and s is ∗-stable.(ii) If t(�x) = ¬s(�x), where s(�x) is a {◦, ∨, ∧, 0, 1}-term and �c ∈ C , then tC(�c) = t(�c�)�u and t is ∗-stable.

(iii) For all variables x1, . . . , xn, y, the term (x1 ◦ · · · ◦ xn) → y is ∗-contracting.(iv) If t1, . . . , tm are ∗-stable (resp., ∗-expanding, ∗-contracting) terms and s(y1, . . . , ym) is a {∧, ∨}-term, then

s(t1, . . . , tm) is ∗-stable (resp., ∗-expanding, ∗-contracting).(v) If t1, . . . , tm are ∗-contracting terms and s(y1, . . . , ym) is a {◦, ∧, ∨}-term, then s(t1, . . . , tm) is ∗-contracting.

(vi) If s is a ∗-contracting term and t is a ∗-expanding term, then s → t is ∗-expanding.

We extend the above results to include the modal operator. In the subsections below, we consider variations of theabove construction for choices of M and M∗ and describe ∗-stable, ∗-contracting and ∗-expanding terms involving themodality f. Larger classes of ∗-stable, ∗-contracting and ∗-expanding terms can then be inferred from Proposition 5.7.The preservation results and FEP for each class are then obtained directly from Theorem 5.6.

We assume here that M is the {◦}-closure of B and M∗ = {a → b : a ∈ M, b ∈ B}. Note that M is also closed under∧ and ∨ and contains 0 and 1. From the definition of f C, we immediately get that f (x), f (1) and f (0) are ∗-stableterms. Thus, by Proposition 5.7, we have the following results.

Lemma 5.8. If s is a {◦, ∨, ∧, 0, 1}-term, then f (s) is ∗-expanding.

Proof. Let t(�x) = f (s(�x)) and �c ∈ C . By Proposition 5.7(i), we have s(�c�) ∈ M and sC(�c) = s(�c�)u , so tC(�c) =f (s(�c�)u�)�u ≥ f (s(�c�))�u = t∗(�c). �

Lemma 5.9. If s is a {◦, ∧, ∨, 0, 1}-term, then f (¬s) is ∗-contracting.

Proof. Let t(�x) = f (¬s(�x)) and �c ∈ C . Then tC(�c) = f C(¬CsC(�c)) = f C(¬C(s(�c�))u) = f C(((s(�c�))u� → 0)�u),where the second equality follows from Proposition 5.7(i). But (s(�c�))u� → 0 ∈ M∗, and by Lemma 5.1(i), we have that((s(�c�))u� → 0)�u ≤ ((s(�c�))u� → 0). Since f C is order-preserving, tC(�c) ≤ f C((s(�c�))u� → 0) = f (((s(�c�))u� →0)�)�u . Since �c� ∈ M and M is closed under {◦, ∧, ∨, 0, 1} it follows that s(�c�) ∈ M and s(�c�) ≤ (s(�c�))u�. But →is order-reversing in the first coordinate and � is order-preserving, so ((s(�c�))u� → 0)� ≤ (s(�c�) → 0)�. Therefore,tC(�c) ≤ f (s(�c�) → 0)�u = t∗(�c�). �


If we consider the special classes of modal MTL-chains considered in Section 4.3, preservation of the followinginequalities follows from the above: f (x) ◦ f (y) � f (x ◦ y), f (1) = 1 and f (x) � x . In addition, the strict conditionx ∧ (¬x) � 0 and n-contraction xn � xn+1 are preserved, as is the involution ¬¬x = x , although for this case it isnecessary to first close B under ¬ (see [13]). Thus, the varieties of LKr -algebras and LKTr -algebras have the FEP foreach L ∈ Logics.

5.3. When M is closed under f

Throughout this subsection we assume that A satisfies f (x) � x and take M to be the {◦, f }-closure of B andM∗ = {a → b : a ∈ M, b ∈ B}. It is immediate that (M1–M3) hold; for (M4), the reverse well-ordering of Mfollows directly from Higman’s theorem [21]: we may consider M as an ordered algebra generated by a finite set B,with operations (◦ and f) compatible with the order and decreasing in all arguments. The reverse well-ordering of M∗then follows. Note that f (x) � x is preserved by the FEP construction since both f (x) and x are ∗-stable terms; that is,f C is decreasing.

Lemma 5.10. If t(x) = ( f (x))n , for any n � 1, and c ∈ C , then tC(c) = (t(c�))u and t is ∗-stable. In addition, ( f (1))n

and ( f (0))n are ∗-stable for any n � 1.

Proof. This is shown by induction on n: if n = 1, then t(x) = f (x) and t is ∗-stable by definition. Moreover, sincec� ∈ M and M is closed under f we have ( f (c�))� = f (c�) and t∗(c) = ( f (c�))u . Now suppose that for some n � 1, theterm s(x) = ( f (x))n is ∗-stable and sC(c) = (s(c�))u . Let t(x) = f (x) ◦ s(x). Then

tC(c) = f C(c) ◦C sC(c)

= f (c�)u ◦C (s(c�))u

= ( f (c�) ◦ s(c�))u

= (t(c�))u = t∗(c).

The final equality follows since c� ∈ M and M’s closure under ◦ and f implies that t(c�) ∈ M , i.e., (t(c�))� = t(c�).The proofs for ( f (1))n and ( f (0))n are similar. �

Lemma 5.11. Any {◦, ∧, ∨, f, 0, 1}-term is ∗-expanding.

Proof. Let s(�x) be a {◦, ∧, ∨, f, 0, 1}-term and �c ∈ C . If s does not contain f, then, by Proposition 5.7(i), s is ∗-stable,and therefore also ∗-expanding for the standard construction and hence also for the modified construction. We justneed to consider the case where s contains f, so let s(�x) = f (t(�x)), where t(�x) is a {◦, ∧, ∨, f, 0, 1}-term and assume,inductively, that t is ∗-expanding. We have sC(�c) = f C(tC(�c)) ≥ f C(t∗(�c)) = f C((t(�c�))�u). Since �c� ∈ M and M isclosed under {◦, ∧, ∨, f, 0, 1} we have (t(�c�))�u = (t(�c�))u . Then sC(�c) ≥ f C((t(�c�))u) = f ((t(�c�))u�)�u . Moreover,since t(�c�) ∈ M we have (t(�c�))u� ≥ t(�c�), by Proposition 5.1(i), and so sC(�c) ≥ f (t(�c�))�u = s∗(�c). �

Lemma 5.12. If t = ¬s, where s is a {◦, ∧, ∨, f, 0, 1}-term, then t is ∗-contracting.

Proof. Let t(�x) = ¬s(�x) and �c ∈ C . By Lemma 5.11, sC(�c) ≥ s∗(�c) and, since ¬C is order-reversing, ¬CsC(�c) ≤¬Cs∗(�c). Furthermore, since �c ∈ M and M is closed under {◦, ∧, ∨, f, 0, 1}, it follows that s(c�) ∈ M , s∗(�c) = (s(�c�))u

and s(�c�) ≤ (s(�c�))u�. Thus, tC(�c) = ¬CsC(�c) ≤ ¬Cs∗(�c) = ((s(�c�))u� → 0)�u ≤ (s(�c�) → 0)�u = t∗(�c). �

By the above results, we have that f (x) � f ( f (x)) and f (x) ◦ f (x) = f (x) are preserved by the above FEPconstruction. Since LS4r - and L!r -algebras have decreasing operators, the varieties of these algebras have the FEP foreach L ∈ Logics. In addition, Lr

�-algebras have the FEP since the identity f (x) ∨ ( f (x) → 0) = 1 is easily seen tobe preserved, as in the proof of Corollary 4.18.

In [4], Ciabattoni et al. investigated the FEP for MTL-algebras and IMTL-algebras. They showed that the subva-rieties of IMTL-, SMTL-, MTL!r - and IMTL!r -algebras have the FEP. Using our construction, we show that all ofthe subvarieties of algebras considered in [4] have the FEP—thus extending their results. Moreover, using our FEPconstruction any subvariety obtained by adding identities preserved by our FEP construction have the FEP.


5.4. Residuated operators

In the previous subsections, the sets of ∗-stable terms excluded those with iterated f’s, such as f ( f (x)). One casein which such terms are ∗-stable is if f is residuated, meaning that there exists a corresponding unary operation h onA such that f (a) ≤ b ⇔ a ≤ h(b) holds for all a, b ∈ A; in this case we call h the residual of f. If f is residuated,then f distributes over all existing joins, i.e., if

∨i∈I ai exists in A, then f (

∨i∈I ai ) = ∨

i∈I f (ai ). Such operators aretherefore a special case of the left-continuous operators considered in Section 4.4. The following properties hold forresiduated operations: x ≤ h( f (x)), f (h(x)) ≤ x and h is order-preserving.

Results regarding the FEP for residuated lattices with additional residuated operators were obtained in [14]. Theseresults specialise to the case of modal MTL-algebras, as we show here. The FEP construction is modified as follows:Set M to be the {◦, f }-closure of B and M∗ the least set containing B and closed under the operations a → x , a ∈ M ,and h.

It is immediate that this choice of M and M∗ satisfy conditions (M1–M3), however (M4) is not generally true,however, it is true in the case that f is decreasing [14]. Thus, we assume that A satisfies f (x) ≤ x . Observe that sinceM is closed under f and ◦, the preservation results of the previous subsection hold.

Lemma 5.13. The operation f C is residuated and its residual is hC(c) = max{d ∈ C : f C(d) ≤ c} = h(c)�u for allc ∈ C . Thus, if b, h(b) ∈ B, then hC(b) = h(b).

Proof. Let c ∈ C . We begin by showing that h(c)�u belongs to {d ∈ C : f C(d) ≤ c}. Since h(c) ∈ M∗

f C(h(c)�u) = f (h(c)�u�)u = f (h(c)�)u ≤ f (h(c))u ≤ cu = c.

Next, suppose d ∈ C such that f C(d) ≤ c, i.e., f (d�)u ≤ c. Then f (d�) ≤ c, hence d� ≤ h(c). By Lemma 5.1, sinced� ∈ M , we have d� ≤ h(c)� hence d = d�u ≤ h(c)�u , which completes the proof of the first statement. Thus, ifb, h(b) ∈ B, then hC(b) = h(b)�u = h(b). �

The usefulness of the residuation property for f comes from the following result, which is then used to describe alarge set of ∗-stable terms.

Lemma 5.14. If a ∈ M , then f C(au) = f (a)u .

Proof. Let a ∈ M . Recall that f C(au) = f (au�)u . By Lemma 5.1, a ≤ au�, hence f (a)u ≤ f (au�)u . Let e =f (a)u ∈ M∗. Then f (a) ≤ e hence a ≤ h(e) by residuation. Then au ≤ h(e) by Lemma 5.1, since h(e) ∈ M∗, soau� ≤ au ≤ h(e). Thus, f (au�) ≤ f (h(e)) ≤ e, so f C(au) = f (au�)u � e = f (a)u . �

Lemma 5.15. If s(�x) is a {◦, ∨, ∧, f, 0, 1}-term and �c ∈ C , then sC(�c) = s(�c�)u and s is ∗-stable.

Proof. If s is a {◦, ∨, ∧, 0, 1}-term, the results follow from Lemma 5.7(i). Inductively, suppose s = f (�x), where t(�x) isa {◦, ∨, ∧, f, 0, 1}-term and tC(�c) = t(�c�)u for all �c ∈ C . For each c ∈ C , c� ∈ M and M is closed under the operationsin {◦, ∨, ∧, f, 0, 1}, so t(�c�) ∈ M and f (t(�c�)) ∈ M . Thus

sC(�c) = f C(tC(�c)) = f C(t(�c�)u)

= f (t(�c�))u (by Lemma 5.14)

= f (t(�c�))�u = s∗(�c). �

5.5. Order-reversing modalities

The standard construction can also be extended to reverse modal MTL-chains and the results obtained in the case ofmodal MTL-chains can be adapted for order-reversing modalities. Assume that M and M∗ are sets satisfying (M1–M4),in particular, we may take M to be the {◦}-closure of B in A and M∗ = {a → b : a ∈ M, b ∈ B}. Let C be the finite


MTL-algebra obtained by the construction in Section 5.1. Extend C with the operation gC on C defined by:

gC(c) = g(c�)�u .

Lemma 5.16. gC is order-reversing.

Proof. For c, d ∈ C with c � d , we have c� � d� hence g(d�) � g(c�), and so gC(d) � gC(d). �

The proofs of the following results are straightforward.

Lemma 5.17. The identity embedding of B into C preserves the operation g, i.e., if b, g(b) ∈ B, then gC(b) = g(b).

Theorem 5.18. For M, M∗ satisfying (M1–M4) the algebra C = 〈C, ◦C, →C, ∧C, ∨C, gC, 0, 1〉 is a finite reversemodal MTL-chain, and the identity map is an embedding of B into C.

Corollary 5.19. The class of reverse modal MTL-chains has the FEP, hence the variety of reverse modal MTL-algebrashas the FEP.

We now investigate preservation theorems that include the order-reversing operation. From the definition of gC itfollows that g(x), g(1) and g(0) are ∗-stable terms. Moreover, by Proposition 5.7 we have the following results.

Lemma 5.20. The terms g(x), g(1) and g(0) are ∗-stable. If s is a {◦, ∧, ∨, 0, 1}-term, then g(s) is ∗-contracting andg(¬s) is ∗-expanding.

Proof. The first statement is immediate from the definition of gC. If s(�x) is a {◦, ∧, ∨, 0, 1}-term and �c ∈ C , then, byProposition 5.7(i)

gC(sC(�c) = gC(s(�c�)u) = g(s(�c�)u�)�u � g(s(�c�))�u,

so g(s) is ∗-contacting. The remaining statement follows similarly from Proposition 5.7(ii). �

Using the above results together with Proposition 5.7 larger classes of ∗-stable, ∗-contracting and ∗-expanding termscan be inferred. From Theorem 5.6 we then obtain preservation results and the FEP for subvarieties of reverse modalMTL-algebras whose corresponding characteristic properties are preserved.

References

[1] J. Girard, Linear logic, Theor. Comput. Sci. 50 (1987) 1–102.[2] M. Baaz, Infinite-valued Gödel logics with 0-1-projections and relativizations, in: P. Hájek (Ed.), Proceedings of Gödel ’96, Logic Foundations

of Mathematics, Computer Science and Physics—Kurt Gödel’s Legacy, Brno 1996, Springer-Verlag, Czech Republic, 1996, pp. 23–33.[3] F. Montagna, Storage operators and multiplicative quantifiers in many-valued logics, J. Logic Comput. 14 (2004) 299–322.[4] A. Ciabattoni, G. Metcalfe, F. Montagna, Algebraic and proof-theoretic characterizations of truth stressers for MTL and its extensions, Fuzzy

Sets Syst. 161 (3) (2010) 369–389.[5] N. Kamide, Normal modal substructural logics with strong negation, J. Philos. Logic 32 (6) (2003) 589–612.[6] H. Ono, Modalities in substructural logics—a preliminary report, in: Proceedings of the 39th MLG Meeting, School of Information Science,

Japan Advance Institute of Science and Technology, Gamagori, Japan, 2005.[7] G. Restall, Modalities in substructural logics, Logique Anal. 35 (1992) 303–321.[8] F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets Syst. 124 (3) (2001) 271–288.[9] N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated lattices: an algebraic glimpse at substructural logics, in: Studies in Logic and the

Foundations of Mathematics, vol. 151, Elsevier, 2007.[10] H. Sahlqvist, Completeness and correspondence in the first and second order semantics of modal logic, in: S. Kanger (Ed.), Proceedings of the

Third Scandinavian Logic Symposium, Studies in Logic 82, North-Holland Publishing Company, 1975, pp. 110–143.[11] B. Jonsson, On the canonicity of Sahlqvist identities, Stud. Logica 53 (1994) 473–491.[12] S. Givant, Y. Venema, The preservation of Sahlqvist equations in completions of Boolean algebras with operators, Algebra Univ. 41 (1) (1999)

47–84.[13] C.J. van Alten, Preservation theorems for MTL-chains, Logic J. IGPL 19 (2011) 490–511.


[14] C.J. van Alten, Completion and finite embeddability property for residuated ordered algebras, Algebra Univ. 62 (2009) 419–451.[15] H.-P. Gumm, A. Ursini, Ideals in universal algebras, Algebra Univ. 19 (1984) 45–54.[16] S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Millennium Edition, Springer-Verlag, New York, USA, 1981.[17] B.A. Davey, H.A. Priestly, Introduction to Lattices and Order, 2nd ed., Cambridge University Press, Cambridge, UK, 2002.[18] M. Theunissen, Y. Venema, MacNeille completions of lattice expansions, Algebra Univ. 57 (2007) 143–193.[19] W.J. Blok, C.J. van Alten, The finite embeddability property for residuated lattices, pocrims and BCK-algebras, Algebra Univ. 48 (3) (2002)

253–271.[20] T. Evans, Some connections between residual finiteness, finite embeddability and the word problem, J. London Math. Soc. s2-1 (1) (1969)

399–403.[21] G. Higman, Ordering by divisibility in abstract algebras, Proc. London Math. Soc. s3-2 (1952) 326–336.

Modal MTL-algebras

Documents

Transcript of Modal MTL-algebras