Logical operations and invariance

Enrique CasanovasUniversity of Barcelona

July 28, 2004. Revised September 8, 2005

Abstract

I present a notion of invariance under arbitrary surjective mappingsfor operators on a relational finite type hierarchy generalizing the so calledTarski-Sher criterion for logicality and I characterize the invariant opera-tors as definable in a fragment of the first-order language. These resultsare compared with those obtained by Feferman and it is argued that fur-ther clarification of the notion of invariance is needed if one wants to useit to characterize logicality.

1 Background

The semantical analysis of logical constants stems from the early work of Tarski,Mautner and Mostowski, and has received a lot of attention recently mainlydue to the contributions of Sher, McGee, van Benthem, and Feferman. Its mainpoint is the proposal that logicality may be understood as invariance under themost general transformations. In [13] Tarski develops the idea of identifyinglogical notions with those notions that are invariant under all permutationsof the universe of discourse. A similar point of view had been previously butindependently maintained by Mautner in [7]. Lindenbaum and Tarski provedin [5] that all logical notions from Principia Mathematica are invariant in thissense.

Mostowski in [9] addressed the problem of characterizing quantifiers. Hedistinguishes between limited and unlimited quantifiers. A limited quantifieris defined on a specific set and it is invariant under all permutations of thisset. An unlimited quantifier determines a limited quantifier on each particularset and it is required to be invariant under all bijections between sets. Thisdistinction is used by Gomez-Torrente in [4] to justify the difference betweenthe Tarski criterion, namely invariance under permutations of the universe, andthe Mostowski criterion, which is invariance under bijections across universes.Note that in order to be able to apply Mostowski’s criterion we have to assumethat the notions and operations we are discussing are defined on every set.

Sher [10] requires that a logical constant be defined over all models, notonly over all sets. Her criterion for logicality includes, among other conditions,

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invariance under isomorphisms, which in fact is a weaker condition than mereinvariance under bijections. This difference between Sher’s and other previousaccounts of logical constancy is frequently neglected. For instance, McGee in [8]and Feferman in [3] identify Sher’s position with the requirement of invarianceunder bijections for logical operations, which is Mostowski’s criterion accordingto Gomez-Torrente’s terminology. For a concise presentation of Sher’s viewssee [11].

The proposal of identifying logical operations and notions with those invari-ant under bijections across domains has received severe criticism. It is generallyacknowledged that invariance is a necessary condition for logicality, but not asufficient one. One reason for this is the observation that all notions formally de-fined in terms of cardinalities turn out to be invariant under bijections, whereastheir logical character is more than dubious in many cases. Moreover bijectionsconnect only sets of the same size, and consequently invariance under bijec-tions does not exclude an arbitrary behavior on different cardinalities. In wordsof Machover in his review of Sher’s book in [6]: “ For example, let C be theclass whose members are all natural numbers belonging to some non-analyticset, as well as all infinite cardinals α (regarded as initial ordinals) such that2ℵα = ℵα+1. Now let Q be a particle of the same syntactic type as the famil-iar standard quantifiers ∃ and ∀, interpreted as follows: in any domain whosecardinality belongs to C, Q is interpreted as ∃, while in any other domain Qis interpreted as ∀. Clearly, Q under its given interpretation satisfies Sher’scriterion, and hence she would have us regard it as a logical particle.”

The defense of the criterion of bijection-invariance has been undertakenmainly by McGee. In [8] he proves, roughly stated, that any bijection-invariantoperation can be defined in an infinitary logical language and maintains that“this confirms Tarski’s thesis that an operation invariant under permutationsis a logical operation” ([8], page 568). To be more precise, McGee shows thatif F is a bijection-invariant operation, then for each cardinal number κ there isa formula ϕk in the infinitary language L∞∞ which characterizes F on sets ofcardinality κ. Therefore McGee’s result does not provide a single formula forF , but a different formula ϕκ for each cardinal number κ.

McGee’s work seems to Feferman “faultless in its execution” but “blatantlyimplausible” ([3] page 32). He presents three basic criticisms of McGee’s ap-proach. In his own words ([3] page 37):

1. The thesis assimilates logic to mathematics, more specifically to set theory.

2. The set-theoretical notions involved in explaining the semantics of L∞∞are not robust.

3. No natural explanation is given by it of what constitutes the same logicaloperation over arbitrary basic domains.

Point 3 is for him the main reason for rejecting McGee’s proposal. As men-tioned above, this was also one of the objections of Machover to Sher’s criterionfor logicality. Feferman proceeds then to present his own ideas, which try, ac-cording to him ([3] page 39), to give an explanation of logical operations “which

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shows how an operation behaves when applied over one domain M0 connectsnaturally with how it behaves over any other domain M ′

0”. The solution is toinvestigate invariance under mappings instead of invariance under bijections.He calls it “homomorphism invariance”, which I think it is an unlucky denomi-nation since there is no structure to be preserved and it is really only a matterof surjective mappings from pure sets to pure sets. Feferman characterizes theoperations (of monadic type) which are invariant in this sense and he findsout that they are exactly the ones which are definable in the λ-calculus with-out equality. In other words, the invariant operations are those which can becharacterized using only pure first-order logic. There are some precedents ofFeferman’s results in Van Benthem [14] but they are of very restricted nature.

In the end Feferman is not completely satisfied with his characterization sinceit excludes equality, and after a brief discussion of the problem of the logicalcharacter of equality he says we should give it “a distinguished role in logiceven if it should not turn out to be logical on its own under some cross-domaininvariance criterion, such as homomorphism” ([3] page 44).

2 Invariance

My approximation to the semantic analysis of logical operations has, in its initialsteps, many points in common with the one developed by Feferman. Clearly,invariance under bijections across universes is not the more general kind ofinvariance one can think of. Equality (or better inequality) is invariant underbijections but not necessarily under arbitrary mappings. In mathematics muchmore general kinds of transformations – which do not respect inequality – aremany times considered the natural ones. It is therefore convenient to investigatemore general kinds of transformations and to find out which are the notions andoperations invariant under them.

Unexpectedly, I arrived at very different conclusions than those of Feferman.Invariant notions and operations in my sense turn out to be only a small frag-ment of all notions and operations definable in first-order logic. In particular,negation, arbitrary conjunctions and universal quantification are not invariant.On the other hand it follows from my results that some particular forms ofequality are invariant. Some part of my work in the last part of this paperhas been devoted to try to understand how we arrive at so distant conclusionsstarting with the same assumptions and very close intuitions.

Feferman’s analysis takes place in a finite type hierarchy of universes similarto the one used first by van Benthem. Just to simplify the proofs, Fefermanreplaces relational types by functional types and replaces relations by theircharacteristic functions. At first sight one can think that his replacement isinnocuous, but in fact it has serious consequences. As I show in section 7,Feferman’s invariance notion does not correspond to invariance under surjectivemappings when translated back into the relational framework. His definitionof invariance seems to be very reasonable in the functional type hierarchy butit is fairly unnatural when restated in the relational setting. As pointed out

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in section 7, it is in fact a different notion of invariance than the one hereinvestigated. For the first levels of the type hierarchy it coincides with invarianceunder preimages of arbitrary mappings. This is a capital point to explain thedifference in conclusions between Feferman’s analysis and mine. If one does notmake the simplification of replacing relations by their characteristic functions,one is forced to deal with the notion of invariance I am discussing here.

The fact that invariance splits into several notions, some stronger than oth-ers, and that accordingly different criteria for logicality appear, should perhapsbe taken as a sign that the logical character of notions and operations from asemantical point of view is a matter of perspective and a matter of degree. Evenoutside the semantical framework, the idea that classifying a notion as a logicalnotion is a question of more or less has been considered by some authors. For in-stance, Byrd discusses in [1] the position that the conditional and the universalquantifier play a prominent role among logical concepts and Warmbrod in [15]proposes a distinction between a secure logical theory with a minimal set ofnotions and an extended setting where a great variety of notions is allowed. Ac-cording to my notion of invariance, existential quantification and disjunction areparticularly strong whereas negation, conjunction and universal quantificationare not. I realize that it is not easy to accept than universal quantification andconjunction are less logical that existencial quantification and disjunction. Butaccording to my results there is a natural sense in which they are less robust.

It is not my purpose here to support any particular point of view concerningthe logical character of abstract operations. My analysis can be used to showthat the semantic characterization based on invariance under all possible trans-formations leads to absurdities. It can also be used to support the view that insome respect some operations are more logical than others. I think that thereis still a need for exploration. Several notions of invariance have appeared andsome other may still be relevant. My research shows that the notion of invari-ance is not as clear as one could have thought. Moreover the previous proposalshave been only partially developed. Feferman has been able to characterize in-variant operations only of a restricted class. They are operations transforminga tuple consisting of n subsets of a set M on a m-ary relation on M . It would beinteresting to see whether his description generalizes to more general operationstransforming tuples of mi-ary relations (for 1 ≤ i ≤ n) onm-ary relations, whichis the most reasonable setting. In this article all the mapping-invariant (in mysense) operations of these more general kinds are investigated. I describe withsome detail these operations and I show they are just the operations definablein a specific fragment of first-order logic. I do not claim that they constitutethe answer to the question of which are the logical operations.

My approach to invariance is influenced by the fact that in algebra themost general kind of transformation usually considered is not isomorphism buthomomorphism. On the contrary to what happens with the relation of being anisomorphic image, the relation of being an homomorphic image is not symmetricand transitive. Thus, one need to construct its symmetric and transitive closurein order to obtain an equivalence relation analogous to isomorphism. Thesetopics are studied in [2] with the main motivation of developing the basic model-

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theoretic aspects of first-order logic without equality. The equivalence relationamong structures obtained as the symmetric and transitive closure of being anhomomorphic image is called relativeness in [2]. “Being relative” is the naturalsubstitute for “being isomorphic” when one does not care of preserving equalitiesand inequalities. In the case we are interested here, only pure sets are present–we do not deal with arbitrary models– and therefore one has to work withbijections in place of isomorphisms and with surjective mappings in place ofhomomorphisms. It is a particular case of the general situation studied in [2],namely the case in which all structures considered have empty similarity type.

Feferman in [3] is also concerned with the equivalence relation characterizedas the symmetric and transitive closure of being an image under an arbitrarysurjective mapping and he chooses the name similarity to refer to it. Thus, asimilarity is a finite composition of surjective mappings and inverses of surjectivemappings and one says that two sets are similar when there is a similaritybetween them. Of course, any two nonempty sets are similar in this sense, butthe important tool is not the similarity relation itself but invariance under everyparticular similarity. Similarity is just relativeness particularized to the case ofstructures which are only pure sets. Here we will follow Feferman’s terminology.Of course, we can apply to the similarity relation all results obtained in [2] forthe relativeness relation. This has been very helpful for my understanding ofthe situation.

I now describe with some detail the content of the paper. We will look atnotions and operations defined simultaneously over any nonempty set. Over anysuch set we will consider the relational and the functional finite type hierarchyof universes and we will show how different transformations of the basic setscan be extended to upper levels. In section 3, after fixing terminology, I definemapping-invariance and I clarify some points concerning invariance under map-pings and under similarities. In section 4, I characterize the mapping-invariantobjects at the lower levels of the relational type hierarchy. In sections 5 and 6,I present my main results: the characterization of all mapping-invariant oper-ators that transform a tuple of relations among individuals of different aritieson a relation among individuals. The particular case of operators transformingm-ary relations on n-ary relations is studied in detail in section 5. Then insection 6 I need only to indicate how to generalize the results from section 5 tothe broader setting.

Sections 7 and 8 are devoted to compare my approach with that of Fefer-man. In section 7, I introduce the finite functional type hierarchy and I provethe equivalence between similarity-invariance and homomorphism-invariance inthe sense of Feferman [3]. In section 8, Feferman’s proposal is presented, and Itranslated it back into the relational setting. In the finite relational type hier-archy I define what I call Feferman invariance and I show that an operator isFeferman invariant if and only if it is invariant in the sense of Feferman whentranslated into the finite functional type framework. Then a last notion of in-variance is introduced, the notion of preimage-invariance and I show that it isthe same as Feferman invariance at the lower levels. This explains why everyfirst-order definable operation is invariant according to Feferman but it is not

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according to my investigation: his transformations of the universe are the re-sult of taking preimages and as is well-known they preserve intersections andcomplements while taking images do not.

The notation is mainly standard. P(X) denotes the power set of X. IfR ⊆Mn and 1 ≤ j ≤ n we will use the notation

fieldj(R) = {a ∈M : (a1, . . . , an) ∈ R for some a1, . . . , an such that aj = a}.

Thus, for example, if R is a binary relation, then n = 2, field1(R) = dom (R),and field2(R) = rng (R). The union of fieldj(R) for all j is field (R). R−1 isused for the inverse of a relation R even in the case that R happens to be a nonnecessarily one-to-one mapping. If R and S are relations then their relationalproduct or composition will be

R ◦ S = {(a, b) : for some c,R(a, c) and S(c, b)}.

If R or S is a mapping we still use this notation, identifying the mapping withits graph. Thus f ◦ g is not defined by f ◦ g(x) = f(g(x)).

3 Mapping-invariance

The relational finite types are generated from the basic type 0 by the rule: ifτ1, . . . , τn are relational finite types, then (τ1, . . . , τn) is a relational finite type.We will call them in short just relational types. If M is a nonempty set and τ arelational type, the τ -universe of M is the set Mτ defined recursively as follows:

1. M0 = M .

2. M(τ1,...,τn) = P(Mτ1 × . . .×Mτn).

When using the notation Mτ , we will always assume that M is nonempty. Thetype τ = (τ1, . . . , τn) where τi = 0 for every i = 1, . . . , n will be denoted by 0n.Hence M0n = P(Mn).

Let f : M → N be a mapping and let τ be a relational type. The mappingf can be extended in a very natural manner to a mapping from the τ -universeMτ in the τ -universe Nτ . The mappings fτ : Mτ → Nτ are defined recursivelyby the following rules:

1. f0 = f .

2. If τ = (τ1, . . . , τn), the mapping fτ : Mτ → Nτ is given by

fτ (a) = {(fτ1(a1), . . . , fτn(an)) : (a1, . . . , an) ∈ a}

for all a ⊆Mτ1 × . . .×Mτn.

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Observe that each fτ is onto if f is onto, and that fτ is one-to-one if f is one-to-one. Sometimes we will write f instead of fτ when the context does not leadto confusion.

Compositions of surjective mappings in every possible direction produce thenotion of similarity. A relation π ⊆ M ×N is a similarity relation between Mand N if and only if for some n ≥ 2 there are sets M1, . . . ,Mn and mappingsf1, . . . , fn−1 such that M = M1, N = Mn and for every i = 1, . . . , n, fi is amapping from Mi onto Mi+1 or it is a mapping from Mi+1 onto Mi, and π isthe relational product or composition R1 ◦ . . .◦Rn−1 where Ri = fi if fi is fromMi onto Mi+1 and Ri = f−1

i if it is from Mi+1 onto Mi.The following is easily proven by induction on the length n of the sequence

of sets M1, . . . ,Mn in the definition of similarity. Moreover it follows from theproof of Proposition 2.6 in [2] since a similarity is just a relativeness relation forthe case of the empty language.

Fact 3.1. A binary relation π ⊆ M × N is a similarity relation between Mand N if and only if for every a ∈ M there is some b ∈ N such that π(a, b)and for every b ∈ N there is some a ∈ M such that π(a, b). In other words, ifdomπ = M and rng π = N .

A similarity relation π between M and N is also naturally extensible to asimilarity relation πτ between the τ -universes Mτ and Nτ . Assume that thereare sets M1, . . . ,Mn and mappings f1, . . . , fn−1 such that M = M1, N = Mn

and for every i = 1, . . . , n, fi is a mapping from Mi onto Mi+1 or it is a mappingfrom Mi+1 onto Mi, and π is the relational product R1◦. . .◦Rn−1 where Ri = fiif fi is from Mi onto Mi+1 and Ri = f−1

i if it is from Mi+1 onto Mi. Then πτis the relational product R1τ ◦ . . .◦Rn−1τ where Riτ = fiτ if fi is from Mi ontoMi+1 and Riτ = fi

−1τ if it is from Mi+1 onto Mi.

Remark 3.2. Let π a similarity relation between M and N . The inducedsimilarity πτ ⊆ Mτ × Nτ can also be defined inductively starting from π0 = πby the rule that if τ = (τ1, . . . , τn) is a relational type, then the similarity πτbetween Mτ and Nτ is given by: πτ (a, b) if and only if

1. for each (a1, . . . , an) ∈ a there is a tuple (b1, . . . , bn) ∈ b such that πτi(ai, bi)

for each i = 1, . . . , n, and

2. for each (b1, . . . , bn) ∈ b there is a tuple (a1, . . . , an) ∈ a such that πτi(ai, bi)

for each i = 1, . . . , n.

We will be concerned with the invariance of objects and operators. Let τbe a relational type. An object of type τ is a function a which associates withevery nonempty set M a corresponding element aM ∈Mτ . We will say that theobject a is mapping-invariant if and only if for every M,N and every surjectivemapping f : M → N , f(aM ) = aN . Equivalently, a is mapping-invariant ifand only if for every M,N , for every similarity relation π between M and N ,π(aM , aN ). We call the object a bijection-invariant if and only if for every M,Nand every bijection f : M → N , fτ (aM ) = aN .

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Let τ1, . . . , τn, τ be relational types. A (τ1, . . . , τn; τ)-ary operator is a func-tion F which for each nonempty set M gives a mapping

FM : Mτ1 × . . .×Mτn→Mτ .

Sometimes we omit the subscript in FM to simplify notation if this is notmisleading. We call F mapping-invariant if for all M,N and every mappingf : M → N from M onto N , for all a1 ∈Mτ1 , . . . , an ∈Mτn

,

fτ (FM (a1, . . . , an)) = FN (fτ1(a1), . . . , fτn(an)).

It is easy to check that a (τ1, . . . , τn; τ)-ary operator F is mapping-invariantif and only if for all M,N and every similarity relation π between M and N ,for all a1 ∈ Mτ1 , . . . , an ∈ Mτn

, and for all b1 ∈ Nτ1 , . . . , bn ∈ Nτn, if πτi

(ai, bi)for all i = 1, . . . , n, then πτ (FM (a1, . . . , an), FN (b1, . . . , bn)).

Following more customary terminology, a (0m1 , . . . , 0mr ; 0n)-ary operatorwill be also called (m1, . . . ,mr, n)-ary.

As mentioned in the introduction, Tarski in [13] talks of invariance underpermutations of a fixed universe and not of invariance under bijections acrossuniverses. We may say that the element b ∈ Mτ is permutation-invariant ifand only if for every permutation f of M , fτ (b) = b. Clearly, if a τ -object a isbijection-invariant, then for every set M , the corresponding element aM ∈ Mτ

is permutation-invariant. On the other hand it is easy to show that for eachset M and each permutation-invariant element b ∈Mτ we can find a bijection-invariant τ -object a such that aM = b. Hence both criteria of invariance coincidewhen applied to a particular universe.

4 Mapping-invariant objects

It is clear that there are no bijection-invariant objects of type 0. This wasobserved by Tarski in [13] for permutation invariance. He also noticed thatthe only permutation-invariant objects of type (0) are the empty set and theuniverse and more generally, that there are only finitely many permutation-invariant objects of type 0n. For instance, the permutation-invariant objects oftype 02 = (0, 0) are the empty relation, the universal relation, the identity andits complement. The same happens for bijection-invariance. It is not difficultto see that the bijection-invariant objects of type 0n are the n-ary relationsfirst-order definable in the language of pure equality. But the defining formulamight be not uniform for all cardinalities.

Fact 4.1. An object a of type 0n is bijection-invariant if and only if for eachcardinal κ there is a first-order formula ϕκ = ϕκ(x1, . . . , xn) in the language ofpure equality such that for each set M of cardinality κ, aM = {(a1, . . . , an) ∈Mn : M |= ϕ(a1, . . . , an)}.

Mapping-invariance is a more restrictive condition. Of course, there is nomapping-invariant object of type 0. Concerning objects of type 0n, the basic

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difference with respect to bijection-invariance is that intersections and com-plements of mapping-invariant relations are not necessarily mapping-invariant.Equality is mapping-invariant, but inequality is not.

Theorem 4.2. An object a of type 0n is mapping-invariant if and only if itis the empty object (aM = ∅ for every set M) or the universe (aM = Mn forevery M) or it is uniformly a union of objects dI for I ⊆ {1, . . . , n}, wheredIM = {(a1, . . . , an) ∈Mn : ai = aj for all i, j ∈ I} for every set M .

Proof. It is clear that all these objects are mapping-invariant. Let a be amapping-invariant object. We will show that a is of the prescribed form. IfaM = ∅ then aN = ∅ for any other set N , since if π is a similarity rela-tion, π(∅, aN ) implies aN = ∅. I claim that if dIM ∩ aM 6= ∅ for a set Msuch that |M | > n, then for any set N , dIN ⊆ aN . From this it will fol-low that a is a union of objects dI . This includes the case aM = Mn, whichcorresponds to |I| ≤ 1. Assume |M | > n and let (a1, . . . , an) ∈ dIM ∩ aMand (b1, . . . , bn) ∈ dIN . We now show that (b1, . . . , bn) ∈ aN . Notice thatπ = {(ai, bi) : i = 1, . . . , n} ∪ ((M r {a1, . . . , an}) × N) is a similarity relationbetween M and N . By mapping invariance of a, there is a tuple (c1, . . . , cn) ∈aN such that π((a1, . . . , an), (c1, . . . , cn)). By construction of π we see that(c1, . . . , cn) = (b1, . . . , bn). Hence (b1, . . . , bn) ∈ aN .

Invariant objects of type ((0)) are closely related to quantifiers. Tarski ob-served that the permutation-invariant objects in this type are the “propertiesconcerning the number of elements of these classes”. Cardinality also plays arole for the mapping-invariant objects of type ((0)), but essentially as upperbound.

Theorem 4.3. An object a of type ((0)) is mapping-invariant if and only if ais the empty object or it is uniformly a union of the following objects:

1. the object e such that for every set M , eM = {∅},

2. the object u such that for every set M , uM = {M},

3. the object p such that for every set M , pM = P(M) r {∅},

4. for some cardinal number κ, the object bκ such that for every set M ,bκM = {A ⊆M : A 6= ∅ and |A| < κ}.

Proof. It is easy to check that all the described objects are mapping-invariant.We prove now that all the mapping-invariant objects are of this kind. If aM = ∅then for any other set N , aN = ∅. Similarly, if ∅ ∈ aM then ∅ ∈ aN for everyN . Now assume aM has nonempty elements different from M and let κ be acardinal such that for each cardinal µ smaller than κ, aM has such elements ofcardinality µ. We prove that in this case, for any set N , bκN ⊆ aN . Let B ⊆ Nbe nonempty and of cardinality strictly less than κ. Choose a set A ∈ aM ofcardinality ≥ |B| and such that A 6= M , choose a surjective mapping f : A→ Bsuch that f(A) = B, and put π = f ∪ ((M r A) × N). It is a similarity

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relation between M and N and hence, by mapping-invariance of a, π(A,C) forsome C ∈ aN . By choice of π it follows that C = B. Thus B ∈ aN . If forarbitrarily large κ there are sets N such that bκN ⊆ aN , then clearly for everyN , aN = P(N) or for every N , aN = P(N) r {∅}. Otherwise, let κ be the leastcardinal such that for some set N , bκN 6⊆ aN . Then for every µ < κ, for everyset N , bµN ⊆ aN . Obviously, κ is a successor cardinal number, say κ = λ+.Moreover A 6∈ aN if A is a subset of N such that A 6= N and |A| ≥ λ. If thereis not a set M such that |M | ≥ λ and M ∈ aM then either aN = bλN for everyN or aN = bλN ∪ {∅} for every N .

Assume now that there is a set M such that |M | ≥ λ and M ∈ aM . We provethat N ∈ aN for every N . From this it will follow that either aN = bλN ∪ {N}for every N or aN = bλN ∪ {∅} ∪ {N} for every N . Consider an arbitrary set N .If |N | < λ, then clearly N ∈ aN . In case |M | = |N | ≥ λ, there is a bijection fbetween M and N and we put π = f . If M , N have different cardinality ≥ λwe may assume that |M | > |N | ≥ λ, we choose A ⊆M and B ⊆ N of the samecardinality ≥ λ such that A 6= M , we choose a bijection f between A and B,and we put π = f ∪ ((M r A) × N). In either case π is a similarity relationbetween M and N and there is a set C ∈ aN such that π(M,C) and it followsthat N = C. Hence N ∈ aN .

The last case to be considered is when there is no set M such that aM hasa nonempty element A 6= M . Then aM ⊆ {∅,M} for every M . It is easy to seethat either aM = ∅ for every M , or aM = {M} for every M , or aM = {∅,M}for every M .

After Theorem 4.3 one might think that all quantifiers ∃<κxϕ(x) are at ourdisposal in constructing mapping-invariant operators and in defining mapping-invariant objects. But this is not the case. It is misleading to restrict thenotion of a quantifier over a set M to a specification of a collection of subsetsof M . This might be enough to quantify formulas with just one free variable.Quantification of a formula with more free variables involves transformation ofn+ 1-ary relations on n-ary relations. In the next section I will characterize allmapping-invariant transformations of this kind. The quantifier ∃≤1 is clearlynot among them.

A similar situation arises with respect to the logical connectives. In a foot-note Corcoran, the editor of [13], explains that during the Buffalo lecture thatoriginated the paper [13] Tarski indicated that truth-functions can be accom-modated in the relational type hierarchy over the universe M by identifying thetruth values T and F with the universe M and the empty set. With this identifi-cation every binary truth-function is an element ofM((0),(0),(0)) and hence we candiscuss its invariance. It is easy to check that all truth-function are mapping-invariant objects in this sense. However this is again misleading. One thingis conjunction acting on sentences and a completely different thing is conjunc-tion acting on formulas with free variables. Conjunction of formulas with freevariables involves the operation of intersection, which is not mapping-invariant.

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5 Characterizing mapping-invariant (m,n)-ary op-erators

This section is devoted to the characterization of all (m,n)-ary mapping-invariantoperators. I start fixing some basic operators and fixing also some generatingoperations to obtain more complex operators. I will show that all mapping-invariant operators are generated with these operations from the basic ones.

The (m,n)-ary operators are the following ones:

1. The constant (m, 1)-ary operators Cm> and Cm⊥ whose actions on R ⊆Mm

are given by

(a) Cm> (R) = M

(b) Cm⊥ (R) = ∅

2. The (m,n ·m)-ary diagonal operator ∆mn such that for any R ⊆Mm,

∆mn (R) = {n× a : a ∈ R}

where n × a is the n-fold concatenation of the tuple a, that is, n × a =(b1, . . . , bn·m) if a = (a1, . . . , am) and bk·m+i = ai for 1 ≤ i ≤ m.

3. The (m,m−1)-ary i-projection operator Πmi (wherem ≥ 2 and 1 ≤ i ≤ m)

such that for any R ⊆Mm,

Πmi (R) = {(a1, . . . , ai−1, ai+1, . . . , am) : (a1, . . . , am) ∈ R}.

4. For any σ ∈ Sym{1, . . . ,m}, the (m,m)-ary permutation operator Pσ suchthat for any R ⊆Mm,

Pσ(R) = {(aσ(1), . . . , aσ(n)) : (a1, . . . , an) ∈ R}.

The generating operations for operators are the following ones:

1. Product. If F is (m,n1)-ary and G is (m,n2)-ary, the product F × G isthe (m,n1 + n2)-operator such that for any R ⊆Mm,

F ×G(R) = F (R)×G(R).

2. Sum. If F and G are (m,n)-ary operators, the sum F ∪G is the (m,n)-aryoperator such that for any R ⊆Mm,

F ∪G(R) = F (R) ∪G(R).

3. Composition. If F is (m,n)-ary and G is (n, k)-ary, the composition of Fand G is the (m, k)-ary operator G ◦ F such that for any R ⊆Mm,

G ◦ F (R) = G(F (R)).

11

Operators generated from projections, diagonals and permutations by composi-tion will be called intern.

Remark 5.1. A (m,n)-ary operator F is intern if and only if there is a mapσ : {1, . . . , n} → {1, . . . ,m} such that for any R ⊆Mm,

F (R) = {(a1, . . . , an) : for some (a′1 . . . , a′m) ∈ R, ai = a′σ(i) for all i = 1, . . . , n}

For the next lemma we need the technical notion of free system. Let R ⊆Mm. We call (M,R) a free system if R is infinite, for all a ∈ M there is atmost one tuple (a1, . . . , am) ∈ R such that a = ai for some i and, finally, for all(a1, . . . , am) ∈ R, ai 6= aj if i 6= j.

Lemma 5.2. Let F be a (m,n)-ary mapping-invariant operator. Let R ⊆Mm

and assume (M,R) is a free system. For any (a1, . . . , an) ∈ FM (R) there are adecomposition

{1, . . . , n} = I1∪ . . . ∪Ikand operators F1, . . . , Fk such that

1. Fl is (m,nl)-ary, where nl = |Il|.

2. If i ∈ Il and ai = aj, then j ∈ Il.

3. Fl is intern if {ai : i ∈ Il} has more than one element or if it has just oneelement a and a ∈ fieldj(R) for some j.

4. If {ai : i ∈ Il} has only one element a and a 6∈ field (R), then Fl =∆1nl◦ Cm> , that is Fl(S) = {nl × a : a ∈ N} for all S ⊆ Nm.

5. (ai : i ∈ Il) ∈ Fl(R).

6. There is a σ ∈ Sym{1, . . . ,m} such that for all S ⊆ Nm, if (N,S) is afree system, then

Pσ(F1(S)× . . .× Fk(S)) ⊆ F (S).

Proof. If a1 ∈ field (R), we choose as I1 a maximal subset of {1, . . . ,m} forwhich 1 ∈ I1 and there is an intern (m,n1)-ary operator G such that (aj : j ∈I1) ∈ G(R), and we put F1 = G. If a1 6∈ field (R), we take I1 = {i : a1 = ai}and we put F1 = ∆1

n1◦ Cm> . In either case (ai : i ∈ I1) ∈ F1(R) and also

j ∈ I1 whenever aj = ai for some i ∈ I1. Now let l be the first element of{1, . . . ,m} r I1. Again, if al ∈ field (R) we choose as I2 a maximal subset of{1, . . . ,m} r I1 for which l ∈ I2 and there is an intern (m,n2)-ary operator Gsuch that (ai : i ∈ I2) ∈ G(R), and we put F2 = G. In case al 6∈ field (R)we set as before I2 = {i : al = ai} and F2 = ∆1

n2◦ Cm> . This procedure

eventually ends and the decomposition is finished. We will show that it verifies6. Assume S ⊆ Nm, (N,S) is a free system, and (b1, . . . , bn) is a tuple such that(bi : i ∈ Il) ∈ Fl(S) for all l = 1, . . . , k. We will prove that (b1, . . . , bn) ∈ F (S).The permutation σ will be the natural enumeration of the sequence sa

1 sa2 . . .

a sk

12

where every sl is the enumeration of Il in increasing order. Let J be the set ofall l ∈ {1, . . . , k} which do not fall under case 4, that is, all l for which Fl isintern. If l ∈ J , we can find a mapping σl : Il → {1, . . . ,m} such that for allT ⊆ Km,

Fl(T ) = {(ci : i ∈ Il) : for some (c′1, . . . , c′m) ∈ T, ci = c′σl(i)

for all i ∈ Il}

and we can find (al1, . . . , aln) ∈ R and (bl1, . . . , b

ln) ∈ S such that ai = alσl(i)

and bi = blσl(i)for all i ∈ Il. For l 6∈ J we choose al and bl such that ai = al

and bi = bl for all i ∈ Il. Let L be the set of all l ∈ {1, . . . , k} r J such thatbl ∈ field (S). For l ∈ L choose now a tuple (bl1, . . . , b

lm) ∈ S such that bl = bli for

some i and choose arbitrary (al1, . . . , alm) ∈ R with the only requirement that

these tuples are all different and also different from the chosen for l ∈ J . Thisis possible because R is infinite.

Let A = M r ({ali : 1 ≤ i ≤ m and l ∈ J ∪ L} ∪ {al : l 6∈ J}) andB = N r ({bli : 1 ≤ i ≤ m and l ∈ J ∪ L} ∪ {bl : l 6∈ L}). Since R is infinite, Aand B are nonempty sets. Let

π = (A×B)∪{(ali, bli) : 1 ≤ i ≤ m and l ∈ J ∪L}∪{(al, bl) : l ∈ {1, . . . , k}rJ}.

It is a similarity relation between M and N and moreover π(R,S). By mapping-invariance of F , π(F (R), F (S)). Hence (c1, . . . , cn) ∈ F (S) for some c1, . . . , cnsuch that π(ai, ci) for all i = 1, . . . , n. By construction, ali 6= al

′

i′ for (l, i) 6= (l′, i′)and also ali 6= al

′for all i, l, l′ and al 6= al

′for all l 6= l′. Hence (c1, . . . , cn) =

(b1, . . . , bn).

Lemma 5.3. Let F be a (m,n)-ary mapping-invariant operator. Assume M isinfinite. For any (a1, . . . , an) ∈ FM (∅) there are a decomposition

{1, . . . , n} = I1∪ . . . ∪Ik

and operators F1, . . . , Fk such that

1. Fl = ∆1nl◦ Cm> , where nl = |Il|.

2. If i ∈ Il and ai = aj, then j ∈ Il.

3. (ai : i ∈ Il) ∈ Fl(∅).

4. There is a σ ∈ Sym{1, . . . ,m} such that for any infinite set N ,

Pσ(F1N (∅)× . . .× FkN (∅)) ⊆ FN (∅).

Proof. It is basically the same reasoning as in the proof of Lemma 5.2 exceptthat intern operators are not needed. Given (b1, . . . , bn) such that (bi : i ∈Il) ∈ FlN (∅) for each l = 1, . . . , k we have to show that (b1, . . . , bn) ∈ FN (∅).Once al and bl are chosen, define A = M r {al : l = 1, . . . , k}, B = N r {bl :l = 1, . . . , k} and π = (A × B) ∪ {(al, bl) : l = 1, . . . , k}. Since M and N

13

are infinite, A and B are nonempty sets and π is a similarity relation be-tween M and N . Clearly π(∅, ∅), and by mapping-invariance π(FM (∅), FN (∅)).Then π((a1, . . . , an), (c1, . . . , cn)) for some (c1, . . . , cn) ∈ FN (∅). It follows that(c1, . . . , cn) = (b1, . . . , bn).

Lemma 5.4. For all R,M such that R ⊆ Mm and R 6= ∅, there are S,N, fsuch that S ⊆ Nm, (N,S) is a free system and f : N → M is a mapping fromN onto M such that f(S) = R.

Proof. Choose an enumeration {(ai1, . . . , aim) : i ∈ I} of R whose index set Iis infinite and choose an enumeration (bi : i ∈ J) of M r field (R) such thatJ ∩ (I × {1, . . . ,m}) = ∅. Let N = J ∪ (I × {1, . . . ,m}) and let f : N → Mbe the mapping defined by f(i) = bi for i ∈ J and f((i, j)) = aij for (i, j) ∈I × {1, . . . ,m}. If S = f−1(R), then (N,S) is a free system, f is surjective andf(S) = R.

Lemma 5.5. Let F be an (m,n)-ary mapping-invariant operator. Let R ⊆Mm,S ⊆ Nm and assume f : N → M is a surjective mapping such that f(S) = R.Then

1. f(FN (S)) = FM (R).

2. If G is also an (m,n)-ary mapping-invariant operator and FN (S) = GN (S),then FM (R) = GM (R).

Proof. 1 is clear, since f is a similarity relation connecting R and S. 2 followsfrom 1 since FM (R) = f(FN (S)) = f(GN (S)) = GM (R).

Theorem 5.6. An (m,n)-ary operator F is mapping-invariant if and only ifthere are two (m,n)-ary operators G, H generated from the basic operators bysum, product and composition and such that:

1. For all M , FM (∅) = GM (∅).

2. For all M and all nonempty R ⊆Mm, FM (R) = HM (R).

Proof. Let us consider first the case of nonempty arguments. By lemmas 5.4and 5.5 we may restrict our attention to free systems. By Lemma 5.2 if (M,R)is a free system, then FM (R) is a union of components, where a component is apermutation of products of intern operators and operators of the form ∆1

nl◦Cm> .

Since there are only finitely many choices for components, it is in fact a finiteunion of them. If we take all components for F arising in all free (M,R), bypoint 6 of Lemma 5.2 we see that in fact their union works for obtaining thevalue of F in all free (M,R). The case of the empty set is similar but usingLemma 5.3 and observing that for any M there are N, f such that N is infiniteand f : N →M is onto.

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Definition 5.1. Let P be a m-ary relation symbol and let ϕ = ϕ(x1, . . . , xn) be afirst-order formula having P as its only extralogical symbol. The (m,n)-operatorGϕ attached to ϕ is defined as follows: for any nonempty M and R ⊆Mm,

GϕM (R) = {(a1, . . . , an) ∈Mn : (M,R) |= ϕ(a1, . . . , an)}.

Definition 5.2. Consider the first-order language L whose only extralogicalsymbol is the m-ary relation symbol P . For convenience we assume that > and⊥ are formulas. The mapping-invariant formulas are all L-formulas generatedwith the following rules:

1. > and ⊥ are mapping-invariant.

2. P (y1, . . . , ym) is mapping-invariant for any distinct variables y1, . . . , ym.

3. ¬∃y1 . . . ymP (y1, . . . , ym) is mapping-invariant for any distinct variablesy1, . . . , ym.

4. If ϕ, ψ are mapping-invariant, then (ϕ ∨ ψ) is mapping-invariant.

5. If ϕ, ψ are mapping-invariant and have no common free variable, then(ϕ ∧ ψ) is mapping-invariant.

6. If ϕ = ϕ(x1, . . . , xn) is mapping-invariant and y is different from all thevariables x1, . . . , xn, then for all i = 1, . . . , n, (ϕ(x1, . . . , xn) ∧ xi = y) ismapping-invariant

7. If ϕ is mapping-invariant, then ∃xϕ is mapping-invariant.

Observe that > and ⊥ are dispensable since we can replace them respec-tively by the mapping-invariant formulas (ψ ∨ ¬ψ) and (ψ ∧ ¬ψ) where ψ =∃y1 . . . ymP (y1, . . . , ym).

Theorem 5.7. An (m,n)-ary operator F is mapping-invariant if and only ifF = Gϕ for some mapping-invariant formula ϕ = ϕ(x1, . . . , xn) in a languagewhose only extralogical symbol is the m-ary relation symbol P .

Proof. It is easy to check that by induction on ϕ that all operators Gϕ formapping-invariant ϕ are mapping-invariant. For the rest we first need to showthat any operator generated by sum, product and composition from the basicones is definable by a mapping-invariant formula. This can be easily done byinduction on the length of the generating sequence of the operator. Now usingTheorem 5.6 we see that there are two mapping-invariant formulas ψ(x1, . . . , xn)and χ(x1, . . . , xn) such that for all M , FM (∅) = GψM (∅) and for all M and allnonempty R ⊆Mm, FM (R) = GχM (R). Let ϕ(x1, . . . , xn) be the formula

(ψ(x1, . . . , xn) ∧ ¬∃yP (y)) ∨ (χ(x1, . . . , xn) ∧ ∃yP (y))

where y = y1, . . . , ym is a tuple of distinct variables. It is a mapping-invariantformula and clearly GϕM (∅) = GψM (∅) while for nonempty R, GϕM (R) = GχM (R).Hence F = Gϕ.

15

6 More general operators

Recall that an (m1, . . . ,mr, n)-ary operator is an operator F which for everynonempty set M gives a mapping

FM : P(Mm1)× . . .× P(Mmr ) → P(Mn).

Sometimes I will call operators with these arities general operators. Here Icharacterize the mapping-invariant general operators. The methods are closelyparallel to the ones used in the previous section. Therefore I will state theresults and give only brief indications about the proofs.

The basic general operators are the basic operators introduced in the previ-ous section and

1. The product. It is the (m,n,m+ n)-ary operator⊗

such that for all M ,and all R ⊆Mm and S ⊆Mn,

⊗M (R,S) = R× S.

2. The sum. It is the (m,m,m)-ary operator⋃

such that for all M , and allR ⊆Mm and S ⊆Mm,

⋃M (R,S) = R ∪ S.

The general composition is the operation which for any (m1, . . . ,ms, n)-ary oper-ator F and any (mi

1, . . . ,miri,mi)-ary operators Fi (where i = 1, . . . , s), gives an

(m11, . . . ,m

1r1 ,m

21, . . . ,m

2r2 , . . . ,m

s1, . . . ,m

srs, n)-ary operator compF1,...,Fs,F such

that for every set M and all subsets Rij ⊆Mmij (i = 1, . . . , s and j = 1, . . . , ri),

compF1,...,Fs,F (R11, . . . , R

srs

) = F (F1(R11, . . . , R

1mr1

), . . . , Fs(Rs1, . . . , Rsmrs

))

A general operator is intern if it is generated from projections, diagonals, andpermutations by general composition.

I generalize now the technical notion of free system to make it useful forgeneral operators. Let R1 ⊆Mm1 , . . . , Rr ⊆Mmr . We say that (M,R1, . . . , Rr)is a general free system if each (M,Ri) is a free system and the Ri have disjointfields.

Lemma 6.1. Let F be a (m1, . . . ,mr, n)-ary mapping-invariant operator. LetR1 ⊆ Mm1 , . . . , Rr ⊆ Mmr , let L ⊆ {1, . . . , r} and assume (M,Ri)i∈L is ageneral free system and Ri = ∅ for i 6∈ L. For any (a1, . . . , an) ∈ F (R1, . . . , Rr)there is a decomposition

{1, . . . , n} = I11 ∪ . . . ∪I1

k1∪ . . . ∪Ir1 ∪ . . . ∪Irkr

and operators F 11 , . . . , F

1k1, . . . , F r1 , . . . , F

rkr

such that

1. F pl is (mp, nl)-ary, where nl = |Ipl |.

2. If i ∈ Ipl and ai = aj, then j ∈ Il.

3. F pl is intern if {ai : i ∈ Ipl } has more than one element or it has just oneelement a and a ∈ fieldj(Rl) for some j.

16

4. If {ai : i ∈ Ipl } has only one element a and a 6∈ field (R), then F pl =∆1nl◦ Cmp

> .

5. (ai : i ∈ Ipl ) ∈ Fpl (Rl).

6. There is a σ ∈ Sym{1, . . . ,m1,m1 +1, . . . ,m1 +m2 + . . .+mr} such thatfor all S1 ⊆ Nm1 , . . . , Sr ⊆ Nmr , if (N,Si)i∈L is a general free systemand Si = ∅ for i 6∈ L, then

Pσ(F 11 (S1)×. . .×F 1

m1(S1)×. . .×F r1 (Sr)×. . .×F rmr

(Sr)) ⊆ F (S1, . . . , Sr).

Proof. The fact that in a general free system (M,Ri)i∈L all the relations Ri havedisjoint fields allows us to adapt the proof of Lemma 5.2 in a straightforwardway to this setting. The details are left to the reader.

Theorem 6.2. Let F be an (m1, . . . ,mr, n)-ary operator. Then F is mapping-invariant if and only if for each L ⊆ {1, . . . , r} there is an (m1, . . . ,mr, n)-aryoperators GL generated from the general basic operators by general compositionand such that for any set M and all R1 ⊆ Mm1 , . . . , Rr ⊆ Mmr such thatRi 6= ∅ if and only if i ∈ L,

F (R1, . . . , Rr) = GL(R1, . . . , Rr).

Proof. General free systems satisfy the corresponding version of Lemma 5.4 andtherefore can be used in the same fashion as free systems were used in the proofof Theorem 5.6. For the case L = ∅ one has to use an infinite set M instead ofa general free system.

Definition 6.1. Let P1, . . . , Pr be relation symbols (where Pi is mi-ary) andlet ϕ = ϕ(x1, . . . , xn) be a first-order formula having P1, . . . , Pr as its onlyextralogical symbol. The (m1, . . . ,mr, n)-operator Gϕ attached to ϕ is definedas follows: for any nonempty M and Ri ⊆Mmi for i = 1, . . . , r,

GϕM (R1, . . . , Rr) = {(a1, . . . , an) ∈Mn : (M,R1, . . . , Rr) |= ϕ(a1, . . . , an)}.

The mapping-invariant formulas of the first order language whose extralogicalsymbols are P1, . . . , Pr are all formulas generated with the rules described in theprevious section except that instead of rules 2 and 3 we now have:

2 Pi(y1, . . . , ymi) is mapping-invariant for all distinct y1, . . . , ymi

for all i =1, . . . , r.

3 ¬∃y1 . . . ymiPi(y1, . . . , ymi

) is mapping-invariant for all distinct y1, . . . , ymi

for all i = 1, . . . , r.

Theorem 6.3. Let F be an (m1, . . . ,mr, n)-ary operator. Then F is mapping-invariant if and only if F = Gϕ for some mapping-invariant formula ϕ =ϕ(x1, . . . , xn) in a language whose only extralogical symbols are the relationalsymbols P1, . . . , Pr (where Pi is mi-ary and corresponds to Ri).

17

Proof. As in the proof of Theorem 5.7, we find a mapping-invariant formulaψL(x1, . . . , xn) for each L ⊆ {1, . . . , r} in such a way that FM (R1, . . . , Rr) =GψL

M (R1, . . . , Rr) if R1 ⊆Mm1 , . . . , Rr ⊆Mmr and Ri 6= ∅ for i ∈ L. The trickfor coding all the ψL in only one mapping-invariant formula ϕ(x1, . . . , xn) isthe same as the one presented in the proof of Theorem 5.7. We take for ϕ thedisjunction of all formulas of the form

ψL(x1, . . . , xn) ∧∧i∈L

∃yPi(y) ∧∧i 6∈L

¬∃yPi(y)

7 The functional type setting

Let us consider now a modification of the finite type hierarchy. We start with thebasic type 0 of individuals and the boolean type b. The finite functional typesare obtained from the types 0 and b by the rule that whenever τ1, . . . , τn, µare finite functional types, then also (τ1, . . . , τn → µ) is a finite functionaltype. We will call them more briefly functional types. Given a nonempty setM , we associate to each functional type τ a corresponding τ -universe Mτ overM . The starting universes are the universe of individuals M0 = M and theboolean universe Mb = {0, 1}. The remainder universes are obtained recursivelyaccording to the rule that if τ = (τ1, . . . , τn → µ) then Mτ is the set of allmappings h : Mτ1 × . . .×Mτn

→Mµ.Relational types, universes, objects, and operators can be represented in the

functional setting by replacing hereditarily every relation by its characteristicfunction. For every relational type τ there is a corresponding functional typeτ∗ defined recursively as follows: 0∗ = 0 and

(τ1, . . . , τn)∗ = (τ∗1 , . . . , τ∗n → b).

Every element a of Mτ corresponds also to an element a∗ of Mτ∗ . This cor-respondence is in fact a bijection between Mτ and Mτ∗ . For a ∈ M0 we justtake a∗ = a. If τ = (τ1, . . . , τn) and a ∈Mτ , we define a∗ as the mapping fromMτ∗1

× . . .×Mτ∗n into {0, 1} such that

a∗(a∗1, . . . , a∗n) = χa(a1, . . . , an)

for all a1 ∈Mτ1 , . . . , an ∈Mτn , where χa is the characteristic function of a as asubset of Mτ1 × . . .×Mτn

.Let τ be a functional type and let f : M → N be a surjective mapping.

In general it is not possible to extend f naturally to a corresponding totalmapping fτ : Mτ → Nτ for any functional type τ as I did for relational types,but nevertheless we can extend it as a partial surjective mapping. If τ = 0we put fτ = f and if τ = b we take for fτ the identity on Mb = {0, 1}. Ifτ = (τ1, . . . , τn → µ), and every fτi

is surjective, we can define fτ as follows.

18

1. Its domain is the subset of Mτ which consists of all mappings

a : Mτ1 × . . .×Mτn→Mµ

such that

(a) a(a1, . . . , an) ∈ dom (fµ) for all a1 ∈ dom (fτ1), . . . , an ∈ dom (fτn)

(b) for all a1 ∈ dom (fτ1), . . . , an ∈ dom (fτn), b1 ∈ dom (fτ1), . . . , bn ∈

dom (fτn), if fτi

(ai) = fτi(bi) for all i = 1, . . . , n, then

fµ(a(a1, . . . , an)) = fµ(a(b1, . . . , bn)).

2. For any such a we define fτ (a) as the mapping from Nτ1 × . . . ,×Nτninto

Nµ such that for all a1 ∈ dom (fτ1), . . . , an ∈ dom (fτn),

fτ (a)(fτ1(a1), . . . , fτn(an)) = fµ(a(a1, . . . , an)).

We check now that fτ is surjective. Let b ∈ Nτ . For any c ∈ Nµ choosea corresponding c′ ∈ dom (fµ) such that fµ(c′) = c. Define a ∈ Mτ as themapping a : Mτ1 × . . .×Mτn

→ Mµ such that for all a1 ∈ Mτ1 , . . . , an ∈ Mτn,

a(a1, . . . , an) = (b(fτ1(a1), . . . , fτn(an)))′. Observe that a ∈ dom fτ . Now,

given b1 ∈ Nτ1 , . . . , bn ∈ Nτntake a1 ∈ Mτ1 , . . . , an ∈ Mτn

with fτi(ai) = bi

for all i = 1, . . . , n, and note that b(b1, . . . , bn) = fµ(a(a1, . . . , an)). Thereforeb = fτ (a).

We can also think of extending a similarity relation π between M and N toevery τ -universe for any functional type τ . The extension πτ will be a similarityrelation between its domain dom (πτ ) ⊆ Mτ and its range rng (πτ ) ⊆ Nτ , sonot necessarily a similarity between Mτ and Nτ . We start by taking π0 = πand by taking as πb the identity in the boolean universe Mb = Nb = {0, 1}. Letτ = (τ1, . . . , τn → µ). We define πτ ⊆ Mτ ×Nτ by stipulating that for a ∈ Mτ

an b ∈ Nτ ,

(∗) πτ (a, b) if and only if for all a1 ∈Mτ1 , . . . , an ∈Mτn , b1 ∈ Nτ1 , . . . , bn ∈ Nτn

if πτi(ai, bi) for i = 1, . . . , n, then πµ(a(a1, . . . , an), b(b1, . . . , bn)).

This last proposal of extensions seems very natural when working with sim-ilarities. For any given similarity π between M and N there is a unique system(πτ )τ a functional type of relations πτ ⊆Mτ ×Nτ which satisfy the defining con-dition (∗) stated above. But if π is a surjective mapping, we obtain the sameextension πτ if we consider it only as a similarity relation. Moreover, if π is a re-lational composition of mappings and inverses of mappings, then the extensionsπτ are nothing more than the corresponding composition of extensions of map-pings. This has not been noticed by Feferman in [3] and therefore he distinguisesbetween what he calls homomorphism-invariance and similarity-invariance. Be-fore proving the equivalence we need to show an easy result according to withevery chain of surjective mappings and inverses of them can be shortened tolength two.

19

Lemma 7.1. If π is a similarity relation between M and N , there is a set Kan surjective mappings f : K →M and g : K → N such that π = f−1 ◦ g.

Proof. Let K be a set of indexes big enough to enumerate π as a set of pairs inthe form

π = {(ai, bi) : i ∈ K}

and define f : K →M by f(i) = ai and g : K → N by g(i) = bi.

Proposition 7.2. Let τ be a functional type.

1. If f : M → N is a surjective mapping, then it is a similarity relationbetween M and N and its extension fτ as a mapping coincides with itsextension as a similarity relation.

2. Assume that M1, . . . ,Mn are sets and f1, . . . , fn−1 are mappings such thatM = M1, N = Mn and for every i = 1, . . . , n, fi is from Mi onto Mi+1

or it is from Mi+1 onto Mi, and π is the relational product R1 ◦ . . .◦Rn−1

where Ri = fi if fi : Mi →Mi+1 and Ri = f−1i if fi : Mi+1 →Mi. Then

πτ is the relational product R1τ ◦ . . . ◦ Rn−1τ , where Riτ is the partialmapping fiτ if fi : Mi →Mi+1 and it is its inverse if fi : Mi+1 →Mi.

Proof. 1. It is easy to check by induction on the complexity of τ that thesystem of partial mappings (fτ )τ a functional type satisfies condition (∗). 2. Thisis proven by induction on n, but using Lemma 7.1 we see that it will be enoughto prove the result in three particular cases:

Case 1. f : K → M , g : K → N are surjective and π = f−1 ◦ g. Weneed only to check that the system (f−1

τ ◦gτ )τ a functional type satisfies condition(∗) and this is done by induction on the complexity of τ . Consider the caseτ = (τ1, . . . , τn → µ). Let a ∈ Mτ and b ∈ Nτ . Let R = f−1

τ ◦ gτ . If R(a, b)then for some c ∈ dom (fτ ) ∩ dom (gτ ) we have fτ (c) = a and gτ (c) = b. LetSi = f−1

τi◦ gτi

and assume Si(ai, bi) for all i = 1, . . . , n. Then for each i there issome ci ∈ dom (fτi) ∩ dom (gτi) such that fτi(ci) = ai and gτi(ci) = bi. Clearlyc(c1, . . . , cn) ∈ dom (fµ) ∩ dom (gµ) and fµ(c(c1, . . . , cn)) = a(a1, . . . , an), andgµ(c(c1, . . . , cn)) = b(b1, . . . , bn). Hence T (a(a1, . . . , an), b(b1, . . . , bn)) for T =f−1µ ◦ gµ. Now we assume the righthand side of (∗) holds for a, b and we show

that R(a, b). We have to find some c ∈ dom (fτ ) ∩ dom (gτ ) with fτ (c) = a andgτ (c) = b. Let ci ∈ Kτi

for each i = 1, . . . , n. If some ci is not in dom (fτi) ∪

dom (gτi), we choose for c(c1, . . . , cn) some arbitrary element of Kµ. If all ci ∈dom (fτi) but some ci 6∈ dom (gτi) we observe that a(fτ1(c1), . . . , fτn(cn)) ∈Mµ

and since fµ is surjective we can define c(c1, . . . , cn) as an element d ∈ dom (fµ)such that fµ(d) = a(fτ1(c1), . . . , fτn

(cn)). The procedure is similar if all ci ∈dom (gτi

) but some ci 6∈ dom (fτi). Assume now ci ∈ dom (fτi

)∩dom (gτi) for all

i = 1, . . . , n. Let Si = f−1τi

◦gτi . Then Si(fτi(ci), gτi(ci)) for all i and we can usethe righthand side of (∗) to make sure that there is some d ∈ dom (fµ)∩dom (gµ)such that fµ(d) = a(fτ1(c1), . . . , fτn

(cn)) and gµ(d) = b(gτ1(c1), . . . , gτn(cn)).

We take as c(c1, . . . , cn) such a d in this last case. It follows that c ∈ dom (fτ )∩dom (gτ ), fτ (c) = a and gτ (c) = b. Hence R(a, b).

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Case 2. f : M → N , g : K → N , h : K → P are surjective and π = f◦g−1◦h.Similar to the previous case. Given a ∈ Mτ and b ∈ Pτ which satisfy therighthand side of (∗) we check that a ∈ dom (fτ ) and we define in a similar wayas before some c ∈ dom (gτ ) ∩ dom (hτ ) such that fτ (a) = gτ (c) and hτ (c) = b.

Case 3. f : N → M , g : K → N , h : K → P are surjective and π =f−1 ◦ g−1 ◦ h. Also similar to the other cases. Now the main point is to startwith a ∈ Mτ and b ∈ Pτ for which the righthand side of (∗) holds and tofind c ∈ dom (gτ ) ∩ dom (hτ ) such that gτ (c) ∈ dom (fτ ), fτ (gτ (c)) = a andhτ (c) = b

In the functional type setting, every operator can be identified with anobject. Therefore it would be enough to consider only objects for all invari-ance purposes. Nevertheless, it is convenient to have both notions (object andoperator) to compare them with objects and operators of the relational typesetting. The definitions of invariance are similar to the ones given in the re-lational case, but types are functional now. Let us say that an object a offunctional type τ is mapping-invariant if for all M,N and every surjective map-ping f : M → N , aM ∈ dom (fτ ) and fτ (aM ) = aN . Feferman in [3] calls thisnotion homomorphism-invariance although the sets M,N are pure sets, withoutany added structure. He calls an object a similarity-invariant if for all M,Nand every similarity relation π between M and N , πτ (aM , aN ). This notion ofinvariance can be easily defined also for (τ1, . . . , τn;µ)-ary operators as I did inthe relational case. The details are left to the reader.

Corollary 7.3. In the functional type hierarchy an object or an operator ismapping-invariant (i.e., homomorphism-invariant in the sense of Feferman) ifand only if it is similarity-invariant.

Proof. By Proposition 7.2.

8 Feferman’s analysis of mapping-invariance

We have seen that every relational type τ can be represented as a functionaltype τ∗ and every a ∈ Mτ has a corresponding element a∗ ∈ Mτ∗ . In a similarway, to any operator F in the relational type setting corresponds an operatorF ∗ in the functional type setting. It is defined by

F ∗M (a∗1, . . . , a∗n) = FM (a1, . . . , an)∗.

Now I raise the question of what kind of invariance on relational types τ isthe correct translation of mapping-invariance on the corresponding functionaltypes τ∗. We will see that it is closely related to preimage-invariance and thatit is just preimage-invariance at the first levels of the type hierarchy. Beforediscussing preimage-invariance I introduce a new version of invariance, which isexactly the relational counterpart of mapping-invariance on functional types.

Let f : M → N be a surjective mapping. I present another way of extendingf to every universe Mτ for any relational type τ . I will call it the Feferman

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extension of f to τ and I will use the notation fFeτ for it. We start, as usual,putting fFe0 = f and define inductively fFeτ for τ = (τ1, . . . , τn). In general it isgoing to be a partial surjective mapping.

1. Its domain consists of all a ⊆ Mτ1 × . . . × Mτn such that for all a1 ∈dom (fFeτ1 ), . . . , an ∈ dom (fFeτn

), b1 ∈ dom (fFeτ1 ), . . . , bn ∈ dom (fFeτn), if

fFeτi(ai) = fFeτi

(bi) for all i = 1, . . . , n, then (a1, . . . , an) ∈ a if and only if(b1, . . . , bn) ∈ a.

2. For any such a, we define fFeτ (a) as

{(fFeτ1 (a1), . . . , fFeτn(an)) : (a1, . . . , an) ∈ a∩(dom (fFeτ1 )×. . .×dom (fFeτn

))}

Lemma 8.1. Let f : M → N be surjective, let τ be a relational type and assumea ∈Mτ . Then

1. a ∈ dom (fFeτ ) iff a∗ ∈ dom (fτ∗)

2. If a ∈ dom (fFeτ ), then fFeτ (a)∗ = fτ∗(a∗).

Proof. It is an easy induction on the complexity of τ .

The notion of invariance for objects and operators in the context of theseextensions fFeτ is defined as in the other cases. We will call it Feferman invari-ance. For instance, an object a of type τ is Feferman invariant if and only iffor any surjective mapping f : M → N , aM ∈ dom (fFeτ ) and fFeτ (aM ) = aN .Invariance for operators is defined in the obvious way.

Proposition 8.2. An operator F is Feferman invariant if and only if its cor-responding operator F ∗ in the functional type setting is mapping-invariant, i.e.,homomorphism-invariant in the sense of Feferman.

Proof. Let F be a (τ1, . . . , τn;µ)-ary operator. Assume F is Feferman invariant.Let f : M → N be surjective and let a1 ∈Mτ1 , . . . , an ∈Mτn

be such that a∗i ∈dom (fτ∗i ) for each i = 1, . . . , n. By Lemma 8.1, ai ∈ dom (fFeτi

) and fFeτi(ai)∗ =

fτ∗i (a∗i ) for every i = 1, . . . , n. By Feferman invariance of F , FM (a1, . . . , an) ∈dom (fFeµ ) and

fFeµ (FM (a1, . . . , an)) = FN (fFeτ1 (a1), . . . , fFeτn(an)).

By Lemma 8.1 F ∗M (a∗1, . . . , a∗n) = FM (a1, . . . , an)∗ ∈ dom (fµ∗) and

fµ∗(F ∗M (a∗1, . . . , a∗n)) = fFeµ (FM (a1, . . . , an))∗.

Hence

fµ∗(F ∗M (a∗1, . . . , a∗n)) = FN (fFeτ1 (a1), . . . , fFeτn

(an))∗ = F ∗N (fτ∗1 (a∗1), . . . , fτ∗n(a∗n)).

The other direction has a similar proof.

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Let f : M → N be a surjective mapping. I propose a last way of extendingf to every universe Mτ for any relational type τ . We well call it the preimageextension of f to τ and we will use the notation fpτ for it. The extension fpτwill not be in general a total mapping from Mτ onto Nτ . It is a mapping butits domain is only a subset of Mτ . As usual, we start with fp0 = f . Now letτ = (τ1, . . . , τn).

1. The domain of fpτ consists of all relations a ⊆Mτ1 ,× . . .×Mτnfor which

there is some b ⊆ Nτ1 × . . .×Nτnsuch that

a = {(a1, . . . , an) ∈ dom (fpτ1)×. . .×dom (fpτn) : (fpτ1(a1), . . . , fpτn

(an)) ∈ b}.

2. For each such a ∈ dom (fpτ ) we set

fpτ (a) = {(fpτ1(a1), . . . , fpτn(an)) : (a1, . . . , an) ∈ a}.

Let F be an operator of relational type (τ1, . . . , τn; τ). We call F preimage-invariant if for any sets M,N , for any surjective mapping f : M → N ,if a1 ∈ dom (fpτ1), . . . , an ∈ dom (fpτn

), then FM (a1, . . . , an) ∈ dom (fpτ ) andfpτ (FM (a1, . . . , an)) = FN (fpτ1(a1), . . . , fpτn

(an)). An object a of relational typeτ is preimage-invariant if for any sets M,N , for any surjective mapping f :M → N , aM ∈ dom (fpτ ) and fpτ (aM ) = aN .

We will concentrate the following discussion on operators, but the resultscan be easily extended also to objects.

Lemma 8.3. If τ = 0n and f : M → N is surjective, then fFeτ = fpτ .

Proof. We show that dom (fFeτ ) = dom (fpτ ) and fFeτ (a) = fpτ (a) for every a ∈dom (fFeτ ). Let a ∈ dom (fFeτ ) and let b = {(f(a1), . . . , f(an)) : (a1, . . . , an) ∈a}. It follows that a = {(a1, . . . , an) : (f(a1), . . . , f(an)) ∈ b} and thereforea ∈ dom (fpτ ). Clearly in this case fFeτ (a) = b = fpτ (a). On the other hand, if a ∈dom (fpτ ) there is some b ⊆ Nn such that a = {(a1, . . . , an) : (f(a1), . . . , f(an) ∈b}. Whenever a1, . . . , an, b1, . . . , bn ∈M , (a1, . . . , an) ∈ a, and f(ai) = f(bi) forevery i = 1, . . . , n, we have that (f(b1), . . . , f(bn)) = (f(a1), . . . , f(an)) ∈ b andtherefore (b1, . . . , bn) ∈ a. This means that a ∈ dom (fpτ ).

Theorem 8.4. An (m1, . . . ,mr, n)-ary operator F is preimage-invariant if andonly if its corresponding operator in the functional type setting F ∗ is mapping-invariant, i.e., homomorphism-invariant in the sense of Feferman.

Proof. By Proposition 8.2 is enough to prove that F is preimage-invariant if andonly if it is Feferman invariant. Let τi = 0mi for every i = 1, . . . , r. Assume F ispreimage-invariant and let f : M → N be surjective. Let R1 ⊆Mm1 , . . . , Rr ⊆Mmr and assume Ri ∈ dom (fFeτi

) for each i = 1, . . . , r. By Lemma 8.3 Ri ∈dom (fpτi

) and fpτi(Ri) = fFeτi

(Ri) for every i = 1, . . . , r. By preimage-invariance,FM (R1, . . . , Rr) ∈ dom (fp0n) and

fp0n(FM (R1, . . . , Rr)) = FN (fpτ1(R1), . . . , fpτr(Rr)).

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By Lemma 8.3 again, FM (R1, . . . , Rr) ∈ dom (fFe0n ) and fFe0n (FM (R1, . . . , Rr)) =fp0n(FM (R1, . . . , Rr)). Therefore

fFe0n (FM (R1, . . . , Rr)) = FN (fFeτ1 (R1), . . . , fFeτr(Rr))

and F is Feferman invariant. The other direction is proven in a similar way.

ACKNOWLEDGMENT

Work partially supported by grant BFM2002-01034 of Spanish MCYT and grant2002SGR 00126 of Catalan DURSI. I thank Xavier Caicedo for correcting somemistaken statements concerning Linsdtrom quantifiers made in a previous ver-sion. I also thank Genoveva Martı for some comments improving the exposition.

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