Filter pairs and natural extensions of logics
aa r X i v : . [ m a t h . L O ] S e p Filter pairs and natural extensions of logics
P. Arndt ∗ , H.L. Mariano † , D.C. Pinto ‡ September 10, 2020
Abstract
We adjust the notion of finitary filter pair, which was coined for cre-ating and analyzing finitary logics, in such a way that we can treat logicsof cardinality κ , where κ is a regular cardinal. The corresponding newnotion is called κ -filter pair. We show that any κ -filter pair gives rise toa logic of cardinality κ and that every logic of cardinality κ comes froma κ -filter pair. We use filter pairs to construct natural extensions for agiven logic and work out the relationships between this construction andseveral others proposed in the literature. Conversely, we describe the classof filter pairs giving rise to a fixed logic in terms of the natural extensionsof that logic. Introduction
In this work we adjust the notion of filter pair from [
AMP ], which wascoined for creating and analyzing finitary logics, in such a way that we cantreat non-finitary logics.Here the cardinality of a logic is the smallest infinite cardinal κ such thatwhenever Γ ⊢ ϕ holds for some formulas, one finds a subset Γ ′ ⊆ Γ of cardinalitystrictly smaller than κ such that Γ ′ ⊢ ϕ . Thus finitary logics are the logics ofcardinality ℵ .In [ AMP ] the notion of finitary filter pair was introduced. The motivatingfact for this definition was that for every finitary logic, with set of formulas
F m , the lattice of theories is an algebraic lattice contained in the powerset,Th ⊆ ℘ ( F m ), and this lattice completely determines the logic. This lattice isclosed under arbitrary intersections and directed unions. The structurality of thelogic means that the preimage under a substitution of a theory is a theory againor, equivalently, that the following diagram commutes for every substitution σ ,seen as an endomorphism of the algebra of formulas: ∗ University of D¨usseldorf-Germany, [email protected] † University of S˜ao Paulo-Brazil, [email protected] ‡ Federal University of Bahia-Brazil, [email protected] m σ (cid:15) (cid:15) Th (cid:31) (cid:127) i / / ℘ ( F m ) F m Th σ − | L O O (cid:31) (cid:127) i / / ℘ ( F m ) σ − O O This says that the inclusion of theories into the power set is a natural transfor-mation, in the sense of category theory.Passing from just the formula algebra to arbitrary Σ-structures (where Σis the signature of the logic), the role of theories can be replaced by the moregeneral notion of filter. The corresponding considerations then apply: preimagesof filters under homomorphisms of Σ-structures are filters again, and this canbe rephrased as saying that the inclusions of filters into the full power sets ofΣ-structures form a natural transformation.Finally, replacing the lattice of filters with a more abstract lattice, we arriveat the notion of finitary filter pair: In [
AMP ] a finitary filter pair over a signa-ture Σ was defined to be a pair (
G, i ), where G : Σ-str op → AL is a functor fromΣ-structures to algebraic lattices and i is a natural transformation from G to thecontravariant power set functor Σ-str op → AL, A ℘ ( A ). The transformation i is required to preserve, objectwise, arbitrary infima and directed suprema.The intuition offered in [ AMP ] about the notion of filter pair was that itis a presentation of a logic, different in style from the usual presentations byaxioms and rules or by matrices. Instead, it is a direct presentation of the latticeof theories as the image of a map of ordered sets, and the required propertiesensure that it really is the theory lattice of a logic. A filter pair presentation canprovide useful structure for analyzing the associated logic; see the last sectionof [
AMP ] for some sketched examples.In the concept of finitary filter pair the cardinality ℵ is hidden in the notionsof directed supremum and algebraic lattice : Recall that a subset S of a poset P is directed if any finite set of elements of S , i.e. any set of cardinality smallerthan ℵ , has a supremum in S . An element is compact if, whenever it is smallerthan or equal to the supremum of a directed set, it is smaller than or equal toone of the members of the directed set. A complete lattice is algebraic if everyelement is a supremum of compact elements. An inspection of the proofs of[ AMP ] shows that it is the condition that i preserves directed suprema, thatimplies that the associated logic is finitary.The notion of κ -filter pair arises by replacing these (implicit) occurrencesof the cardinal ℵ in the definition of filter pair by a regular cardinal κ in anappropriate way, see Definition 2.1. Doing this, one can show in a similar wayas that of [ AMP ] that κ -filter pairs give rise to logics of cardinality κ (Prop.2.5) and that vice versa every logic of cardinality κ arises from a κ -filter pair(Theorem 2.14).More precisely, one can associate to a given logic l = ( F m Σ ( X ) , ⊢ ) a filterpair F ( l ). Conversely, for every set of variables Y one obtains from a filter pair( G, i ) a logic L Y ( G, i ). We show in Thm. 2.14 that L X ( F ( l )) = l , so that in2articular every logic arises from a filter pair.In considering κ -filter pairs one is led to the topic of natural extensions.A natural extension of a logic l = ( F m Σ ( X ) , ⊢ ) to a set of variables Y is aconservative extension to F m Σ ( Y ) which has the same cardinality as l . Thisnotion appears in some proofs of transfer theorems in Abstract Algebraic Logic.Clearing up some misconceptions from the literature, Cintula and Noguera [ CN ]showed that a certain proposed construction of a natural extension could fail.They gave sufficient conditions for the existence and uniqueness of natural ex-tensions, and asked whether there always exists a unique natural extension toa given set of variables. Shortly after, Pˇrenosil [ Pˇre ] gave two constructions ofnatural extensions of logics whose cardinality is a regular cardinal κ , a maximaland a minimal one. He also gave further results on proposed solutions from theliterature, and showed that there can be several different natural extensions ofa logic, answering the uniqueness question in the negative.After fixing notation and recalling preliminaries in Section 1 and introducing κ -filter pairs in Section 2, we take up the discussion of natural extensions inSection 3.In Corollary 3.5 we show that for Y ⊇ X the logic L Y ( F ( l )) is a naturalextension of l , thus giving an alternative proof for the existence of naturalextensions for logics of regular cardinality.In Remark 3.2 we explain where the regularity assumption, left implicit byPˇrenosil, enters. In Corollary 3.7 we give the next best construction availablefor a logic of singular cardinality.In the literature one finds tentative constructions of consequence relations ⊢ LS , ⊢ SS , ⊢ − and ⊢ + κ of which the first one can fail to be structural, the secondone can fail to satisfy idempotence and the last are the two are the minimal,resp. maximal, natural extensions found by Pˇrenosil in [ Pˇre ]. We summarizethe definitions and the known interrelations between these proposed solutionsat the beginning of Section 3.We identify the natural extension given by the filter pair F ( l ) with Pˇrenosil’sminimal one, and complete the picture painted by Cintula, Noguera and Pˇrenosilin the following result. Theorem (Theorem 3.14)
Given sets X ⊆ Y of variables and a logic l =( F m Σ ( X ) , ⊢ ) we have the following inclusions between the associated relationson F m Σ ( Y ) : ⊢ LS ⊆ ⊢ SS ⊆ ⊢ − = ⊢ L Y ( F ( l )) ⊆ ⊢ + κ The second relation is the structural closure of the first one and the third is theidempotent closure of the second one.
While the topic of natural extensions has some technical importance, welargely agree with Pˇrenosil that in considering a particular logic for a concretepurpose one can, in probably all cases, just endow it from the beginning withsufficiently many variables to escape the questions about existence and unique-ness. 3he topic of natural extensions does, however, come up in connection withfilter pairs in a new and different way: not only do filter pairs offer an immediateconstruction of natural extensions, they also give a solution to the “reverseengineering” question of parametrizing all filter pairs which present a givenlogic:As mentioned above, a filter pair can be seen as a presentation of a logic,different from the usual kind of presentation by rules and axioms. Thus it is nosurprise that different filter pairs can give rise to the same logic. Indeed, thelogic with set of variables X associated to a filter pair ( G, i ) is only determinedby the image of i F m Σ ( X ) (which is the lattice of theories of the logic), andthus one obtains, for example, the same logic if one modifies the filter pair byprecomposing i : G → ℘ with a natural surjection G ′ ։ G .It is then a natural question, how many different filter pairs there are pre-senting a fixed logic if one requires the natural transformation i to be objectwiseinjective, i.e. if one only considers so-called mono filter pairs . Since a filter paircomes together with a choice of natural extensions to all sets of variables, andby results of Pˇrenosil there can be different such choices, there can be morethan one such mono filter pair. We discuss this issue in Section 4 and arrive ata more accurate point of view of a filter pair as being a presentation of a logictogether with a family of natural extensions to all sets of variables.As a byproduct of the discussion we obtain the following new result: Theorem (Corollary 4.9)
The set of natural extensions of a logic with respectto a fixed set of variables, ordered by deductive strength, is a complete lattice.
Since finitary logics allow a unique natural extension to every infinite set ofvariables, the appearance of this lattice of natural extensions is a genuinely newaspect for κ -filter pairs, not present for finitary filter pairs. In this section, we recall the basic definitions and results on logic, closure op-erators and complete lattices and their relative versions associated to an infinitecardinal κ . Definition 1.1
A signature is a sequence of pairwise disjoint sets
Σ = (Σ n ) n ∈ N .The set Σ n is called the set of n -ary connectives. For a set X we denote by F m Σ ( X ) the absolutely free algebra over Σ generated by X , also called the setof formulas with variables in X .A consequence relation is a relation ⊢⊆ ℘ ( F m Σ ( X )) × F m Σ ( X ) , on a signa-ture Σ = (Σ n ) n ∈ N , such that, for every set of formulas Γ , ∆ and every formula ϕ, ψ of F m Σ ( X ) , it satisfies the following conditions: ◦ Reflexivity : If ϕ ∈ Γ , Γ ⊢ ϕ ◦ Cut : If Γ ⊢ ϕ and for every ψ ∈ Γ , ∆ ⊢ ψ , then ∆ ⊢ ϕ Monotonicity : If Γ ⊆ ∆ and Γ ⊢ ϕ , then ∆ ⊢ ϕ ◦ Structurality : If Γ ⊢ ϕ and σ is a substitution , then σ [Γ] ⊢ σ ( ϕ )The notion of logic that we consider is the following: Definition 1.2
A logic is a triple (Σ , X, ⊢ ) where Σ is a signature, X is a infinite set of variables and ⊢ is a consequence relation on F m Σ ( X ) . We oftenwrite a logic as a pair ( F m Σ ( X ) , ⊢ ) , with the datum of the signature and theset of variables combined into that of the formula algebra. Note that for the considerations in this article the set of variables needs tobe part of the definition of logic.
Definition 1.3 A closure operator in a set A is a function c : P ( A ) → P ( A ) that is inflationary, increasing and idempotent. We denote by ( C ( A ) , ≤ ) theposet of closure operators in A ordered setwise by inclusion. We will freely switch between the two formulations of a logic as a consequencerelation on the set of formulas and as a certain closure operator on that set.The properties of Def. 1.1 translate to the operator C ⊢ : Γ
7→ { ϕ | Γ ⊢ ϕ } beingincreasing, idempotent, order preserving and structural, respectively. Definition 1.4
Let Σ be a signature, l = (Σ , X, ⊢ ) be a logic and M a Σ -algebra. • A subset T of F m Σ ( X ) is an l -theory if for every Γ ∪ { ϕ } ⊆ F m Σ ( X ) such that Γ ⊢ ϕ , if Γ ⊆ T then ϕ ∈ T . Equivalently, an l -theory is a ⊢ -closed subset of F m Σ ( X ) . • A subset F of M is an l -filter on M if for every Γ ∪ { ϕ } ⊆ F m Σ ( X ) suchthat Γ ⊢ ϕ and every valuation (i.e. Σ -homomorphism) v : F m Σ ( X ) → M , if v (Γ) ⊆ F then v ( ϕ ) ∈ F . • A pair ( M, F ) , where F is an l -filter on M , is called an l -matrix . • We denote the collection of all l -filters on M by F i l ( M ) and ι l ( M ) : F i l ( M ) ֒ → ℘ ( M ) denotes the inclusion. Note that, by structurality, a subset T ⊆ F m Σ ( X ) is an l -theory iff it is an l -filter. Definition 1.5
Let l = (Σ , X, ⊢ ) , l ′ = (Σ , X ′ , ⊢ ′ ) be logics over a signature Σ and let t : F m Σ ( X ) → F m Σ ( X ′ ) be a Σ -homomorphism. t : l → l ′ is a translation (respectively, a conservative translation )whenever for each Γ ∪ { ϕ } ⊆ F m Σ ( X ) : Γ ⊢ ϕ ⇒ t (Γ) ⊢ t ( ϕ ) (respectively, Γ ⊢ ϕ ⇔ t (Γ) ⊢ t ( ϕ ) ). I.e. σ ∈ hom Σ ( F m Σ ( X ) , F m Σ ( X )). otation 1.6 For a cardinal κ , we write P <κ (Γ) := { Γ ′ ⊆ Γ | | Γ ′ | < κ } for theset of subsets of cardinality smaller than κ . Definition 1.7
Let ⊢ ⊆ P ( A ) × A be a relation between subsets and elementsof a given set A . • Let κ be an infinite cardinal. The relation ⊢ is κ -ary if for every subset Γ ∪ { ϕ } ⊆ A if Γ ⊢ ϕ then there exists Γ ′ ∈ P <κ (Γ) and Γ ′ ⊢ ϕ . • The cardinality of a relation ⊢ is the smallest infinite cardinal κ suchthat ⊢ is a κ -ary relation. Definition 1.8
Given a relation ⊢ between subsets and elements of a set, its κ -ary part , ⊢ κ , is defined by Γ ⊢ κ ϕ : ⇔ ∃ Γ ′ ∈ P <κ (Γ) with Γ ′ ⊢ ϕ . Definition 1.9 If l = ( F m Σ ( X ) , ⊢ ) is a logic, such that ⊢ = ⊢ κ , then l will becalled a κ -logic . This means that l is a logic of cardinality ≤ κ . Remark 1.10
A logic ( F m Σ ( X ) , ⊢ ) is a κ -logic if and only if its associatedclosure operator satisfies C ⊢ (Γ) = S Γ ′ ∈ P <κ (Γ) C ⊢ (Γ ′ ) . In general, such a closureoperator is called a κ -ary closure operator. Remark 1.11
Let l = (Σ , X, ⊢ ) be a logic. Then the following are equivalent: • The κ -ary part of l is a logic and id : (Σ , X, ⊢ κ ) → (Σ , X, ⊢ ) is a conser-vative translation. • The logic l is κ -ary. Example 1.12
For every infinite cardinal κ and any set of variables X , thereis a logic of cardinality κ over X .There is nothing new to add in the case κ = ℵ . If κ > ℵ , consider asignature Σ by setting Σ := { c α | α < κ } and Σ n := ∅ for n ≥ - we just haveconstant symbols and variables - thus F m Σ ( X ) = Σ ∪ X .Define a logic over F m Σ ( X ) by taking the closure operator on F m Σ ( X ) generated by the rules:Let γ < κ be a limit ordinal, γ > . Then { c α +1 | α < γ } ⊢ c γ . Thus, foreach Γ ⊆ F m Σ ( X ) : C ⊢ (Γ) = Γ ∪ { c γ | γ is a limit ordinal, < γ < κ and ∀ α < γ , c α +1 ∈ Γ } .This determines a logic with cardinality exactly κ . Since c γ , for any limitordinal γ < κ that is not a cardinal can be derived by a minimal (non-trivial)set of hypotheses with cardinality equal to card ( γ ) < κ , the cardinality of thelogic is ≥ κ . It is also clear that cardinality of the logic is ≤ κ , since that is thecardinality of the language. Recall that an infinite cardinal κ is called regular if the union of fewer than κ sets of cardinality less than κ has cardinality less than κ again. An infinitecardinal that is not regular is called singular .6 emark 1.13 Let κ be a regular cardinal. Recall the following notions fromlattice theory: • A subset S of a partially ordered set P is called κ -directed if every subsetof S of cardinality strictly smaller than κ has an upper bound in S . An exampleis given by the collection of all subsets of a set which have cardinality smallerthan κ . • In a partially ordered set P , an element x ∈ P is called κ - small if forevery κ -directed subset D ⊆ P one has x ≤ sup D iff ∃ d ∈ D : x ≤ d (e.g. thefinite sets in a power set are ℵ -small, a.k.a. compact). • A κ -presentable lattice , or κ -algebraic lattice , is a complete lattice suchthat every element is the supremum of the κ -small elements below it (e.g. powersets are κ -presentable for every κ ). We will denote the category of κ -presentablelattices and all order preserving functions by L κ . Definition 1.14
For each infinite cardinal κ , we denote by reg ( κ ) the leastregular cardinal ≥ κ . Note that if κ is a singular cardinal, then reg ( κ ) = κ + ,thus, in general, reg ( κ ) ∈ { κ, κ + } . Fact 1.15
1. The κ -ary part of a relation that is reflexive (resp. monotonous,resp. structural concerning some set of endofunctions) has the same prop-erty and always is a κ -ary relation.2. If κ is a regular cardinal, then the κ -ary part of a relation that determinesa closure operator still determines a closure operator. In particular the κ -ary part of a logic is a κ -logic. Proof: (1) The inflationary, increasing and structural properties are easy tosee. The κ − ary property: Γ ⊢ κ ϕ , then there is Γ ′ ∈ P <κ (Γ) such that Γ ′ ⊢ ϕ .Then Γ ′ ⊢ κ ϕ .(2) idempotency or cut:Suppose that ∆ ⊢ κ φ and Γ ⊢ κ ∆. Then let ∆ ′ ∈ P <κ (∆) such that ∆ ′ ⊢ ϕ and for each δ ∈ ∆ ′ let Γ δ ∈ P <κ (Γ) such that Γ δ ⊢ δ . Since κ is regular , takeΓ ′ := S { Γ δ : δ ∈ ∆ ′ } , then Γ ′ ∈ P <κ (Γ) and Γ ′ ⊢ ∆ ′ . Thus Γ ′ ⊢ ϕ and Γ ⊢ κ ϕ . (cid:3) By Fact 1.15, if κ is a regular cardinal, then the κ -ary part of the consequencerelation of a logic is a logic again. For a singular cardinal κ this can fail: Example 1.16
Let κ be a singular cardinal, and let M i , i ∈ I be a family ofpairwise disjoint sets with | I | < κ , | M i | < κ for all i ∈ I and | S i ∈ I M i | = κ .Consider the signature with Σ := S i ∈ I M i ` I ` {∗} and Σ n = ∅ for n = 0 .For an enumerable set X of variables, consider the consequence relation ⊢ on F m Σ ( X ) generated by the rules M i ⊢ i ( i ∈ I ) , and I ⊢ ∗ By idempotence, this consequence relation satisfies S i ∈ I M i ⊢ ∗ ,, but no propersubset allows this conclusion. Therefore it has cardinality > κ . In fact it has ardinality = κ + , the successor of κ , because that is the cardinality of the lan-guage.The κ -ary part ⊢ κ of ⊢ contains all of our generating rules, but not the rule S i ∈ I M i ⊢ ∗ , so it fails to satisfy idempotence and is not a logic. Remark 1.17
The different behaviour of regular and singular cardinals withrespect to κ -ary parts, and also when taking closures, leads to various regularityassumptions in our results, but also for example in the construction of naturalextensions of logics – see Remark 3.2 for the latter point. Last, we will recall some notions and results on general closure operators.
Remark 1.18 On general closure operators and complete lattices:
Re-call that, for every set X : • A subset I ⊆ P ( X ) is an intersection family iff it is closed under ar-bitrary intersections (with the convention that empty intersection = X ).This is the same as the complete lattices ( I, ≤ ) such that the inclusion ι : I ֒ → P ( X ) preserves arbitrary infima. We denote by ( I ( X ) , ⊆ ) theposet of all intersection families in X , ordered by inclusion. • It is a well-known result that the mappings below are well defined andprovide a natural anti-isomorphism between the posets ( I ( X ) , ⊆ ) and ( C ( X ) , ≤ ) : I ∈ I ( X ) c I : P ( X ) → P ( X ) , c I ( A ) = T { C ∈ I : A ⊆ C } c ∈ C ( X ) I c = { A ∈ P ( X ) : c ( A ) = A } The key points to establish these are: c ( A ) is the least I c -closed above A and c ( T i ∈ I A i ) ⊆ T i ∈ I c ( A i ) . Remark 1.19
Given a regular cardinal κ , the above correspondence restricts to κ -ary closure operators (Notation: C κ ( X ) ) and the κ -presentable lattices ( I, ≤ ) such that the inclusion ι : I ֒ → P ( X ) preserves arbitrary infima and κ -directedunions (Notation: I κ ( X ) ): The key point to show this is that c ( S i ∈ I A i ) = S i ∈ I c ( A i ) for every κ -directed union (not only c ( A ) = S A ′ ∈ P <κ ( A ) c ( A ′ ) , asin definition). The κ -compact elements of I c are exactly the c ( A ) , for each A ∈ P <κ ( X ) . Note that { c ( A ′ ) : A ′ ∈ P <κ ( A ) } is a κ -directed family of closedsubsets, thus x / ∈ S A ′ ∈ P <κ ( A ) c I ( A ′ ) entails x / ∈ c I ( A ) . We present below the explicit calculation of the infs and the relevant supsin the posets ( C κ ( X ) , ⊆ ) that will be useful in Section 4. Fact 1.20
Calculation of non-empty infimum of a non-empty family in { c t : t ∈ T } ⊆ C κ ( X ) (that corresponds to a non-empty supremum in I κ ( X ) ):- if A ∈ P <κ ( X ) , c ( A ) := T t ∈ T c t ( A ) ;- if B ∈ P ( X ) , c ( B ) := S B ′ ∈ P <κ ( B ) c ( B ′ ) .The key point here (to show idempotence) is realize that if A ∈ P <κ ( X ) and D ∈ P <κ ( c ( A )) then, for all t ∈ T , c t ( D ) ⊆ c t ( A ) . op = inf of the empty family in C κ ( X ) (that corresponds to bottom = emptysup in I κ ( X ) ): c ⊤ ( A ) = X, ∀ A ∈ P ( X ) , I ⊤ = { X } bottom = sup of the empty family in C κ ( X ) (that corresponds to top = emptyinf in I κ ( X ) ): c ⊥ ( A ) = A, ∀ A ∈ P ( X ) , I ⊥ = P ( X ) Calculation of a κ -directed sup of a upward κ -directed family in { c i : i ∈ ( I, ≤ ) } ⊆ C κ ( X ) (that corresponds to a downward κ -directed inf in I κ ( X ) ):- if A ∈ P <κ ( X ) , c ( A ) := S i ∈ I c i ( A ) ;- if B ∈ P ( X ) , c ( B ) := S B ′ ∈ P <κ ( B ) c ( B ′ ) . Fact 1.21
Under the notation and hypothesis above, the poset inclusion ( C κ ( X ) , ⊆ ) ֒ → ( C ( X ) , ⊆ ) has a right adjoint. I.e.: Let c be a closure operator.Define c ( κ ) ( A ) = S A ′ ∈ P <κ ( A ) c ( A ′ ) , A ∈ P ( X ) . Then c ( k ) ∈ C κ ( X ) , c ( κ ) ≤ c and, for each c ′ ∈ C κ ( X ) such that c ′ ≤ c , we have c ′ ≤ c ( κ ) .The only non-trivial part of the verification is to show that c ( κ ) is an idem-potent operator: this follows in the same vein of the construction of κ -directedsups in the poset ( C κ ( X ) , ⊆ ) that we have described above. κ -Filter pairs In this section we introduce the notion of κ -filter pair, discuss some basicproperties and show how a κ -filter pair gives rise to a κ -logic (see Def. 1.9)and how a logic of cardinality κ gives rise to a reg ( κ )-filter pair (where reg ( κ )denotes the regularization, Def. 1.14). From now on all infinite cardinals appearing in the article areassumed to be regular, unless explicitly mentioned.Definition 2.1
Let Σ be a signature. A κ -filter pair is a pair ( G, i ) where G : Σ -str op → L κ is a contravariant functor from the category of Σ -structuresto the category of κ -presentable lattices and i = ( i M ) M ∈ Σ − str is a collection oforder preserving functions i M : G ( M ) → ( ℘ ( M ); ⊆ ) with the following proper-ties: For any A ∈ Σ − str , i A preserves arbitrary infima (in particular i A ( ⊤ ) = A ) and κ -directed suprema. Given a morphism h : M → N the following diagram commutes: M h (cid:15) (cid:15) G ( M ) i M / / ℘ ( M ) N G ( N ) G ( h ) O O i N / / ℘ ( N ) h − O O emark 2.2 Condition 2. says that i is a natural transformation from G tothe functor ℘ : Σ − str op → L κ sending a Σ -structure to the power set of itsunderlying set and a homomorphism of Σ -structures to its associated inverseimage function. The first class of examples of κ -filter pairs will be established in Theorem2.12, towards which we will work in items 2.8 to 2.11.But first we explain how one can think of a κ -filter pair as a presentation ofa logic of cardinality ≤ κ . Remark 2.3 A κ -presentable lattice is equivalent to a small locally κ -presentablecategory. Condition 1. then says that each i M , seen as a functor, is accessibleand preserves limits. By [ AR , Thm. 1.66] it has a left adjoint, i.e. it is partof a (covariant) Galois connection. We thus get a closure operator on ℘ ( M ) (corresponding to the unit of the adjunction) and a kernel operator (or coclosureoperator) on G ( M ) (corresponding to the counit). We will prove below that theclosure operator on F m Σ ( X ) , the absolutely free Σ -structure over a set X , hascardinality ≤ κ and is structural and hence gives rise to a κ -logic. This will bethe logic associated to the filter pair. We will now spell this out in less categorytheoretical terms. Recall that an order preserving function f : P → Q between posets is rightadjoint to a function g : Q → P (and g is left adjoint to f ) if the followingrelation holds for all p ∈ P, q ∈ Q : g ( q ) ≤ p ⇔ q ≤ f ( p ), i.e. if f and g forma (covariant) Galois connection. In this case the composition f ◦ g is a closureoperator on Q and g ◦ f a coclosure operator (or kernel operator) on P and f , g restrict to a bijection between the (co)closed elements. The (co)closed elementsare exactly those elements in the image of f , resp. g , since from the adjunctionproperties it follows that f ◦ g ◦ f = f and g ◦ f ◦ g = g .It is easy to see that any f : P → Q that has a (automatically unique) leftadjoint, preserves all the infima existing in P . Moreover: Theorem 2.4 [ Tay , Thm. 3.6.9]
Let f : P → Q be a function between completeposets that preserves arbitrary infima. Then f has a left adjoint g : Q → P , givenby g ( q ) := inf { p ∈ P | q ≤ f ( p ) } . Of course, there is a dual result concerning increasing functions that have aright adjoint or preserve suprema.By the theorem above, the maps i M forming the natural transformation ofa κ -filter pair ( G, i ) have left adjoints j M (since they preserve arbitrary infima).From this we have the closure operator i M ◦ j M on each Σ-structure M . Inparticular for a set X there is a closure operator on F m Σ ( X ). This defines alogic: Proposition 2.5
Let ( G, i ) be a κ -filter pair and X be a set. For the Σ -structure F m Σ ( X ) let j F m Σ ( X ) be the left adjoint to i F m Σ ( X ) . Then the clo-sure operator C G := i F m Σ ( X ) ◦ j F m Σ ( X ) defines a logic of cardinality at most κ ( κ -logic) on F m Σ ( X ) . roof: By the axioms of a κ -filter pair, i F m Σ ( X ) preserves κ -directed suprema.Since j F m Σ ( X ) is a left adjoint it preserves arbitrary suprema. Hence the closureoperator C G := i F m Σ ( X ) ◦ j F m Σ ( X ) preserves κ -directed suprema. Since any set S ∈ ℘ ( F m Σ ( X )) is κ -directed union of its subsets of cardinality smaller than κ ,we have that C G ( S ) = S S ′ ⊆ S, | S ′ | <κ C G ( S ′ ).It remains to show structurality. Let σ ∈ hom ( F m Σ ( X ) , F m Σ ( X )) andΓ ∪ { ϕ } ⊆ F m Σ ( X ) such that ϕ ∈ C G (Γ) (i.e. Γ ⊢ G ϕ in the associatedconsequence relation). Then we need to show σ ( ϕ ) ∈ C G ( σ (Γ)).We have σ (Γ) ⊆ C ( σ (Γ)) = i ( j ( σ (Γ))) and therefore Γ ⊆ σ − i ( j ( σ (Γ))).Since the naturality square F m Σ ( X ) σ (cid:15) (cid:15) G ( F m Σ ( X )) i / / ℘ ( F m Σ ( X )) F m Σ ( X ) G ( F m Σ ( X )) G ( σ ) O O i / / ℘ ( F m Σ ( X )) σ − O O commutes, we have σ − ( i ( j ( σ (Γ)))) = i ( G ( σ )( j ( σ (Γ)))), so σ − ( i ( j ( σ (Γ)))) isin the image if i and therefore closed. Hence applying the closure operator C G to the inclusion Γ ⊆ σ − i ( j ( σ (Γ))) yields ϕ ∈ C G (Γ) ⊆ C G ( σ − i ( j ( σ (Γ)))) = σ − i ( j ( σ (Γ))). Now applying σ yields σ ( ϕ ) ∈ i ( j ( σ (Γ))) = C G ( σ (Γ)). (cid:3) Definition 2.6
For a filter pair F = ( G, i ) and a set X we will denote the logicobtained from Prop. 2.5 by L X ( F ) . Remark 2.7
More generally, and with the same proof, for every Σ -structure A one obtains an abstract logic in the sense of [ BBS ], given by the closureoperator i A ◦ j A .A different description of the consequence relation of this abstract logic is D ⊢ A a iff f or every z ∈ G ( A ) , if D ⊆ i A ( z ) then a ∈ i A ( z ) . The proof is the same as that of [
AMP , Prop. 2.4].
We now show that every κ -logic comes from a κ -filter pair (whenever κ is aregular cardinal). Let l = (Σ , X, ⊢ ) be a κ -logic.For the following, recall that F i l ( A ) denotes the collection of all filters on A (Def. 1.4). Lemma 2.8
An arbitrary intersection of filters is a filter again. In particular
F i l ( A ) is a complete lattice. Proof:
That filters are closed under intersection is immediate from the def-inition. Thus the subset
F i l ( A ) ⊆ ℘ ( A ) has arbitrary infima and hence is acomplete lattice. (cid:3) Lemma 2.9
The inclusion i A : F i l ( A ) ֒ → ℘ ( A ) preserves arbitrary infima and κ -directed suprema. roof: The statement about infima is Lemma 2.8. For the statement aboutsuprema we need to show that a κ -directed union of filters is a filter.Let ( F i ) i ∈ I be a κ -directed system of filters. Let Γ ∪ { ϕ } ⊆ F m Σ ( X ) suchthat Γ ⊢ l ϕ and v : F m Σ ( X ) → A be a morphism satisfying v (Γ) ⊆ S i ∈ I F i .Since l is of cardinality ≤ κ , there is Γ ′ ⊆ Γ with | Γ ′ | < κ such that Γ ′ ⊢ ϕ . Everyelement γ ∈ Γ ′ is in some F γ and all these F γ are contained in some F j , sincethe system is κ -directed. Since F j is a filter, we have that v ( ϕ ) ∈ F j ⊆ S i ∈ I F i .This shows the claim. (cid:3) Lemma 2.10
Let A be a Σ -structure. Then F i l ( A ) is a κ -presentable lattice. Proof:
Completeness has been stated in Lemma 2.8. In particular, given anarbitrary subset S ⊆ A one can form the filter generated by S by setting ¯ S := T F ∈ F i l ( A ) , S ⊆ F F . The operation ¯( − ) is evidently a closure operation on ℘ ( A )(indeed it is the closure operation coming from the adjunction of the filter pair).It remains to show that F i l ( A ) is κ -presentable, i.e. that every F ∈ F i l ( A )is a κ -directed supremum of κ -small elements.The filters ¯ S generated by subsets S ⊆ A with | S | < κ are κ -small ele-ments: Indeed, if ¯ S ⊆ W i ∈ I F i for some κ -directed system ( F i ) i ∈ I , then also S ⊆ W i ∈ I F i = S i ∈ I F i (the latter equality follows from Lemma 2.9, and is ex-plicitly shown in the proof there). Hence each of the less than κ many elementsof S is in some F i , hence all are simultaneously in some F j (because ( F i ) i ∈ I is a κ -directed system), i.e. S ⊆ F j , hence ¯ S ⊆ ¯ F j = F j , the latter equality holdingbecause F j is a filter.Now we claim that every F ∈ F i l ( A ) can be written as F = _ F ′ ⊆ F, | F ′ | <κ ¯ F ′ = [ F ′ ⊆ F, | F ′ | <κ ¯ F ′ . Indeed, since for every element f ∈ F the singleton subset { f } occurs in theindex of the supremum, the inclusion ⊆ holds. On the other hand for every F ′ occurring in the index of the supremum we have ¯ F ′ ⊆ ¯ F = F , hence theinclusion ⊇ holds. (cid:3) Lemma 2.11
Preimages of filters under homomorphisms of Σ -structures arefilters again. Proof:
Let f : A ′ → A be homomorphism of Σ-structures and F ⊆ A a filter.To see that f − ( F ) ⊆ A ′ is a filter again, consider Γ ∪ { ϕ } ⊆ F m Σ ( X ), and ahomomorphism v : F m Σ ( X ) → A ′ such that v (Γ) ⊆ f − ( F ). Then ( f ◦ v )(Γ) = f ( v (Γ)) ⊆ F , hence, since F is a filter and f ◦ v a homomorphism, f ( v ( ϕ )) ∈ F ,so v ( ϕ ) ∈ f − ( F ). (cid:3) Denote by i the collection of the inclusions i A : F i l ( A ) ֒ → ℘ ( A ). Theorem 2.12
Let l be a κ -logic. Then ( F i l ( − ) , i ) is a κ -filter pair. Proof:
By Lemmas 2.10 and 2.11
F i l is a well defined functor from Σ-structuresto κ -presentable lattices. It is clear that i is a natural transformation. Theremaining condition for a κ -filter pair is ensured by Lemma 2.9. (cid:3) efinition 2.13 We denote the filter pair of Proposition 2.12 by F ( l ) := ( F i l ( − ) , i ) and call it the canonical filter pair of the logic l . The next theorem says that passing from a logic to a filter pair as in Prop.2.12 and then back to a logic as in Prop. 2.5 gives back the same logic.
Theorem 2.14
Let l = ( F m Σ ( X ) , ⊢ ) be a logic. Then the closure operator i F m Σ ( X ) ◦ j F m Σ ( X ) on F m Σ ( X ) coming from the filter pair F ( l ) is equal tothe closure operator associated to the consequence relation ⊢ . In other words, L X ( F ( l )) = l . Proof:
The closure operator on ℘ ( A ), for a Σ-structure A , associated to thefilter pair ( F i l , i ) is exactly the operator ¯( − ) from the proof of Lemma 2.10,which sends a set to the smallest filter containing it. This is true in particular for A = F m Σ ( X ). The closure operator on F m Σ ( X ) associated to the consequencerelation ⊢ is the operator which sends a set to the smallest theory containing it.It thus suffices to show that the filters on the algebra F m Σ ( X ) are exactly thetheories of the logic l .Let F ⊆ F m Σ ( X ) be a filter for l . Let Γ ∪ { ϕ } ⊆ F such that Γ ⊢ ϕ . ThenΓ = id (Γ), so by the filter property ϕ = id ( ϕ ) ∈ F .On the other hand let T ⊆ F m Σ ( X ) be a theory. Let Γ ∪ { ϕ } ⊆ F suchthat Γ ⊢ ϕ and let σ : F m Σ ( X ) → F m Σ ( X ) be a homomorphism such that σ (Γ) ⊆ T . Then from substitution invariance we get σ (Γ) ⊢ σ ( ϕ ) and hence,since T is a theory, σ ( ϕ ) ∈ T . (cid:3) For the following statement we depart from our standing assumption thatthe cardinal κ is regular. Theorem 2.15
The canonical filter pair F ( l ) of a logic l = ( F m Σ ( X ) , ⊢ ) ofcardinality ≤ κ – where κ is allowed to be singular – is a reg ( κ ) -filter pair. Proof:
By hypothesis l is a κ -logic, thus it is also a reg ( κ )-logic. Now applyTheorem 2.12. (cid:3) The basic discussions of this section do not yet provide a justification forthe introduction of filter pairs. The only example we have given so far is thefilter pair
F i l of all filters, and this may seem to be a rather convoluted way ofintroducing a logic. Also, it would at this point be a natural question, whetherwe should not demand the maps comprising the natural transformation i tobe injective – Theorem 2.14 shows that this can always be arranged. A shortanswer to this is that the utility of filter pairs ultimately lies in semantic con-siderations. For example, congruence filter pairs, are filter pairs for which thefunctor G associates to each Σ-structure the lattice of congruences relative tosome quasivariety. It turns out that a logic admits a presentation by a congru-ence filter pair if and only if it admits an algebraic semantics in the sense ofBlok-Pigozzi, and that the transformation i can be chosen to be injective if andonly if the logic is algebraizable. Thus injectivity of i can become a meaningfuladditional information and should not be made part of the definition.13hat said, we leave these semantic aspects for upcoming works and in theremainder of the article explore, how much different choices of filter pair presen-tations there still are if we do demand i to be injective. The answer is intimatelyrelated with the notion of natural extension. Definition 3.1
For sets X ⊆ Y of variables, a natural extension of a logic l = ( F m Σ ( X ) , ⊢ l ) to F m Σ ( Y ) is a logic ( F m Σ ( Y ) , ⊢ ) which is a conservativeextension of l with the same cardinality as l . One reason for studying natural extensions in Abstract Algebraic Logic isthat some proofs of transfer theorems, that are central in it, require the existenceof extensions of logics to bigger sets of variables.We begin this section by listing four tentative constructions of natural ex-tensions and summarizing the results on them and their interrelations. In thecontext of constructing a natural extension of a logic l of cardinality κ – where κ is a regular cardinal – the following relations between subsets and elementsof F m Σ ( Y ) have been defined in the literature:(a) ⊢ LS ( Lo´s-Suszko), defined byΓ ⊢ LS ϕ iff there are an automorphism v : F m Σ ( Y ) → F m Σ ( Y ) and Γ ′ ⊆ Γ and ϕ s.t. v (Γ ′ ∪ { ϕ } ) ⊆ F m Σ ( X ) and v (Γ ′ ) ⊢ l v ( ϕ )(b) ⊢ SS (Shoesmith-Smiley), defined byΓ ⊢ SS ϕ iff there are Γ ′ ∪ ϕ ′ ⊆ F m Σ ( X ) and v : X → F m Σ ( Y ) s.t.v (Γ ′ ) ⊆ Γ , v ( ϕ ′ ) = ϕ and Γ ′ ⊢ l X ϕ ′ (c) ⊢ − (Pˇrenosil), the smallest consequence relation on F m Σ ( Y ) satisfying therules Γ ⊢ − ϕ whenever Γ ∪ { ϕ } ⊆ F m Σ ( X ) and Γ ⊢ l ϕ .(d) ⊢ + κ (Pˇrenosil), defined as the κ -ary part (see Def. 1.8) of the relation ⊢ + ,given byΓ ⊢ + ϕ iff σ (Γ) ⊢ l σ ( ϕ ) f or every substitution σ : F m Σ ( Y ) → F m Σ ( X )The Lo´s-Suszko relation ⊢ LS is a conservative extension of ⊢ l to F m Σ ( Y )which satisfies monotonicity and reflexivity [ Pˇre , Prop. 16] and is clearly κ -ary,but may fail to satisfy structurality [ Pˇre , Prop. 18].While the Shoesmith-Smiley relation ⊢ SS was for a while thought to alwaysyield a natural extension, this was shown not to be the case in general byCintula and Noguera. It is always a conservative extension of ⊢ l that satisfies14onotonicity, structurality, reflexivity and is κ -ary [ CN , Lem. 2.4], but it mayfail to satisfy the Cut rule (i.e. idempotence) [ CN , Prop. 2.8].The Lo´s-Suszko relation is always contained in the Shoesmith-Smiley relationand they coincide if either | X | < | Y | or card l ≤ | X | [ CN , Lem. 2.7] [ Pˇre ,Prop. 15] or if the Lo´s-Suszko relation actually is a logic [
Pˇre , Thm. 17]. Sincestructurality can fail for ⊢ LS but not for ⊢ SS , they need not coincide in general.In view of their results Cintula and Noguera asked whether a logic alwayshas a natural extension to a given bigger set of variables. For logics of regularcardinality κ , Pˇrenosil gave an affirmative answer: both ⊢ − and ⊢ + κ are alwaysnatural extensions of l , with ⊢ − being the minimal and the ⊢ + κ the maximal one[ Pˇre , Prop. 7, Cor. 6].Furthermore, Cintula and Noguera showed that whenever | X | = | Y | or card ( l X ) ≤ | X | + (where ( − ) + denotes the successor of a cardinal), there isa unique natural extension and that it is given by the Shoesmith-Smiley re-lation [ CN , Thm 2.6]. They asked whether there is always a unique naturalextension, which Pˇrenosil showed not to be the case [ Pˇre , Prop. 19, Prop. 20].
Remark 3.2
We now elucidate the assumption of the regularity of κ . In con-structing the minimal and the maximal natural extensions L − := (( F m Σ ( Y ) , ⊢ − )) and L + κ := ( F m Σ ( Y ) , ⊢ + κ ) , Pˇrenosil leaves implicit the assumption that thecardinality of the logic in question is a regular cardinal.The logic L + κ is explicitly defined by taking the κ -ary part of another logic,and thus by Fact 1.15 exists for regular κ , but it is not clear whether it exists ingeneral for singular κ , see Example 1.16.The construction of the logic L − does not involve taking the κ -ary part ofanother logic, but the proof that the resulting logic is κ -ary (the proof of [ Pˇre ,Prop. 7]) uses the κ -ary part and a priori again only works for regular κ .Further, in [ Pˇre , Cor. 8], Pˇrenosil characterizes L − as the logic over thelanguage with the enhanced set of variables generated by the rules of the originallogic. This is another construction that does not involve taking the κ -ary partof a logic, but a closure process like generating a logic from rules is the kindof thing where one often passes from a cardinal to its regularization (see Def.1.14) as it happens e.g. in Example 1.16. So it is not clear that this offers away around the regularity assumption.As it stands, it thus remains an open problem whether every logic of singularcardinality has a natural extension. What we show below about the singular case,is the next best thing, namely that Pˇrenosil’s construction gives a conservativeextension of cardinality at most the successor of the cardinality of the originallogic.The existence of natural extensions in this remaining open case is, however,a problem of no practical importance. Singular cardinals are rare (the smallestone is ℵ ω ) and logics of singular cardinality are to our knowledge unheard of inconcrete applications.We merely wish to point out this state of affairs, in order to explain theappearance of the regularity assumptions in this work. n the following we keep the standing assumption that all occurringlogics have regular cardinality. We start by shedding some more light on the connection between the Lo´s-Suszko relation and the Shoesmith-Smiley relation. As we just remarked, boththese relations are monotonous and reflexive and the former can fail to be struc-tural, while the latter is always structural. Since relations that are monotonous,reflexive and structural are closed under arbitrary intersections, there is a small-est such relation containing ⊢ LS , which we call its structural closure. Proposition 3.3
The Shoesmith-Smiley relation ⊢ SS is the structural closure ofthe Lo´s-Suszko relation ⊢ LS . Proof:
Denote by (cid:13) the structural closure of the Lo´s-Suszko relation, i.e. theintersection of all monotonous, reflexive and structural relations containing ⊢ LS .Since by the above remarks the Shoesmith-Smiley relation ⊢ SS occurs in thisintersection, we have (cid:13) ⊆ ⊢ SS .For the opposite inclusion note that by taking the inverse of the automor-phism v in the definition of the Lo´s-Suszko relation, one arrives at the descriptionΓ ⊢ LS ϕ iff there are an automorphism v : F m Σ ( Y ) → F m Σ ( Y ) and Γ ′ ∪ { ϕ ′ } ⊆ F m Σ ( X ) s.t. Γ ′ ⊢ l ϕ ′ , v ( ϕ ′ ) = ϕ and v (Γ ′ ) ⊆ ΓThis says that the pairs (Γ , ϕ ) with Γ ⊢ LS ϕ are exactly the images under F m Σ ( Y )-automorphisms of pairs (Γ ′ , ϕ ′ ) with Γ ′ ⊢ l ϕ ′ . The structural closure (cid:13) the contains all images under F m Σ ( Y )- endo morphisms of pairs (Γ ′ , ϕ ′ ) withΓ ′ ⊢ l ϕ ′ . But this says exactly that ⊢ SS ⊆ (cid:13) . (cid:3) It follows that if ⊢ LS is already structurally closed, it coincides with ⊢ SS . Inparticular this implies Pˇrenosil’s result that if the Lo´s-Suszko relation is alreadya logic, then so is the Shoesmith-Smiley relation and the two coincide [ Pˇre ,Thm. 17].As stated, the question of whether there always exists a natural extensionhas been answered by Pˇrenosil, with his two constructions. Next we show,how natural extensions are also easily obtained through the language of filterpairs. We show that these natural extensions coincide with Pˇrenosil’s minimalones and complete the picture by relating the Lo´s-Suszko and Shoesmith-Smileyrelations to this one.
Theorem 3.4
Let Σ be a signature, ( G, i ) a filter pair over Σ and X, Y setswith X ⊆ Y . Then the induced inclusion F m Σ ( X ) → F m Σ ( Y ) is a conservativetranslation L X ( G, i ) → L Y ( G, i ) . Proof:
Denote the inclusion by σ : F m Σ ( X ) → F m Σ ( Y ). Choose a map˜ τ : Y → F m Σ ( X ) such that ˜ τ | X = id X , thus the induced homomorphism Remember that our sets of variables are infinite, in particular X = ∅ . : F m Σ ( Y ) → F m Σ ( X ) is a left inverse of σ , i.e. τ ◦ σ = id F m Σ ( X ) . We thenhave the following diagram (which is commutative if one deletes the j X , j Y ): F m Σ ( X ) (cid:127) _ σ (cid:15) (cid:15) G ( F m Σ ( X )) i X / / ℘ ( F m Σ ( X )) j X o o F m Σ ( Y ) τ (cid:15) (cid:15) (cid:15) (cid:15) G ( F m Σ ( Y )) i Y / / G ( σ ) O O ℘ ( F m Σ ( Y )) σ − O O j Y o o F m Σ ( X ) G ( F m Σ ( X )) id i X / / G ( τ ) O O ℘ ( F m Σ ( X )) j X o o τ − O O id Note that σ − ( Z ) = Z ∩ F m Σ ( X ).Abbreviating l X := L X ( G, i ) and l Y := L Y ( G, i ), we need to show that forΓ ∪ { ϕ } ⊆ F m Σ ( X ) we haveΓ ⊢ l X ϕ iff Γ ⊢ l Y ϕ “ ⇒ ” Suppose that Γ ⊢ l X ϕ . We need to show Γ ⊢ l Y ϕ , i.e. ϕ ∈ i Y j Y (Γ).Since ϕ ∈ F m Σ ( X ), this is equivalent to ϕ ∈ i Y j Y (Γ) ∩ F m Σ ( X ) = σ − i Y j Y (Γ) = i X G ( σ ) j Y (Γ)where the last equality holds because of naturality.Since Γ ⊆ i Y j Y (Γ), and again since Γ ⊆ F m Σ ( X ), we have Γ ⊆ i Y j Y (Γ) ∩ F m Σ ( X ) = i X G ( σ ) j Y (Γ). Since Γ ⊢ l X ϕ , every set in the image of i X thatcontains Γ also contains ϕ , so ϕ ∈ i X G ( σ ) j Y (Γ) = i Y j Y (Γ) ∩ F m Σ ( X ) ⊆ i Y j Y (Γ).“ ⇐ ” Suppose that Γ ⊢ l Y ϕ . We know that Γ ⊆ i X j X (Γ). Since τ ◦ σ =id F m Σ ( X ) , this implies Γ ⊆ τ − i X j X (Γ) = i Y G ( τ ) j X (Γ) (the equality againcoming from the naturality square). Since Γ ⊢ l Y ϕ , every set in the image of i Y that contains Γ also contains ϕ . As ϕ ∈ F m Σ ( X ), it follows that ϕ ∈ i Y ( G ( τ )( j X (Γ))) ∩ F m Σ ( X ) = σ − ( i Y ( G ( τ )( j X (Γ))))= i X ( G ( σ )( G ( τ )( j X (Γ)))) = i X ( j X (Γ)) (cid:3) Corollary 3.5
Let ( G, i ) be a κ -filter pair, X a set and suppose that card L X ( G, i ) = κ . Then for every Y ⊇ X the logic L Y ( G, i ) is a naturalextension of L X ( G, i ) . Proof:
We know from Theorem 3.4 that L Y ( G, i ) is a conservative extension of L X ( G, i ). Since L Y ( G, i ) is presented by a κ -filter pair, we have card L Y ( G, i ) ≤ κ . Finally, since by hypothesis card L X ( G, i ) = κ , for every cardinal ρ < κ thereare formulas Γ ∪{ ϕ } ⊆ F m Σ ( X ) such that Γ ⊢ L X ( G,i ) ϕ and for no subset Γ ′ ⊆ Γ17ith | Γ ′ | < ρ one has Γ ′ ⊢ L X ( G,i ) ϕ . As L Y ( G, i ) is a conservative extension of L X ( G, i ), we also have for no subset Γ ′ ⊆ Γ with | Γ ′ | < ρ that Γ ′ ⊢ L Y ( G,i ) ϕ ,showing that card L Y ( G, i ) ≥ κ , and hence card L Y ( G, i ) = κ . (cid:3) Corollary 3.6 (Pˇrenosil [Pˇre])
Let
X, Y be sets, X ⊆ Y . Then every logicover F m Σ ( X ) has a natural extension to F m Σ ( Y ) . Proof:
We know from Theorem 2.14 that every logic of cardinality κ can bepresented by a κ -filter pair. Hence the claim follows from Corollary 3.5. (cid:3) Our results so far, for a logic singular cardinality, do not give a naturalextension, but the next best thing:
Corollary 3.7
Let
X, Y be sets, X ⊆ Y . Then every logic of singular cardi-nality κ over F m Σ ( X ) has a conservative extension to F m Σ ( Y ) of cardinalityat most κ + . Proof:
By Thm. 2.15 the logic can be presented by a κ + -filter pair ( G, i ). ByThm. 3.4 the logic L Y ( G, i ) is a conservative extension, and, coming from a κ + -filter pair, it has cardinality < κ + . (cid:3) We proceed to pin down the precise relationships between the several (tenta-tive) constructions of natural extensions. As Cintula and Noguera proved, theonly thing that can fail with Shoesmith-Smiley’s tentative definition of a natu-ral extension is idempotence. Next we show, in Proposition 3.10 below, that ifone takes Shoesmith-Smiley’s relation ⊢ SS and forces it to be idempotent, oneobtains our consequence relation on L Y ( G, i ). To show this we review somefacts about idempotent hulls.
Construction 3.8
Consider a set M and an increasing, monotonous operation E : ℘ ( M ) → ℘ ( M ) . There is a smallest idempotent operation C : ℘ ( M ) → ℘ ( M ) which is bigger than E in the setwise order, i.e. satisfying E ( X ) ⊆ C ( X ) forall X ∈ ℘ ( M ) . One can construct it by iterating the operation E until nothingchanges anymore:For an ordinal number i we define inductively E i +1 ( X ) := E ( E i ( X )) fora successor ordinal, and E i ( X ) := S j | M | . Since ℘ ( M ) does not contain chains of strict inclusions indexed by the ordinal I , we have E I ( X ) = E I +1 ( X ) . Now we de-fine the operator C : ℘ ( M ) → ℘ ( M ) by C ( X ) := E I ( X ) = S i
Let X ⊆ M . Then C ( X ) is the smallest E -closed subset of M containing X . Proof: C ( X ) is E -closed by the observation E ( C ( X )) = C ( X ) from above. If Y is another E -closed set containing X , then C ( X ) = S i Let E, C : ℘ ( F m Σ ( Y )) → ℘ ( F m Σ ( Y )) be the operationsgiven by E (Γ) := { ϕ | Γ ⊢ SS ϕ } and C (Γ) := { ϕ | Γ ⊢ L Y ( F ( l )) ϕ } , respec-tively. Then the operation C is the idempotent hull of E . Proof: By definition we have that ϕ ∈ E (Γ) iff ∃ Γ ′ ∪ { ϕ ′ } ⊆ F m Σ ( X ) and v : F m Σ ( X ) → F m Σ ( Y ) such that Γ ′ ⊢ ϕ ′ , v (Γ ′ ) ⊆ Γ and v ( ϕ ′ ) = ϕ .The operator C on the other hand is the the closure operator of the logic L Y ( F ( l )), and thus by definition associates to a set Z ⊆ F m Σ ( Y ) the smallest l -filter containing Z .In other words, by definition of l -filter, ϕ ∈ C (Γ) means that ϕ is con-tained in the smallest set Z of formulas on the variables Y that contains Γ andthat, whenever there are Γ ′ ∪ { ϕ ′ } ⊆ F m Σ ( X ) s.t. Γ ′ ⊢ l X ϕ ′ and a morphism v : F m Σ ( X ) → F m Σ ( Y ) such that v (Γ ′ ) ⊆ Z then also v ( ϕ ′ ) ∈ Z . The lattercondition is exactly the condition of being E -closed, hence the claim followsfrom Lemma 3.9. (cid:3) Lemma 3.11 There is an inclusion ⊢ SS ⊆ ⊢ − . Proof: Remember the definition of ⊢ − as the smallest consequence relation on F m Σ ( Y ) satisfying the rules Γ ⊢ − ϕ whenever Γ ∪ { ϕ } ⊆ F m Σ ( X ) and Γ ⊢ l ϕ .Let Γ ⊢ SS ϕ . By definition there are Γ ′ ∪ ϕ ′ ⊆ F m Σ ( X ) and v : X → F m Σ ( Y ) s.t. v (Γ ′ ) ⊆ Γ , v ( ϕ ′ ) = ϕ and Γ ′ ⊢ l ϕ ′ .By definition Γ ′ ⊢ l ϕ ′ implies Γ ′ ⊢ − ϕ ′ . Choose any extension ˜ v of v to allof Y . Then ˜ v (Γ ′ ) = v (Γ) ⊆ Γ and ˜ v ( ϕ ′ ) = v ( ϕ ′ ) = ϕ and since ⊢ − is structuraland monotonous, we have Γ ⊢ − ϕ . (cid:3) With this we can start tying together all the different relations consideredin this section. Corollary 3.12 The idempotent hull of the Shoesmith-Smiley relation is theminimal natural extension ⊢ − . Proof: Apply the idempotent hull construction to both sides of the inclusionof Lemma 3.11. Then we obtain an inclusion between consequence relations.The left hand side becomes a natural extension of the initial logic l by Prop.3.10 (namely the natural extension coming from the canonical filter pair of l )and the right hand side does not change. Since the right hand side is the minimal natural extension of l , we also have the opposite inclusion. (cid:3) orollary 3.13 The natural extension ⊢ L Y ( F ( l )) of Corollary 3.5, obtainedfrom the canonical filter pair of l , is the minimal natural extension ⊢ − . Proof: Immediate from Corollary 3.12 and Proposition 3.10. (cid:3) We summarize the results so far in the following theorem. Theorem 3.14 Given sets X ⊆ Y of variables and a logic l = ( F m Σ ( X ) , ⊢ ) we have the following inclusions between the associated relations ⊢ LS ⊆ ⊢ SS ⊆ ⊢ − = ⊢ L Y ( F ( l )) ⊆ ⊢ + κ where the second relation is the structural closure of the first one and the thirdis the idempotent closure of the second one. Proof: The first inclusion has been noted in [ Pˇre , Thm. 17], the statementabout structural closure is Proposition 3.3. The second inclusion is Lemma 3.11and the statement about the idempotent hull is Corollary 3.12. The equality isCorollary 3.13. The final inclusion follows from Pˇrenosil’s result that ⊢ − is theminimal and ⊢ + κ the maximal natural extension. [ Pˇre , Prop. 7, Cor. 6] (cid:3) As stated at the beginning of the section, uniqueness of natural extensionsholds only under certain cardinality restrictions. One can deduce this result inthe language of filter pairs by directly proving the independence of the notionof filter from the choice of natural extensions. In this, Cintula and Noguera’scardinality conditions show up for the same reasons as they do in their originalwork. Proposition 3.15 Let X, Y be sets, X ⊆ Y , and l X = (Σ , X, ⊢ ) , l Y = (Σ , Y, ⊢ ′ ) logics such that l Y is a natural extension of l X . Suppose that either | X | = | Y | or card ( l X ) ≤ | X | + (where ( − ) + denotes the successor of a cardinal). Then asubset F of a Σ -structure A is an l X -filter iff it is an l Y -filter. Proof: Let A be a Σ-structure and F ⊆ A .An l Y -filter is an l X -filter: Indeed, let F be an l Y -filter, Γ ∪ { ϕ } ⊆ F m Σ ( X )such that Γ ⊢ l X ϕ , and v : F m Σ ( X ) → A a valuation with v (Γ) ⊆ F . We needto show that v ( ϕ ) ∈ F .Choose a map g : Y → F m Σ ( X ) such that g ( x ) := x for x ∈ X . Thisinduces a homomorphism ˆ g : F m Σ ( Y ) → F m Σ ( X ) and hence a valuation ( v ◦ ˆ g ) : F m Σ ( Y ) → A . We have ( v ◦ ˆ g )(Γ) = v (Γ) ⊆ F and hence, since F is an l Y -filter, ( v ◦ ˆ g )( ϕ ) ∈ F . Since v ◦ ˆ g coincides with v on F m Σ ( X ) this means v ( ϕ ) ∈ F .An l X -filter is an l Y -filter: Let F be an l X -filter, Γ ∪ { ϕ } ⊆ F m Σ ( Y ) suchthat Γ ⊢ l Y ϕ , and v : F m Σ ( Y ) → A a valuation with v (Γ) ⊆ F . We need toshow that v ( ϕ ) ∈ F .Choose Γ ′ ⊆ Γ with | Γ ′ | < card ( l Y ). Since card ( l Y ) = card ( l X ) ≤ | X | + ,we have that | Γ ′ | ≤ | X | and also | Γ ′ ∪ { ϕ ′ }| ≤ | X | , since X is infinite. Sinceevery formula of Γ ′ ∪ { ϕ ′ } only has finitely many variables, we have that the set20 ar (Γ ′ ∪ { ϕ ′ } ) of variables ocurring there has cardinality ≤ | X | . Hence we canchoose functions τ : Y → Y and σ : Y → Y such that τ maps V ar (Γ ′ ∪ { ϕ ′ } )injectively to X and ( σ ◦ τ ) | V ar (Γ ′ ∪{ ϕ ′ } ) = id . As usual we keep the notations τ, σ for the induced maps on the formula algebra.We then have τ (Γ) ∪ { τ ( ϕ ) } ⊆ F m Σ ( X ). By substitution invariance of l Y we have τ (Γ) ⊢ l Y τ ( ϕ ) and, since l Y is a conservative extension of l X , also τ (Γ) ⊢ l X τ ( ϕ ).Then w := v ◦ σ | F m Σ ( X ) : F m Σ ( X ) → F m Σ ( Y ) → A is a valuation with w ( τ (Γ)) = v ( σ ( τ (Γ))) = v (Γ) ⊆ F . Since F is an l X -filter, we have v ( ϕ ) = v ( σ ( τ ( ϕ ))) = w ( τ ( ϕ )) ∈ F . (cid:3) Corollary 3.16 (Cintula, Noguera) Under the cardinality restrictions ofProposition 3.15, natural extensions are unique. Proof: Let l X be a logic with set of variables X and l Y a natural extension of l X with set of variables Y . By Proposition 3.15 we have F i l X ( A ) = F i l Y ( A )for any Σ-structure A and hence the equality of filter pairs F ( l X ) = F ( l Y ). ByTheorem 2.14 l Y = L Y ( F ( l Y )). Therefore l Y = L Y ( F ( l Y )) = L Y ( F ( l X )) is thenatural extension of Corollary 3.5. (cid:3) Remark 3.17 We now have a second proof of Theorem 2.14: By Corollary 3.5,in the special case X = Y we obtain that L X ( F ( l )) is a natural extension of l .Of course l is also a natural extension of itself and the cardinality conditions ofCorollary 3.16 are satisfied, so L X ( F ( l )) = l . This shows that, given Cintulaand Noguera’s uniqueness result, Corollary 3.5 is in fact a generalization ofTheorem 2.14. We have seen in Theorem 3.4 that a κ -filter pair can be regarded as a pre-sentation of a family of logics over all sets of variables, all of which are naturalextensions of each other. In this final section we consider the collection of pos-sible choices of such families of natural extensions of a fixed base logic.Throughout the section we fix a regular cardinal κ , a signature Σ, an infiniteset X and a logic l = ( F m Σ ( X ) , ⊢ )We consider the collection FP Σ of all filter pairs ( G, i ) such that G : Σ − Str op → Lat . We can give this collection the structure of a category by defininga morphism ( G ′ , i ′ ) → ( G, i ) to be a natural transformation t : G → G ′ (note theopposite direction!) such that the following triangle of natural transformationscommutes: G i (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ t / / G ′ i ′ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ ℘ 21n fact we will be more interested in FP l , the full subcategory of FP Σ suchthat L X ( G, i ) = l .We have introduced the reversal of arrows in the definition of FP l , FP Σ ,because in this way morphisms of filter pairs induce translations between theirassociated logics in the same direction : Indeed, a map of logics induces, bytaking preimage, a map in the opposite direction between the theory lattices.In particular passage to a stronger logic, means restriction to a smaller theorylattice, which is reflected in the anti -isomorphism between the poset of sublat-tices of powerset lattices and the poset of closure operators fromSection 1.Here is an overview of how directions of morphisms correspond to each other:logics and translations: l / / l ′ closure operators: C l / / C l ′ theory lattices: T h ( l ) T h ( l ′ ) o o filter pairs: F ( l ) / / F ( l ′ )Here the last reversal of the arrow is purely formal; literally such an arrowis given by lattice maps in the opposite direction.It is probably helpful, in the following, to keep in mind that one can eitherthink of lattice inclusions and revert arrows or think directly in terms of closureoperators and maintain the direction of arrows – whichever provides a betterunderstanding. Remark 4.1 • It would also be a natural choice to demand an inclusion i ⊆ i ′ ◦ t instead of the equality i = i ′ ◦ t , but for the current discussionthis would only add redundancy. • The categories FP l , FP Σ can be seen as (non-full) subcategories of thecategory of all κ -filter pairs, that generalizes the category of all finitaryfilter pairs introduced in [ AMP ]. The first observation is that the category FP l has a initial object. Lemma 4.2 Let ( G, i ) be a κ -filter pair and X a set. Then for every Σ -structure M and a ∈ G ( M ) the set i M ( a ) ∈ ℘ ( M ) is a filter for the logic L X ( G, i ) . Proof: Consider a Σ-structure M and an element a ∈ G ( M ). Let Γ ∪ ϕ ⊆ F m Σ ( X ) be such that Γ ⊢ L X ( f ) ϕ and let σ : F m Σ ( X ) → M be a morphismsuch that σ (Γ) ⊆ i M ( a ). We need to show that σ ( ϕ ) ∈ i M ( a ).22y the commutativity of the naturality square below, we have σ − ( i M ( a )) = i F m Σ ( X ) ( G ( σ )( a )) for every a ∈ G ( M ): F m Σ ( X ) σ (cid:15) (cid:15) G ( F m Σ ( X )) i Fm Σ( X ) / / ℘ ( F m Σ ( X )) M G ( M ) G ( σ ) O O i M / / ℘ ( M ) σ − O O By definition of L X ( f ) the hypothesis Γ ⊢ L X ( f ) ϕ means that ϕ is containedin every set in the image of i F m Σ ( X ) that contains Γ. Thus, sinceΓ ⊆ σ − σ (Γ) ⊆ σ − ( i M ( a )) = i F m Σ ( X ) ( G ( σ )( a )) , we also have ϕ ∈ i F m Σ ( X ) ( G ( σ )( a )) = σ − ( i M ( a )). Applying σ yields σ ( ϕ ) ∈ i M ( a ). (cid:3) Proposition 4.3 The filter pair F ( l ) is the initial object of the category FP l .In other words, for every filter pair f = ( G ′ , i ′ ) such that L X ( f ) = l there is aunique morphism from F ( l ) to f . Proof: We really construct a terminal object in the opposite category: Lemma4.2 says that for every M ∈ Σ − Str , i M [ G ′ ( M )] ⊆ F i L X ( f ) ( M ). These inclusionsform a natural transformation t : G ′ → F i l fitting into a commutative trianglewith the inclusion i : F i L X ( f ) ( M ) ⊆ ℘ ( M ) of l -filters into all subsets. Theuniqueness of t simply follows from the fact that i is objectwise injective. (cid:3) One may ask about further structure or properties of the categories FP Σ , FP l .This would lead to a discussion which is best carried out in the context of generalmorphisms of filter pairs, and is left for a later work. For now we concentrateinstead on mono filter pairs , i.e. filter pairs ( G, i ) such that i A is injective (i.e.a mono morphism) for every A ∈ Σ-str. The full subcategory of FP Σ (respec. FP l ) whose objects are mono filter pairs will be denoted by FP mono Σ (respec. FP monol ). One sees immediately that this category is actually a pre-ordered class , because if both i A and i ′ A in the defining triangle for morphisms areinjective, then t A is unique and is injective too, for each A ∈ obj (Σ − Str ).Another subcategories that are natural to consider are FP incl Σ and FP incll ,where the maps i A and i ′ A (and thus t A ) are in fact inclusions , A ∈ obj (Σ − Str ).Obviously FP incl Σ , FP incll are partially ordered classes and, moreover, FP incl Σ ≃ FP mono Σ and FP incll ≃ FP monol . Dealing directly with FP incl Σ , FP incll turnseasier all the calculations, in fact, is easier to deal first with (arbitrary) infsand (set-sized) sups in FP incl Σ – described “coordinatewise” from the resultson intersection families recalled in Section 1 – and then provide the adaptionsneeded to calculate infs and sups in FP incll . But a direct calculation is providedbelow: Proposition 4.4 The partially ordered class equivalent to FP monol admits setsized suprema of nonempty sets. roof: Let ( G r , i r ) r ∈ R where R is a set. Consider C rA the closure operator over A ∈ Σ − str determined by ( G r , i r ) as, for each M ⊆ A , C rA ( M ) = \ a ∈ G r ( A ) { i rA ( a ) | M ⊆ i rA ( a ) } The closed sets of C rA are exactly the image of i rA . Define the operator C A as, for each subset M of A , C A ( M ) := [ N ⊆ M, | N | <κ \ { N ⊆ X ⊆ A | X = C rA ( X ) ∀ r ∈ R } It is easy to check that C A is a closure operator. Notice that for each subset N ⊆ M such that | N | < κ , C A ( N ) = T { N ⊆ X ⊆ A | C rA ( X ) = X ∀ r ∈ R } .Then C A ( M ) = S N ⊆ M, | N | <κ C A ( N ). Proving the κ -arity of C A . Now we provethat C A is the sup of ( C rA ) r ∈ R .Let M ⊆ A and N ⊆ M where | N | < κ such that. Notice that C rA ( N ) ⊆ X for each subset N ⊆ X ⊆ A such that C r ′ A ( X ) = X for all r ′ ∈ R . So, C rA ( N ) ⊆ C A ( N ). Since C A is κ -ary, we have that C rA ( M ) ⊆ C A ( M ). Thus C rA ≤ C A . Now, let C be a κ -ary closure operator over A such that C rA ≤ C for all r ∈ R . Let N ⊆ A such that | N | < κ . Let X be a subset of A containing N such that C ( X ) = X . Since C rA ≤ C for all r ∈ R , we have that C A ( N ) = T { X ⊇ N | C rA ( X ) = X ∀ r ∈ R } ⊆ T { X ⊇ N | C ( X ) = X } = C ( N ).Since C A and C are κ -ary closure operator, then C A ≤ C . This proves that C A = W r ∈ R C rA .Define the application G : Σ − str → L κ such that G ( A ) is the κ -lattice of C A -closed sets. For a morphism f : A → B of Σ − str , G ( f ) := f − . Firstnotice that for any r ∈ R , and F closed set of C rB , then f − ( F ) is a closed setof C rA . Since C A is the sup of C rA for all r ∈ R , then, for F closed set of C B , C A ( f − ( F )) = W r ∈ R C rA ( f − ( F )) = f − ( F ). Thus f − ( F ) is a closed set of C A . This proves the functoriality of G and that ( G, i ) is a mono κ -filter pair.We have constructed the closure operator of G at each Σ-structure as supre-mum of the closure operators of the G r . This induces inclusions of the theorylattices in the opposite directions, G ( M ) ֒ → G r ( M ) for all M ∈ Σ-Str, and onereadily sees that these form a natural transformation. By the reversal of arrowsin FP monol , this means ( G r , i ) ≤ ( G, i ) for all r ∈ R . As ( G, i ) was constructedas a pointwise supremum it is a supremum. (cid:3) Remark 4.5 If FP monol is equivalent to a set, then by Prop. 4.4 and Prop. 4.3it is equivalent to a complete lattice. In this case we also have a terminal objectand arbitrary infima. If FP monol has an terminal object, i.e. a mono κ -filter pair ( H, j ) presenting l into which all other filter pair in FP monol map then we can give a concretedescription of the values of this filter pair on free algebras. Lemma 4.6 Let X ⊆ Y ⊆ Z be sets of variables and l = ( F m Σ ( X ) , ⊢ l ) a logic.Consider the maximal natural extension l + ,Zκ = ( F m Σ ( Z ) , ⊢ + ,Zκ ) of l to the set f variables Z . Then L Y ( F ( l + ,Zκ )) = l + ,Yκ , i.e. the restriction of the maximalextension to F m Σ ( Z ) down to F m Σ ( Y ) is again the maximal extension. Proof: We know that l + ,Z is a conservative extension of l , so the l + ,Z -filters on F m Σ ( X ) are exactly the l -theories, i.e. L X ( F ( l + ,Zκ )) = l . Thus by Corollary3.5 L Y ( F ( l + ,Zκ )) is a natural extension of l with set of variables Y . Since l + ,Yκ is the strongest such extension, we have Γ ⊢ L Y ( F ( l + ,Zκ )) ϕ ⇒ Γ ⊢ + ,Yκ ϕ .For the opposite implication suppose Γ ∪ { ϕ } ⊆ F m Σ ( Y ) are such thatΓ ⊢ + ,Yκ ϕ . Then by definition of the maximal natural extension, there is a Γ ′ ⊆ Γsuch that | Γ ′ | ≤ κ and for every substitution σ : F m Σ ( Y ) → F m Σ ( X ) we have σ (Γ ′ ) ⊢ l σ ( ϕ ).We need to show that Γ ⊢ L Y ( F ( l + ,Zκ )) ϕ . Since by Theorem 3.4 l + ,Zκ = L Z ( F ( l + ,Zκ )) is a conservative extension of L Y ( F ( l + ,Zκ )), this is equivalent toshowing Γ ⊢ + ,Zκ ϕ , i.e. to showing that there is a Γ ′ ⊆ Γ such that | Γ ′′ | ≤ κ and for every substitution σ : F m Σ ( Z ) → F m Σ ( X ) we have σ (Γ ′′ ) ⊢ l σ ( ϕ ). Forthis we can simply take Γ ′′ := Γ ′ and observe that every such substitution canbe restricted to a substitution σ : F m Σ ( Y ) → F m Σ ( X ), and then we know that σ (Γ ′ ) ⊢ l σ ( ϕ ). (cid:3) Proposition 4.7 Let l := (Σ , X, ⊢ ) be a logic of cardinality κ . Suppose that ( H, j ) is a terminal filter pair in FP monol . Then H is determined on the ab-solutely free algebras F m Σ ( Y ) as follows: it takes the value H ( F m Σ ( Y )) = F ( l + κ )( F m Σ ( Y )) , the set of filters of the maximal natural extension l + κ to F m Σ ( Y ) . Proof: We know from Corollary 3.5 that H ( F m Σ ( Y )) is the set of filters of anatural extension. It is the strongest natural extension l + κ that has the fewestfilters, so if there exists a mono filter pair with the values F ( l + κ )( F m Σ ( Y )), forevery set Y , then these are necessarily also the values of the initial one. (cid:3) While, as illustrated by Prop. 4.7, the possible values on free algebras aresharply restricted once one knows the logic represented by a mono filter pair, itis harder to say something about non-free algebras.For obtaining a precise statement disregarding the non-free algebras, weconsider a variant of the notion of κ -filter pair: a free κ -filter pair is a pair ( G, i )where G : Σ-str opfree → L κ is a functor from the category of free Σ-structures andall endomorphisms to the category of κ -presentable lattices, and i is a naturaltransformation exactly as in the definition of κ -filter pair. Every κ -filter pairhas an underlying free κ -filter pair, given by restricting the functor part fromall Σ-structures to just free Σ-structures, and clearly the associated logics onlydepend on this.For a fixed logic l of cardinality κ , we have the categories free- FP l andfree- FP monol and the restriction functors FP l → free- FP l , respec. FP monol → free- FP monol which forget about the values at non-free algebras. Of course thelatter is still a pre-ordered class. The map FP monol → free- FP monol is a quotientmap, which identifies two mono filter pairs if their values agree for free algebras.With our final result we give a description of the pre-ordered class free- FP monol :25 heorem 4.8 Let l := (Σ , X, ⊢ ) be a logic of cardinality κ and Z be a set ofcardinality κ .Then the pre-ordered class free- FP monol is equivalent to the poset of naturalextensions of l to F m Σ ( Z ) , ordered by deductive strength, and both are equivalentto complete lattices. Proof: Denote the lattice of natural extensions of l to F m Σ ( Z ), ordered bydeductive strength, by NatExt Z ( l ).The claimed equivalence is given by the map L Z : free- FP monol → NatExt Z ( l ) that sends a free mono filter pair presenting l to the associated logic with set of variables Z .It is clear that the map is order preserving, since having more filters meanspresenting a weaker logic (and the inclusions of lattices become morphisms inthe opposite direction in free- FP monol ). The map L Z : free- FP monol → NatExt Z ( l ) is surjective :Let l ′ be a natural extension of l to F m Σ ( Z ). The filter pair F ( l ′ ) is a mono filterpair. Since l ′ is a conservative extension of l by assumption and l ′ = L Z ( F ( l ′ ))(Thm. 2.14) is also a conservative extension of L X ( F ( l ′ )) by Theorem 3.4, wehave L X ( F ( l ′ )) = l . So F ( l ′ ) ∈ FP monol , and thus for its restriction to freealgebras we have F ( l ′ ) ∈ free- FP monol . By Theorem 2.14 L Z ( F ( l ′ )) = l ′ , whichshows surjectivity. The map L Z : free- FP monol / ∼ = → NatExt Z ( l ) is injective :We show that for a filter pair ( G, i ) presenting the logic l , the value L Z ( G, i )completely determines the values of the filter pair on free algebras F m Σ ( Y ).Indeed, for a set Y of lower cardinality than Z the consequence relation of L Y ( G, i ) is simply the restriction from F m Σ ( Z ) to F m Σ ( Y ) by Theorem 3.4,so the filters are determined up to isomorphism by those on F m Σ ( Z ). Onthe other hand, for a set Y of bigger cardinality than Z , the logic L Y ( G, i )will be a natural extension of L Z ( G, i ) by Corollary 3.5, but the latter has a unique natural extension by Cor. 3.16, so this is also completely determined by L Z ( G, i ).We thus have an isomorphism of partially ordered classes L Z : free- FP monol / ∼ = ∼ = → NatExt Z ( l ). But since there is only a set of natu-ral extensions, by Remark 4.5 both are complete lattices. (cid:3) Corollary 4.9 The poset of natural extensions of a logic l is a complete lattice. We conclude by remarking that the reults of this article suggest to view a κ -filter pair as a presentation of a logic together with a chosen family of naturalextensions . In fact, the notion of free mono filter pair captures precisely that.The view of finitary filter pairs as presentations of a logic, suggested in[ AMP ], remains as valid as before: by Cintula and Noguera’s uniqueness result,Cor. 3.16, for a finitary logic there is a unique natural extension to every setof variables, and hence the lattices of Theorem 4.8 are trivial. Thus this is agenuinely new aspect arising for κ -filter pairs.26 Final remarks The main interest in κ -filter pairs is that they can be used to treat infinitarylogics along the lines of [ AMP ]. Most results of loc. cit. carry over, and theprospects listed for finitary filter pairs in the final section of loc. cit continueto be make sense and be interesting. The extra flexibility of allowing logics ofhigher cardinalities can be used to speak about logics which have an algebraicsemantics in generalized quasivarieties, and similar criteria to those mentioned[ AMP ][Sect. 4.2] for being truth-equational or algebraizable can be established.Furthermore, one can generalize the notion of κ -filter to that of C - κ -filterpair, by considering pairs ( G, i ) where G : C op → L κ is a functor from a sub-category C ⊆ Σ-str to the category of κ -presentable lattices, and i is a naturaltransformation exactly as in the definition of κ -filter pair. Taking for C the sub-category of Σ-Str consisting of a single free algebra and all its endomorphisms,a C - κ -mono-filter pair is exactly the same thing as a logic of cardinality ≤ κ .Taking for C the subcategory consisting of all free algebras and homomorphisms,as done in the end of Section 4, gives back exactly the notion of logic togetherwith a family of natural extensions. Taking a non-full subcategory C ⊆ Σ-strgives a notion of logic which is structural only with respect to a certain class ofsubstitutions.More generally one can allow C to be any concrete category. A practicalexample of this is the notion of Horn filter pair : Here one takes C to be categoryof models of a Horn theory. A presentation of a logic by a Horn filter pair, thenis the same as a semantics in this class of models. The special case where onehas no relation symbols recovers the notion of algebraic semantics of [ BP ]. Inthe setting of Horn filter pairs it is possible to prove a bridge theorem relatingthe Craig interpolation property with the amalgamation property in the classof models – this generalizes the well-known bridge theorem for algebraizablelogics. We explore these directions in upcoming works.Finally, by [ AMP , Thm 3.9], the association of the canonical filter pair F ( l )to a logic l can be extended to a functor, exhibiting the category of logics andtranslations as a full co-reflexive subcategory of the category of all filter pairs,endowed with a natural notion of morphism. In general, the category of filterpairs seems to be a convenient setting for the long-term project laid out in[ MaPi1 ], [ MaPi2 ], of establishing local-global principles in logic, setting up arepresentation theory of logics and giving applications in remote algebraization. References [ AMP] P. Arndt, H. L. Mariano, D. C. Pinto, Finitary Filter Pairs andPropositional Logics , South American Journal of Logic , 257-280, 2018. 27 AR] J. Ad´amek, J. Rosick´y, Locally Presentable and AccessibleCategories , Lecture Notes Series of the LMS , CambridgeUniversity Press, Cambridge, Great Britain, 1994.[ BBS] S. Bloom, D. Brown, R. Suszko, Some theorems on abstract log-ics , Algebra i Logika , 274-280, 1970.[ BP] W. J. Blok, D. Pigozzi, Algebraizable logics , Memoirs of theAMS , American Mathematical Society, Providence, USA,1989.[ CN] P. Cintula, C. Noguera, A note on natural extensions in abstractalgebraic logic , Studia Logica, , 815-823, 2015.[ Cz] J. Czelakowski, Protoalgebraic logics , Trends in Logic—Studia Logica Library , Kluwer Academic Publishers, xii+452pp., 2001.[ LS] J. Lo´s, R. Suszko, Remarks on sentential logics , IndagationesMathematicae , 177-183, 1958.[ MaPi1] H. L. Mariano, D. C. Pinto, Representation the-ory of logics: a categorial approach , arXiv preprint,http://arxiv.org/abs/1405.2429, 2014.[ MaPi2] H. L. Mariano, D. C. Pinto, Algebraizable Logics and a functorialencoding of its morphisms , Logic Journal of the IGPL ,524-561, 2017.[ MP] M. Makkai, R. Par´e, Accessible categories: The Founda-tions of Categorical Model Theory , Contemporary Mathe-matics , American Mathematical Society, Providence, USA,1989.[ Pˇre] A. Pˇrenosil, Constructing natural extensions of propositional log-ics , Studia Logica , 1179-1190, 2016.[ SS] D. J. Shoesmith, T. J. Smiley, Deducibility and many-valuedness ,Journal of Symbolic Logic , 610-622, 1971.[ Tay] P. Taylor,