aa r X i v : . [ m a t h . C T ] F e b Are locally finite MV-algebras a variety?
Marco Abbadini a , Luca Spada a, ∗ a Dipartimento di Matematica, Universit`a degli Studi di Salerno, Piazza Renato Caccioppoli, 2, 84084, Fisciano (SA), Italy
Abstract
We answer Mundici’s problem number 3 (D. Mundici.
Advanced Lukasiewicz calculus.
Trends in Logic Vol.35. Springer 2011, p. 235):
Is the category of locally finite MV-algebras equivalent to an equational class?
We prove:1. The category of locally finite MV-algebras is not equivalent to any finitary variety.2. More is true: the category of locally finite MV-algebras is not equivalent to any finitely-sorted finitaryquasi-variety.3. The category of locally finite MV-algebras is equivalent to an infinitary variety; with operations of atmost countable arity.4. The category of locally finite MV-algebras is equivalent to a countably-sorted finitary variety.Our proofs rest upon the duality between locally finite MV-algebras and the category of “multisets” byR. Cignoli, E. J. Dubuc and D. Mundici, and categorical characterisations of varieties and quasi-varietiesproved by J. Duskin, J. R. Isbell, F. W. Lawvere and others. In fact no knowledge on MV-algebras is needed,apart from the aforementioned duality.
Keywords:
MV-algebra, multisets, algebraic category, infinitary variety
1. Introduction
MV-algebras were introduced in [10] to serve as algebraic semantics for the many-valued Lukasiewiczpropositional logic. They are defined as algebras of the form h A, ⊕ , ¬ , i such that h A, ⊕ , i is a commutativemonoid, the unary operation ¬ satisfies ¬¬ x = x and ¬ ⊕ x = ¬
0, and the following axiom holds: ¬ ( ¬ x ⊕ y ) ⊕ y ) = ¬ ( ¬ y ⊕ x ) ⊕ x ). In their over fifty years of history, MV-algebras have found surprising applicationsin many fields of mathematics (see, e.g., [8, 11, 20, 21, 25, 26]); the reader is referred to [13] for their basictheory and to [24] for more advanced topics. Examples of MV-algebras are given by the real interval [0 , x ⊕ y := min { x + y, } and ¬ x := 1 − x, (1)as well as its subalgebra of rational numbers, and the ones generated by any rational number; the latter alltake the form S n := { , n , n . . . , n − n , } for some n ∈ N + := { , , , . . . } . It is important to notice at thispoint that the MV-operations ⊕ and ¬ always allow to define a lattice order by setting x ≤ y if and only if ¬ ( ¬ x ⊕ y ) ⊕ y = y. ∗ Corresponding author
Email addresses: [email protected] (Marco Abbadini), [email protected] (Luca Spada)
Preprint submitted to Elsevier February 25, 2021 n MV-algebra is called a chain if it is linearly ordered under the above-defined order. The finite MV-chains are exactly the algebras S n for n ∈ N + and every finite MV-algebra is a direct product of finitechains [13, Proposition 3.6.5]. This simple observation provides an intuitive entryway to a rich front inMV-algebras: their duality theory. Indeed, any finite MV-chain is completely characterised by the naturalnumber n as above. Thus, every finite MV-algebra can be recovered from a finite set in which every elementhas attached a natural number different from 0, i.e. from a finite multiset . We call these numbers thedenominators (somewhere else called multiplicities) of the elements, for reasons that will be made clearin Remark 2.5. Furthermore, a homomorphism exists from an MV-chain S m to an MV-chain S n if andonly if m divides n . Therefore, we shall consider maps between multisets that decrease —in the order ofdivisibility— denominators, more precisely: if ( S , d : S → N + ) and ( S , d : S → N + ) are two finitemultisets, a function f : S → S is said to decrease denominators if, for any s ∈ S , d ( f ( s )) divides d ( s ).This correspondence is in fact a categorical equivalence between the category MV f of finite MV-algebras,together with their homomorphisms, and the category MS f of finite multisets and functions that decreasedenominators.As it often happens, the “toy” duality described above hints at a more interesting one that is obtained bytaking, on the one hand, the completion of MV f under all directed co-limits —called ind - MV f — and, on theother, the completion of MS f under all co-directed limits —called pro - MS f — (see [18, Chapter 6] for moredetails and applications of this technique). Indeed, one immediately obtains a formal categorical duality: ind - MV f ≃ ( pro - MS f ) op . Recall that, as a general concept in universal algebra, an algebra A is called locallyfinite if every finitely generated subalgebra of A is finite. In any variety V , the locally finite algebras areexactly the direct limits (= directed co-limits) of finite algebras in V . Thus, ind - MV f is equivalent to thecategory MV lf of locally finite MV-algebras and it remains to provide a more familiar description of pro - MS f .This is done in [12], where the authors prove that a certain category MS of “multisets” (with possibly infiniteunderlying set and possibly infinite denominator function) is the pro-completion of the category of finitemultisets (see Definition 2.3 for details).As a matter of fact, there are MV-algebras which are not locally finite, the MV-algebra [0 ,
1] withthe operations defined in (1) being a prime example, for its subalgebras generated by any of its irrationalelements are infinite. Therefore, locally finite MV-algebras are a proper subclass of MV-algebras. The classof locally finite MV-algebras is easily seen to be closed under homomorphic images, subalgebras and finiteproducts. However, it is not closed under arbitrary products. Nonetheless, the category of locally finiteMV-algebras has all products in the categorical sense and they can be described as a certain subalgebrasof the classical algebraic product [12, Theorem 5.4]. Driven by these considerations, in one of the elevenproblems at the end of [24], D. Mundici asks whether the category of locally finite MV-algebras is equivalentto an equational class. We point out that, if one such equivalence exists, it is necessarily not concrete,because the underlying set functor from MV lf to Set does not preserve products.We answer Mundici’s question using categorical results that give precise descriptions of the categoriesthat are equivalent to varieties of algebras. It turns out that Mundici’s question can be answered both in thenegative and in the positive, depending on restrictions on the language that one wants to assume. The studyof abstract characterisations of varieties and quasi-varieties has a long history in category theory, startingwith the works of M. Barr, J. Duskin, W. Felscher, J. R. Isbell, F. W. Lawvere and F. E. J. Linton in the1960s. The book [5] is a great source of information and provides an updated account on the subject.We conclude this section by briefly describing our results and the techniques used to prove them.Section 2 includes the definition and some basic information on the category MS . A presentation of theresults about algebraic categories needed in the paper is given.In Section 3 we prove that the category MV lf of locally finite MV-algebras is not equivalent to anyvariety of finitary algebras (Corollary 3.17). Our argument proceeds along the following lines. Every varietyof finitary algebras admits a “finitely generated regular generator” (see Definition 2.20): the free algebraover one generator (see Theorem 2.26). We show that MV lf does not admit any finitely generated regulargenerator by proving a dual statement for MS : MS does not admit any finitely co-generated regular co-generator. Indeed we prove that, in MS ,(T1) finitely co-generated objects have a finite underlying set (Proposition 3.1);2T2) regular co-generators have an infinite underlying set (Corollary 3.14).Items (T1) and (T2) together imply that no object of MS is simultaneously finitely co-generated and aregular co-generator, thus proving that the opposite category MV lf does not admit any finitely generatedregular generator. We conclude that MV lf is not equivalent to any variety of finitary algebras. Since everyquasi-variety of finitary algebras admits a finitely generated regular generator, too, the argument aboveimplies the stronger fact that MV lf is not equivalent to any quasi-variety of finitary algebras, either.In fact, our results are even more general, entailing that also a finitary language with finitely many sortsdoes not afford a (quasi-)equational axiomatisation: MV lf is not equivalent to any finitely sorted quasi-variety of finitary algebras (Theorem 3.16). The ideas are similar: Every finitely sorted quasi-variety offinitary algebras admits a regularly generating finite set of finitely generated objects (these objects are —roughly speaking— one for each sort: they correspond to the free algebra over an element placed in thatparticular sort). We have:(T3) a regularly co-generating set either has infinitely many objects, or contains an object whose underlyingset is infinite (Theorem 3.12).Items (T1) and (T3) together imply that MS admits no regularly co-generating finite set of finitely co-generated objects; thus MV lf admits no regularly generating finite set of finitely generated objects, therefore MV lf is not equivalent to any finitely sorted quasi-variety of finitary algebras.In Section 4 we exhibit a countable family of regular injective multisets that is a regularly co-generatingabstractly co-finite set. Using once more the categorical characterisation outlined in Theorem 2.26, wededuce that MV lf is equivalent to a quasi-variety of finitary algebras in a language with countably manysorts (Proposition 4.4). At this point we note that the product of the aforementioned family is a regularinjective regular co-generator of MS . We deduce that MV lf is equivalent to a quasi-variety of algebras in alanguage with operations of at most countable arity (Proposition 4.6).In Section 5, we improve the results of the previous section by studying the (internal) equivalence relationsof MV lf (see Definition 2.24 for a short reminder of their definition). Working once more in the dual category MS , we prove that all reflexive relations in MV lf are effective equivalence relations; thus MV lf is a Mal’tsevcategory (i.e. every reflexive relation is an equivalence relation) and every equivalence relation in MV lf iseffective. Using once more the abstract characterisation of Theorem 2.26 we deduce that MV lf is equivalentto a variety in a language:(T4) with one sort and operations of at most countable arity (Theorem 5.4), and(T5) with countably many sorts and operations of finite arity (Theorem 5.5).Finally, in the Appendix we offer some supplementary results on the topology of the supernatural numbers(to be defined in Section 2) and on internal relations in MV lf . Despite the fact that these results are notneeded in the proofs outlined above, we think that they might be interesting in the study of the category MS .
2. PreliminariesNotation 2.1.
We write N for the set of natural numbers { , , , . . . } , N + for { , , , . . . } and P for theset of prime numbers. Throughout the paper, we assume all categories to be locally small. Unless otherwisestated, varieties and quasi-varieties possibly admit operations of infinite arity in their signatures. Finally, ascustomary, we use the prefix “co-” for the dual of a categorical concept, obtained by switching the direction ofany arrow involved in the definition; we depart from this notation only when the dual notion is traditionallyindicated with a different name (e.g., monic and epic).As customary, we call Stone space any topological space that is compact, Hausdorff and has a basis ofclopen subsets. We denote by
Stone the category of Stone spaces with continuous maps. In the paper weonly assume some basic knowledge on Stone spaces and Boolean algebras, for more information the readeris referred to [15]. 3 .1. Locally finite MV-algebras
In [12] the authors give a description of pro - MS f in terms of “possibly infinite multisets”. It turnsout that infinity in pro - MS f appears in two different aspects: infinite cardinality of the underlying set andinfinitely-valued denominators. Definition 2.2. A supernatural number is a function ν : P −→ { , , . . . , ∞} . Upon writing ν ≤ µ iff ν ( p ) ≤ µ ( p ) for each p ∈ P , the set of supernatural numbers forms a complete latticedenoted by N . We say that ν is finite iff ∞ does not belong to the range of ν and ν ( p ) is nonzero only forfinitely many p .Regarding ν as assigning exponents to prime numbers and using the Unique Factorisation Theorem,we may assign to each number n ∈ N + a corresponding finite supernatural number ν n . In particular, ν constantly takes value 0 on every prime number. The one-one correspondence n ↔ ν n identifies thesub-lattice of finite supernatural numbers with the lattice ( N + , div), where a div b if and only if a divides b .We equip N with the topology induced by the following open basis: U n := { ν ∈ N | ν ≥ ν n } for n ∈ N + (2)or equivalently by the open sub-basis of all sets of the form U p,k = { ν ∈ N | ν ( p ) > k } , for p ∈ P and k ∈ N . This definition differs from the one used in [12] —which probably contains a typo— but agrees with the onein [24, Section 8.4]. We postpone to the Appendix a discussion on the various possible formulations andtheir equivalences.
Definition 2.3.
We define the category MS of multisets . Objects are pairs ( X, ζ ), where X is a Stone space,and ζ is a continuous map from X to N , equipped with the topology defined in (2). If ( X, ζ X ) and ( Y, ζ Y )are two multisets, an arrow between them is a continuous function f : X → Y such that,for every x ∈ X, ζ X ( x ) ≥ ζ Y ( f ( x )) . (3)When more that one multiset is involved in an argument we write ζ X to make it clear that we refer tothe map associated to the multiset X . Theorem 2.4 ([12, Theorem 6.8]) . The categories MV lf and MS are dually equivalent.Remark . The above duality is part of a larger duality between semisimple MV-algebras and closedsubspaces of the hyper-cubes [0 , κ , for κ that ranges among cardinals. See [22, 23] and in particular [9,Remark 10] where the duality of Theorem 2.4 is framed in the setting of [22]. Here we limit ourselvesto notice that every multiset can be homeomorphically embedded into some power of [0 ,
1] in such a waythat the denominator function ζ equals the least common multiple of the denominators of the (necessarilyrational) points in the image of the embedding. For more information about this kind of embeddings and fora topological characterisation of the maps from a compact and Hausdorff space into N that are concretelyrepresentable as “denominator maps” we refer the reader to [1].For the reasons sketched in Remark 2.5, if ( X, ζ ) is a multiset we call ζ ( x ) the denominator of x , forevery x ∈ X , and ζ is called denominator map . When there is no danger of confusion, we write X insteadof ( X, ζ ). Since ( N + , div) embeds into N , we also say that f decreases denominators when it satisfy theproperty described in (3). Further, we say that a function f : X → Y preserves denominators whenever, forevery x ∈ X , we have ζ X ( x ) = ζ Y ( f ( x )). Notice that every locally finite MV-algebra is semisimple. .2. Properties of the category MS In this subsection we investigate some basic categorical constructions in MS . The following crucial remarkand the ensuing Theorem 2.7 greatly simplify calculations.The category MS has a natural forgetful functor U into the category Stone of Stone spaces which simplyforgets denominator maps. Vice versa, there are at least two natural denominator maps that can be attachedto any Stone space X to make it into a multiset: the denominator map ζ , where ζ ( x )( p ) := 0 for every x ∈ X and p ∈ P , and the denominator map ζ ∞ , where ζ ∞ ( x )( p ) := ∞ for every x ∈ X and p ∈ P . Thisrelation between the two categories resembles the one between Set and
Top in which one can endow anyset with the discrete or the indiscrete topology. In this subsection we shall see that this similarity can beformally stated by proving (see Theorem 2.7 below) that the forgetful functor U : MS → Stone is topological —to be defined in the next paragraph. From this fact we shall derive many consequences about limits,colimits, epic and monic arrows, some of which were already observed in [12].We recall basic notions and results concerning topological functors. For more details, we refer to [3,Chapter 21]. Given a faithful functor G : A → X , a family of arrows { f i : A → A i } i ∈ I in A is called G -initial provided that, for each arrow h : G ( B ) → G ( A ) in X , if for every i ∈ I there exists an arrow g i : B → A i in A with G ( g i ) = G ( f i ) ◦ h , then there exists an arrow h : B → A in A such that G ( h ) = h (in loose terms, h is an A -arrow whenever all compositions G ( f i ) ◦ h are so). BA A i ∃ ! h g i f i G ( B ) G ( A ) G ( A i ) G ( h )= h G ( g i ) G ( f i ) Furthermore, we say that a family of arrows { f i : A → A i } i ∈ I in A is a lift of { f i : X → G ( A i ) } i ∈ I if G ( A ) = X and G ( f i ) = f i . Finally, a faithful functor G : A → X is called topological (see e.g. [3, Definition21.1]) provided that every class-indexed family of arrows { f i : X → GA i } i ∈ I in X has a unique G -initial lift. Lemma 2.6.
Let X be a topological space, and let { f i : X → N } i ∈ I be a family of continuous functions.Then, the function W i ∈ I f i : X → N that maps x to W i ∈ I f i ( x ) is continuous.Proof. A proof, which can also be found in [12, Lemma 3.4], runs as follows. For p ∈ P and k ∈ N we have _ i ∈ I f i ! − [ U p,k ] = ( x ∈ X | _ i ∈ I f i ( x ) ∈ U p,k ) = ( x ∈ X | _ i ∈ I f i ( x )( p ) > k ) = { x ∈ X | ∃ i ∈ I s.t. f i ( x )( p ) > k } = [ i ∈ I { x ∈ X | f i ( x )( p ) > k } = [ i ∈ I f − i [ U p,k ] . We let U denote the forgetful functor from MS to Stone . Theorem 2.7.
Every family of arrows { f i : X → U ( X i , ζ X i ) } i ∈ I in Stone admits a unique U -initial lift,namely { ¯ f i : ( X, ζ X ) → ( X i , ζ X i ) } i ∈ I , where, for each x ∈ X , ζ X ( x ) := _ i ∈ I ζ X i ( f i ( x )) , and, for each i ∈ I , ¯ f i is just the function f i . Therefore the functor U : MS → Stone is topological. roof. By Lemma 2.6, (
X, ζ X ) is a multiset. Notice that each f i : X → X i decreases denominators withrespect to ζ X i and the newly defined ζ X , thus it corresponds to an arrow of multisets, denoted by ¯ f i , so U ( ¯ f i ) = f i . Hence the family { f i : X → U ( X i , ζ X i ) } i ∈ I has a lift { ¯ f i : ( X, ζ X ) → ( X i , ζ X i ) } i ∈ I . Let usprove that it is U -initial. Let ( C, ζ C ) be a multiset and h : C → X be an arrow in Stone . Suppose thateach composite U ( ¯ f i ) ◦ h : C → X i decreases the denominators, i.e., for every i ∈ I , ζ X i ( U ( ¯ f i )( h ( c ))) ≤ ζ C ( c ). Therefore, ζ X ( h ( c )) = W i ∈ I ζ X i ( f i ( h ( c ))) ≤ ζ C ( c ). This shows that h decreases denominators, so { ¯ f i : ( X, ζ X ) → ( X i , ζ X i ) } i ∈ I is U -initial. Uniqueness follows from the easily verifiable fact that the functor U : MS → Stone has the property that an iso f in MS is an identity whenever U ( f ) is an identity, see [3,Proposition 21.5] . Corollary 2.8.
A family of arrows { f i : ( X, ζ ) → ( X i , ζ X i ) } i ∈ I in MS is U -initial if and only if, for every x ∈ X , we have ζ X ( x ) = W i ∈ I ζ X i ( f i ( x )) . In particular, an arrow of multisets f : ( X, ζ X ) → ( Y, ζ Y ) is U -initial if and only if it preserves denominators.Proof. A family of arrows { f i : ( X, ζ ) → ( X i , ζ X i ) } i ∈ I in MS is a clearly lift of { U ( f i ) : X → U ( X i , ζ X i ) } i ∈ I .By the description and the uniqueness of U -initial lifts (Theorem 2.7), { f i : ( X, ζ ) → ( X i , ζ X i ) } i ∈ I is U -initialif and only if we have ζ X ( x ) = W i ∈ I ζ X i ( x ). Corollary 2.9.
The category of multisets is complete and co-complete.Proof.
By [3, Theorem 21.16.(1)], if G : A → X is topological, then A is (co-)complete if and only if X is(co-)complete. The claim hence follows from the fact that Stone , being dually equivalent to the variety ofBoolean algebras, is complete and co-complete.
Remark . Every topological functor G : A → X has a left adjoint L ′ (the discrete functor) and a rightadjoint R ′ (the indiscrete functor), which are full embeddings satisfying G ◦ L ′ = 1 X and G ◦ R ′ = 1 X [3,Proposition 21.12]. The left adjoint L to the forgetful functor U : MS → Stone maps a Stone space S to themultiset ( S, ζ ∞ ), where ζ ∞ ( x )( p ) = ∞ for every x ∈ S and p ∈ P . The right adjoint R to U maps a Stonespace S to the multiset ( S, ζ ), where ζ ( x )( p ) = 0 for every x ∈ S and p ∈ P . The functor R : Stone → MS is a full embedding, and U is left adjoint to R ; it follows that Stone is reflective in MS . For similar reasons, Stone is co-reflective in MS . Since the forgetful functor U : MS → Stone has both a right and left adjoint,the functor U preserves both limits and co-limits. This entails that limits in the category MS can easily bedescribed in terms of limits in the category of Stone spaces. Lemma 2.11.
Let D : I → MS be a diagram. Then L = { l i : ( X, ζ X ) → D ( i ) } i ∈ I is a limit of D if and onlyif U ( L ) is a limit of U ◦ D and, for every x ∈ X , we have ζ X ( x ) = W i ∈ I ζ D ( i ) ( l i ( x )) .Proof. The forgetful functor U : MS → Stone is faithful and preserves all limits by Remark 2.10. Therefore, L is a limit of D if and only if U ( L ) is a limit of U ◦ D and L is U -initial [3, Proposition 13.15]. ByCorollary 2.8, L is U -initial if and only if, for every x ∈ X , we have ζ X ( x ) = W i ∈ I ζ D ( i ) ( l i ( x )). Remark . We recall that limits in
Stone are the same as in
Top . This is guaranteed by the followingfacts. The category
Stone is a reflective full subcategory of the category of compact Hausdorff spacesand continuous maps: the reflector assigns to each compact Hausdorff space the space of its connectedcomponents [7, Proposition 5.7.12]. Furthermore, the category of compact Hausdorff spaces is a reflectivefull subcategory of the category
Top of topological spaces and continuous maps: the reflector assigns to eachtopological space its Stone- ˇCech compactification (see [18, Chapter IV, Section 2]). Hence, the category
Stone is a reflective full subcategory of
Top . Therefore, the forgetful functor from
Stone to Top preservesand reflects limits.A characterisation of U -final lifts (i.e. the dual notion of U -initial lifts) and co-limits in MS in terms ofco-limits in Stone is also available, although in general it is not as explicit as for U -initial lifts and limits. A functor with this property is called amnestic, see [3, Definition 3.27]. emma 2.13. Every family of arrows { f i : U ( X i , ζ X i ) → X } i ∈ I in Stone admits a unique U -final lift, whichis { ¯ f i : ( X i , ζ X i ) → ( X, ζ X ) } i ∈ I , where ζ X is the greatest continuous function from X to N such that, forevery x ∈ X , every i ∈ I and every y ∈ X i such that l i ( y ) = x , we have ζ X ( x ) ≤ ζ D ( i ) ( y ) . For I finite, wehave ζ X ( x ) = ^ i ∈ I,y ∈ D ( i ): l i ( y )= x ζ D ( i ) ( y ) . Proof.
From Theorem 2.7, we have that every family of arrows { l i : U ( X i , ζ X i ) → X } i ∈ I in Stone admits aunique U -final lift —see the Topological Duality Theorem [3, Proposition 21.9]— and this provides the liftdescribed in the statement. We prove that, when I is a finite set, the function ζ X is precisely ζ X ( x ) = ^ i ∈ I,y ∈ X i : l i ( y )= x ζ X i ( y ) . (4)Verifying this statement amounts to prove that the so-defined ζ X is continuous, since the remaining condi-tions are easily seen to hold. We prove that the preimage under ζ X of a closed set is closed: for every p ∈ P and k ∈ N we have ζ − X [ N \ U p,k ] = ζ − X [ { ν ∈ N | ν ( p ) ≤ k } ]= { x ∈ X | ζ X ( x ) ∈ { ν ∈ N | ν ( p ) ≤ k }} = { x ∈ X | ζ X ( x )( p ) ≤ k } = x ∈ X | ^ i ∈ I,y ∈ X i : l i ( y )= x ζ X i ( y ) ( p ) ≤ k = x ∈ X | ^ i ∈ I,y ∈ X i : l i ( y )= x ζ X i ( y )( p ) ≤ k = { x ∈ X | ∃ i ∈ I, ∃ y ∈ X i : l i ( y ) = x, ζ X i ( y )( p ) ≤ k } = [ i ∈ I { x ∈ X | ∃ y ∈ X i : l i ( y ) = x, ζ X i ( y )( p ) ≤ k } = [ i ∈ I (cid:8) x ∈ X | ∃ y ∈ ζ − X i [ N \ U p,k ] : l i ( y ) = x (cid:9) = [ i ∈ I l i (cid:2) ζ − X i [ N \ U p,k ] (cid:3) , (5)which is closed since I is finite.It can be proved, although we do not need this result in this paper, that, in general, when I is infinite,(4) fails to define the denominator map of the U -final lift. Lemma 2.14.
Let D : I → MS be a diagram. Then L = { l i : D ( i ) → ( X, ζ X ) } i ∈ I is a co-limit of D if andonly if U ( L ) is a co-limit of U ◦ D and ζ X is the greatest continuous function from X to N such that, forevery x ∈ X , every i ∈ I and every y ∈ X i such that l i ( y ) = x , we have ζ X ( x ) ≤ ζ D ( i ) ( y ) . For I finite, theconditions on ζ X are satisfied by ζ X ( x ) = ^ i ∈ I,y ∈ D ( i ): l i ( y )= x ζ D ( i ) ( y ) . Proof.
The forgetful functor U : MS → Stone is faithful and preserves all co-limits by Remark 2.10. There-fore, L is a co-limit of D if and only if U ( L ) is a co-limit of U ◦ D and L is U -final [3, Proposition 13.15].Using the characterisation of U -final lifts available from Lemma 2.13, we have the needed result.7y the previous lemma, co-limits in MS are built on co-limits in Stone . Moreover, this construction ismost explicit in the case of finite co-limits. Hence, it might be useful to recall the following property of finiteco-limits in
Stone . Lemma 2.15.
The forgetful functor from
Stone to Set reflects finite co-limits.Proof.
Let us denote with |−| the forgetful functor from
Stone to Set . Let D : I → Stone be a finite diagram,and let L = { l i : D ( i ) → X } i ∈ I be a co-cone in Stone such that {| l i | : | D ( i ) | → | X |} i ∈ I is a co-limit in Set . We claim that a subset Z of X is closed if and only if, for every i ∈ I , the set l − i [ Z ] is a closedsubset of D ( i ). The left-to-right implication follows from continuity of l i for each i ∈ I . Let us prove theconverse direction. Suppose that, for every i ∈ I , the set l − i [ Z ] is closed. By the Closed Map Lemma,the set l i [ l − i [ Z ]] = Z ∩ l i [ D ( i )] is closed. Since {| l i | : | D ( i ) | → | ( X, ζ X ) |} i ∈ I is a co-limit in Set , we have X = S i ∈ I l i [ D ( i )]. Therefore, we have Z = Z ∩ X = Z ∩ [ i ∈ I l i [ D ( i )] = [ i ∈ I Z ∩ l i [ D ( i )];this set is closed because it is a union of finitely many closed sets. This proves our claim. Since {| l i | : | D ( i ) | →| X |} i ∈ I is a co-limit in Set and, by the previous claim, the topology on X is final, the diagram { l i : D ( i ) → X } i ∈ I is a co-limit in Top . Since
Stone fully embeds in
Top , the latter is also a co-limit in
Stone .The fact that U is topological helps characterising some general categorical concepts in MS that will playimportant rˆoles in the rest of the paper.Recall that an arrow m is extremal monic if it is monic and whenever m = g ◦ e , where e is epic, then e is iso. The dual concept defines extremal epic arrows. Also recall that an arrow m : A → B is called regularmonic if there exists a pair of parallel arrows f, g : B → C for which m is an equaliser, i.e., f ◦ m = g ◦ m andfor every arrow n : D → B with the same property there exists a unique arrow u such that n = m ◦ u . Dually,an arrow m : B → A is called regular epic if it is a co-equaliser of a pair of parallel arrows f, g : C ⇒ B . Lemma 2.16.
The following equivalences hold for any arrow in
Stone .1. regular monic ⇔ extremal monic ⇔ monic ⇔ injective;2. regular epic ⇔ extremal epic ⇔ epic ⇔ surjective;3. iso ⇔ bijective.Proof. We start with some general considerations. Recall that in every category any regular monic arrowis extremal monic and every extremal monic is monic. Similarly, in every category any regular epic arrowis extremal epic and extremal epic arrows are epic. Furthermore, notice that since the category BA ofBoolean algebras with homomorphisms is a variety of algebras, in BA monic arrows are precisely the injectivehomomorphisms, and regular epic arrows are precisely the surjective homomorphisms. Applying Stoneduality between Stone and BA [19, Theorem 8.2], we obtain that in Stone every epic arrow is surjective andevery injective arrow is regular monic.Now, regarding item 1 it is easy to see that in
Stone every monic arrow is injective, so the proof of item 1is settled.To prove item 2, recall from Remark 2.12 that
Stone is a reflective full subcategory of
Top . Since theinclusion functor of any reflective full subcategory reflects regular epimorphisms (see [3, Exercise 7F.(c)] fora list of properties satisfied by the inclusion functor of reflective full subcategories), the forgetful functorfrom
Stone to Top reflects regular epimorphisms. Recall that the regular epimorphisms in
Top are preciselythe topological quotient maps [3, Examples 7.72.(2)]. We deduce that a topological quotient map betweenStone spaces is a regular epic arrow in
Stone . Recall the Closed Map Lemma: a continuous function froma compact space to a Hausdorff space is closed. It follows that a surjective continuous map between Stonespaces is a topological quotient map. Therefore, in
Stone , every surjective arrow is regular epic. Thus,item 2 is settled.Finally, an arrow is iso if and only if it is extremal monic and epic; item 3 follows.8 emma 2.17.
Let f : X → Y be an arrow in MS .1. f is monic ⇔ f is injective.2. f is regular monic ⇔ f is extremal monic ⇔ f is injective and preserves denominators.3. f is epic ⇔ f is surjective.4. f is regular epic ⇔ f is extremal epic ⇔ f is surjective and, for every y ∈ Y , we have ζ Y ( y ) = V x ∈ X : f ( x )= y ζ X ( x ) .5. f is iso ⇔ f is bijective and preserves denominators.Proof. Items 1 and 3 are immediate consequences of the facts that topological functors preserve and reflectmonic and epic arrows [3, Proposition 21.13.(1)] and that, by Lemma 2.16, in
Stone monic arrows areprecisely the injective ones and epic arrows are precisely the surjective ones. Furthermore, if G : A → X istopological, then an arrow in A is an extremal (resp. regular) monic arrow if and only if it is G -initial and anextremal (resp. regular) monic arrow in X [3, Proposition 21.13.(2)]. By Corollary 2.8, the arrow f is initialif and only if it preserves denominators. Item 2 follows. Similarly, for a topological functor G : A → X , anarrow in A is an extremal (resp. regular) epic arrow if and only if it is G -final and an extremal (resp. regular)epic arrow in X [3, Proposition 21.13.(3)]. By Lemma 2.13, an arrow f in MS is U -final if and only if wehave ζ Y ( y ) = V x ∈ X : f ( x )= y ζ X ( x ). Item 4 follows. Since an arrow is an iso if and only if it is extremal monicand epic [3, Proposition 7.66], we have item 5. Remark . Given a multiset X , a regular monic arrow in MS with codomain X is, up to an iso, a closedsubspace of X with the induced denominator map. In this section we recall (Theorem 2.26 below) a well-known characterisation of those categories whichare equivalent to some (quasi-)variety of algebras.We warn the reader that we admit infinitary algebras, i.e. algebras with operations of infinite arity, andwe allow large signatures (i.e. a class, rather than a set). In particular, we work with a large signature Σwhich is the union of the classes of κ -ary operations, for κ cardinal.A quasi-equation is a (universally quantified) formula ^ i ∈ I ( u i = v i ) = ⇒ ( u = v ) , where I is a (possibly infinite) set, and u i , v i are, for i ∈ I ∪ { } , terms over a given set of variables.Following [2], a class A of Σ-algebras is called a quasi-variety of Σ -algebras (resp. a variety of algebras )if 1. the class A can be presented by a class of quasi-equations (resp. equations), and2. the class A has free algebras (equivalently, for each cardinal κ , the class A has only a set of isomorphismclasses of algebras on κ generators).We now provide the background needed to state the characterisation of varieties and quasi-varieties ofalgebras. Definition 2.19.
A set of objects G is generating provided that for each pair f , f : K ⇒ K ′ of distinctparallel arrows there exists an object G ∈ G and an arrow g : G → K such that f ◦ g = f ◦ g [4, 0.6].In a category with co-products, a set of objects G is generating if and only if, for every object A ,the canonical arrow, obtained via an application of the universal property of co-products to the co-cone { h : G → A } G ∈G , h ∈ hom( G,A ) , X G ∈G , h ∈ hom( G,A ) G → A (6)is epic. 9 efinition 2.20. A set of objects G is regularly generating if the hom-functors hom( G, − ) (for G ∈ G )collectively reflect regular arrows. In categories with co-products, this is equivalent (see [2, Section 5.1]) tothe fact that the canonical quotient in (6) is regular epic. As a special case, we have that an object G is a regular generator if the hom-functor hom( G, − ) reflects regular epic arrows. As observed in [2, Section 1.1],if the object G has co-powers, this is equivalent to the following condition: for every object A , the canonicalarrow X hom( G,A ) G → A is regular epic. Definition 2.21.
Recall that an object P is called regular projective if the hom-functor hom( P, − ) preservesregular epics. In other words, for any arrow f : P → B and every regular epic arrow g : A _ B , the arrow f factors through g , i.e., there exists h such that the following diagram commutes. ABP gfh
Finally, a set of objects G = { G s | s ∈ S } is abstractly finite if every arrow from an object G s of G to aco-product of objects in G factors through a finite sub-co-product. As a particular case, an object G is called abstractly finite if every arrow from G to a co-power of G factors through a finite sub-co-power. Analogously,we say that an object is abstractly countable if every arrow from G to a co-power of G factors through anat most countable sub-co-power. Definition 2.22.
Following [14, 6.1], we say that an object A in a category C is finitely presentable if thecovariant hom-functor hom ( A, − ) : C → Set preserves filtered co-limits. Explicitly, this means that if I is afiltered category and D : I → C is a functor with co-limit co-cone { b i : D ( i ) → B } i ∈ I , then, for every arrow f : A → B in C , the following two conditions are satisfied.(F) There is g : A → D ( i ) such that f = b i ◦ g .(E) For any g ′ , g ′′ : A → D ( j ) such that f = b j ◦ g ′ = b j ◦ g ′′ , there is d jk : j → k such that D ( d ij ) ◦ g ′ = D ( d ij ) ◦ g ′′ . A BD ( k ) D ( j ) D ( d ij ) fg ′ g ′′ b k b j Similarly, after [14, 6.1], we say that A is finitely generated if hom ( A, − ) : C → Set preserves filtered co-limitsof diagrams ( P i , p ij ) all of whose transition arrows p ij are monic arrows in C . Lemma 2.23 ([27, Proposition 2.4]) . If G is an abstractly finite, regular projective, regular generator, then G is finitely generated. Vice versa, if G is finitely generated and has co-powers, then G is abstractly finite. efinition 2.24. Let C be a category with finite limits and A an object of C . An (internal) equivalencerelation on A is a subobject h p , p i : R A × A satisfying the following properties: reflexivity there exists an arrow d : A → R in C such that the following diagram commutes; A RA × A h A , A i ∃ d h p ,p i symmetry there exists an arrow s : R → R in C such that the following diagram commutes; R RA × A ∃ s h p ,p i h p ,p i transitivity if the left-hand diagram below is a pullback square in C , then there is an arrow t : P → R suchthat the right-hand diagram commutes. P RR A π π y p p P RA × A h p ◦ π ,p ◦ π i ∃ t h p ,p i Definition 2.25.
Recall that the kernel pair of an arrow f : X → Y is the pair of arrows R ⇒ X in thepullback of f along itself: R XX Y. y ff It is folklore that every kernel pair is an equivalence relation. An equivalence relation h p , p i : R A × A is effective if there exists an arrow q : A → S such that p , p : R ⇒ A is the kernel pair of q .For varieties and quasi-varieties of algebras, the definition of equivalence relation given above coincideswith the usual notion of congruence, while the effective equivalence relations in quasi-varieties are the so-called relative congruences.We have finally collected all necessary background to recall some well-known characterisations of varietiesand other classes of algebras. Theorem 2.26.
Let C be a (locally small) category.1. C is equivalent to a quasi-variety of algebras if and only if(a) C is co-complete, and(b) C has a regular projective regular generator.2. C is equivalent to a quasi-variety of finitary algebras if and only if(a) C is co-complete, and(b) C has an abstractly finite, regular projective regular generator.3. C is equivalent to a many-sorted quasi-variety of finitary algebras if and only if a) C is co-complete, and(b) C has an abstractly finite, regularly generating set of regular projective objects.4. All items above remain true if we replace “quasi-variety” with “variety” on the left side of the equiva-lence and add the condition “(c) every equivalence relation in C is effective” on the right one.Proof. Item 2, with the additional requirement that C has equalisers, was proved by Isbell in [17]. However,Ad´amek [2] proved that such an assumption can be dropped (and proved a characterisation stronger thanthe one presented here). Items 1 and 3 can also be found in [2]. Finally, item 4 holds because in every varietyequivalence relations are effective and, vice versa, if in a quasi-variety equivalence relations are effective,then it is a variety (see e.g., [4, Corollary 3.25]). Remark . In the proof of item 3 in Theorem 2.26 (see [2]), the number of objects in the regularlygenerating set corresponds to the number of sorts.In the proof of item 1 in Theorem 2.26 (see [2]), every operation in the variety depends on at mostcountably many coordinates if and only if the regular projective regular generator G is abstractly countable,i.e., every morphism from G to a co-power of G factors through an at most countable sub-co-power.
3. The case of a finite language
In this section we prove that the category MV lf of locally finite MV-algebras is not equivalent to anyquasi-variety in a language with finitary operations and finitely many sorts; the other weaker results statedin the introduction will follow at once. To achieve so we shall prove:1. every finitely co-generated object in MS has a finite underlying set;2. every regularly co-generating set G contains either infinitely many objects or an object whose under-lying set is infinite.Items 1 and 2 together imply that MS admits no regularly co-generating finite set of finitely co-generatedobjects, and thus the dual statement holds for MV lf . Then, an application of Theorem 2.26 leads to thedesired results. We begin by settling item 1. Proposition 3.1.
For a multiset X the following are equivalent:1. X is a finite set and every element of X has finite denominator;2. X is finitely co-presentable in MS ;3. X is finitely co-generated in MS .Proof. [Item 1 ⇒ item 2] This follows from the fact that MS is the pro-completion of the category of multisetswith finite underlying set and finite denominators (see [18, Lemma VI.1.8, p. 231]).[Item 2 ⇒ item 3] This holds trivially.[Item 3 ⇒ item 1] Let X be a finitely co-generated multiset. By [12, Theorem 5.1], every multiset,hence a fortiori X , is a limit of a co-filtered system F : D → MS of multisets, each with finite underlyingset and finite denominators, and such that the transition arrows are epic arrows. For each D ∈ D , let f D : X → F ( D ) be the corresponding limit arrow. Since X is finitely co-generated, by (the dual version of)(F) in Definition 2.22, there exists D ∈ D such that the identity 1 X : X → X factors through f D : X → F D via a certain arrow g : F D → X . Since F D has finite cardinality and g is surjective, X must also havefinite cardinality. Since F D has finite denominators and g decreases denominators, the multiset X has finitedenominators, too.We now address item 2 at the beginning of the section.12 efinition 3.2. Dualising Definition 2.19, we have that G is a set of co-generators in MS if one of the twofollowing equivalent conditions are satisfied:(i) For each pair of distinct arrows f , f : X ⇒ X ′ , there exists G ∈ G and an arrow g : X ′ → G suchthat g ◦ f = g ◦ f .(ii) The canonical arrow X → Y G ∈G , t ∈ hom( X,G ) G (7)is monic.The main tool of this section is Theorem 3.12, where we prove that a set of objects G in MS is a regularlyco-generating set of objects if and only if for every n ∈ { } ∪ { p k | p ∈ P , k ∈ N + } , there exists G ∈ G withtwo distinct elements of denominator ν n and ν , respectively. Notation 3.3.
We write for the multiset ( { , } , ζ ) where { , } is endowed with the discrete topologyand ζ (0) = ζ (1) = ν . Lemma 3.4.
The multiset is a co-generator in MS .Proof. We check that item (i) in Definition 3.2 holds. Let f , f : X ⇒ X ′ be a pair of distinct arrows in MS .Then, there exists x ∈ X such that f ( x ) = f ( x ). By the separation properties of Stone spaces it follows thatthere exists a clopen subset C of X ′ such that f ( x ) ∈ C and f ( x ) / ∈ C . Consider the characteristic function1 C of C defined from X ′ to : this function is clearly continuous, it decreases denominators because thedenominator of the points of is the bottom of the lattice N , and we have 1 C ( f ( x )) = 1 = 0 = 1 C ( f ( x )),so 1 C ◦ f = 1 C ◦ f . Remark . Recall that the projection arrows from a product are jointly monic, i.e., if f, g : X → Q i ∈ I Y i are distinct arrows, then there exists i ∈ I such that π i ◦ f = π i ◦ g (This is a consequence of the fact thateach cone on { Y i } i ∈ I factors in a unique way through Q i ∈ I Y i ). Lemma 3.6.
A set of objects G in MS is co-generating if and only if there exists G ∈ G that has at leasttwo distinct points of denominator ν .Proof. Suppose G is a co-generating set. Then, the canonical arrow h : → Q G ∈G , t ∈ hom( ,G ) G is monic;hence, by Lemma 2.17, it is injective. It follows that h (0) = h (1), hence there are two distinct maps f and g from the one-element multiset ( {∗} , ζ ∗ ), where ζ ∗ ( ∗ ) = ν , into the product. By Remark 3.5, there exist G ∈ G and a projection π into G such that π ( h (0)) = π ( h (1)). Therefore, G has two distinct elements; since π ◦ h decreases denominators, both elements have denominator ν .For the converse direction, let us assume that there exists G ∈ G that has at least two distinct points g and g with denominator ν . We define a map t : → G by setting t (0) := g and t (1) := g . The map t isclearly continuous, denominator-decreasing and injective, hence a monic arrow by Lemma 2.17. Recall that is a co-generator (Lemma 3.4); so, for all distinct arrows f , f : X ⇒ X ′ , there exists an arrow g : X ′ → such that g ◦ f = g ◦ f . Since t is monic, the latter inequality holds if and only if t ◦ g ◦ f = t ◦ g ◦ f . Thus, G is a co-generator, as well. We conclude that G is a co-generating set, for it contains a co-generator.As the next step we characterise regularly co-generating sets of objects in MS . Dualising Definition 2.20we say that a set of objects G in MS is regularly co-generating if, for every object X in MS , the canonicalarrow in (7) is regular monic. By Lemma 2.17, an arrow f : X → Y in MS is regular monic if and only if itis monic and it preserves denominators. Lemma 3.6 gives already a characterisation of the sets G for whichthe map (7) is monic. So, we now focus on the remaining condition, i.e. preservation of denominators. Weneed some preliminary results.Given X and Y multisets, and given x ∈ X and y ∈ Y , a necessary condition for the existence of anarrow f : X → Y such that f ( x ) = y is that ζ X ( x ) ≥ ζ Y ( y ). The following result establishes a partialconverse. 13 emma 3.7. Let X and Y be multisets, and let x ∈ X , y ∈ Y be such that ζ X ( x ) ≥ ζ Y ( y ) . Suppose that ζ Y ( y ) is finite, and that Y has an element of denominator ν . Then, there exists an arrow of multisets f : X → Y such that f ( x ) = y .Proof. Let x and y be as in the statement. Since ζ Y ( y ) is finite, the set { ν ∈ N | ν ≥ ζ Y ( y ) } is open. Itfollows that the set { z ∈ X | ζ X ( z ) ≥ ζ Y ( y ) } = ζ − X [ { ν ∈ N | ν ≥ ζ Y ( y ) } ] is open because ζ X is continuous,so it can be written as a union of clopen sets; let C be one among those, which contains x . Notice that,for any z ∈ C , we have ζ X ( z ) ≥ ζ Y ( y ). Let y be an element of Y of denominator ν given by hypothesis.Then, we define a function f : X −→ Yz ( y if z ∈ C ; y otherwise.The function f is continuous because C is clopen; moreover, a case inspection immediately shows that f decreases denominators. So, f is an arrow of multisets, and f ( x ) = y because x ∈ C .The elements of the form ν p k play a special role in N , and the following two lemmas capture their mainproperties. Recall that, in a complete lattice L , an element x is said to be completely join-irreducible if, forevery subset J ⊆ L , the condition x = W J implies x ∈ J . In this definition J is allowed to be empty, so theleast element of L is not completely join-irreducible. Lemma 3.8.
Every ν ∈ N is the supremum in N of the elements under ν of the form ν p k , for p ∈ P and k ∈ N + .Proof. Let ν be an arbitrary element of N and let S ν := { ν p k | ν p k ≤ ν, p ∈ P , k ∈ N + } . Obviously ν is anupper-bound for S ν . Suppose µ is another upper-bound for S ν and µ < ν . Then, for every p ∈ P , we have µ ( p ) ≤ ν ( p ), and there exists p ∈ P such that µ ( p ) < ν ( p ). Thus, the supernatural number ¯ µ which agreeswith µ on every prime different from p and attains the value µ ( p ) + 1 on p is strictly greater than µ butstill in S ν , for ¯ µ ≤ ν . This contradicts the fact that µ is an upper-bound, and the proof is concluded. Lemma 3.9.
The completely join-irreducible elements of N are precisely the elements of the form ν p k , for p ∈ P and k ∈ N + .Proof. Let ν be a completely join-irreducible element of N . By Lemma 3.8, ν ∈ N is the supremum of theset { ν p k | ν p k ≤ ν, p ∈ P , k ∈ N + } ; so, since ν is completely join-irreducible, ν must belong to this set, hencehave the form ν p k for some p ∈ P and k ∈ N + .For the converse implication, fix p ∈ P and k ∈ N + . If ν p k = W J , then J is linearly ordered and containsonly elements of the form ν p j with j ≤ k , hence J is finite. Therefore, ν p k = W J if and only if ν p k ∈ J . So,every element of the form ν p k is completely join-irreducible. Notation 3.10.
For any n ∈ N + we let D n denote the multiset whose underlying Stone space is a two-pointdiscrete space { , } , with ζ (0) := ν and ζ (1) := ν n . Notice that, with our previous notation, coincideswith D . Lemma 3.11.
Let G be a set of multisets. The following are equivalent.(i) For every multiset X , the canonical arrow from X to Q G ∈G , t ∈ hom( X,G ) G preserves denominators.(ii) For every p ∈ P and k ∈ N + , there exists G ∈ G that has at least one point of denominator ν p k andone point of denominator ν .Proof. Throughout the proof, let us denote by ζ Q the denominator map of Q G ∈G , t ∈ hom(D pk ,G ) G .14uppose item (i) holds. Let p ∈ P and k ∈ N + . Consider the canonical arrow h : D p k → Y G ∈G , t ∈ hom(D pk ,G ) G. We have ν p k = ζ D pk (1) item (i) = ζ Q ( h (1)) Lemma 2.11 = _ G ∈G , t ∈ hom(D pk ,G ) ζ G ( t (1)) . By Lemma 3.9, the element ν p k is completely join-irreducible; therefore, there exist G ∈ G and t ∈ hom(D p k , G ) such that ζ G ( t (1)) = ν p k . Furthermore, ζ G ( t (0)) = ν . Since p and k were arbitrary, thisproves (i) ⇒ (ii).Let us prove the converse implication: suppose item (ii) holds. Let X be a multiset and h be thecanonical arrow from X to Q G ∈G , t ∈ hom( X,G ) G . Then, ζ Q ( h ( x )) = _ G ∈G , t ∈ hom( X,G ) ζ G ( t ( x ))= _ G ∈G , t ∈ hom( X,G ) _ ν pk ≤ ζ G ( t ( x )) ν p k (Lemma 3.8)= _ { ν p k | ∃ G ∈ G ∃ t ∈ hom( X, G ) s.t. ν p k ≤ ζ G ( t ( x )) } = _ { ν p k | ν p k ≤ ζ G ( x ) } (Lemma 3.7)= ζ G ( x ) . Thus, the map h preserves denominators.Combining Lemmas 3.6 and 3.11 we obtain a characterisation of regularly co-generating sets of objectsin MS . Theorem 3.12.
A set of objects G in MS is regularly co-generating if and only if, for every n ∈ { } ∪ { p k | p ∈ P , k ∈ N + } , there exists G ∈ G with two distinct elements, one having denominator ν n and the otherone ν .Proof. A set of objects G in MS is regularly co-generating if and only if the canonical arrow h from X to Q G ∈G , t ∈ hom( X,G ) G is regular monic. By Lemma 2.17, an arrow in MS is regular monic if and only if it isinjective and denominator-preserving. By Lemma 3.6, the map h is injective if and only if there exists G ∈ G that has at least two distinct points of denominator ν . By Lemma 3.11, the map h preserves denominatorsif and only if, for every p ∈ P and k ∈ N + , there exists G ∈ G that has at least one point of denominator ν p k and one point of denominator ν . Corollary 3.13.
Every regularly co-generating set in MS has either infinitely many objects or an objectwhose underlying set is infinite. If we replace the notion of co-generating set with the notion of co-generator, we obtain the followingresult.
Corollary 3.14.
An object G is a regular co-generator in MS if and only if, for every n ∈ { } ∪ { p k | p ∈ P , k ∈ N + } , there exists g ∈ G that has denominator ν n . In particular, the underlying set of a regularco-generator in MS is infinite. As a consequence of Theorem 3.12 and Corollary 3.13, we have the following.
Proposition 3.15.
The category MV lf of locally finite MV-algebras has no regularly generating finite set offinitely generated objects. roof. We prove the dual statement. By Proposition 3.1, every finitely co-generated object in MS has finiteunderlying set. By Corollary 3.13, no finite set of multisets with finite underlying set is a regularly co-generating set in MS . Thus, MS has no regularly co-generating finite set of finitely co-generated objects. Theorem 3.16.
The category of locally finite MV-algebras is not equivalent to any finitely-sorted quasi-variety of finitary algebras.Proof.
From Proposition 3.15 we know that the category MV lf has no regularly generating finite set of finitelygenerated objects. Now we use the characterisation of many-sorted quasi-varieties of finitary algebras (item 3in Theorem 2.26), together with Lemma 2.23. By Remark 2.27, the number of objects in the regularlygenerating set corresponds to the number of sorts. It follows that MV lf is not equivalent to a finitely-sortedquasi-variety of finitary algebras. Corollary 3.17.
The category of locally finite MV-algebras is not equivalent to any variety of finitaryalgebras.
4. The case of an infinite language
In this section we shall prove that MV lf is equivalent to a quasi-variety of algebras if one allows eithera language with operations of at most countable arity or a language with countably many sorts. The keypoint is to prove that the objects D n defined in the previous section are all regular injective.Dualising the notion of regular projective, we obtain the notion of regular injective. Explicitly, an object X is called regular injective if, for every every regular monic arrow g : B ֒ → A and every arrow f : B → X ,there exists an arrow h : A → X such that the following diagram commutes. BA Xg fh
It is known that in a Stone space X , if A ⊆ B ⊆ X , A is closed and B is open, then there exists a clopensubset C of X such that A ⊆ C ⊆ B . The following lemma extends this result to a family of pairs of openand closed sets, in such a way that the C i form a partition of X . Lemma 4.1.
Let X be a Stone space, let K , . . . , K n be disjoint closed subsets of X and let Z , . . . , Z n bean open cover of X such that, for every i ∈ { , . . . , n } , we have K i ⊆ Z i . Then X can be partitioned intoclopen subsets C , . . . , C n such that, for every i ∈ { , . . . , n } , we have K i ⊆ C i ⊆ Z i .Proof. Without loss of generality we can assume that for all i = j the sets K i and Z j are disjoint; indeed,if this is not the case, then one can replace, for all j ∈ { , . . . , n } , the set Z j with the smaller Z j \ S i ∈{ ,...,n }\{ j } K i , which satisfies this additional hypothesis. We now prove the statement of the lemma byinduction on n ; the case n = 1 is trivial. Let n ≥
2, and suppose that the property holds for n −
1. Let L n := X \ ( Z ∪ · · · ∪ Z n − ); since Z , . . . , Z n cover X , we have L n ⊆ Z n . Moreover, by hypothesis, we have K n ⊆ Z n . Therefore, K n ∪ L n ⊆ Z n , the set K n ∪ L n is closed, and Z n is open. Hence, there exists a clopensubset C n of X such that K n ∪ L n ⊆ C n ⊆ Z n . Since C n ⊆ Z n , it follows from our additional hypothesisthat the set C n is disjoint from K , . . . , K n − . So, Y := X \ C n is a Stone space, the sets K , . . . , K n − are closed and disjoint subsets of Y , the sets Z ∩ Y , . . . , Z n − ∩ Y are an open cover of Y and, for every i ∈ { , . . . , n − } , we have K i ⊆ Z i ∩ Y . Then, by inductive hypothesis, the space Y can be partitioned intoclopen subsets C , . . . , C n − such that, for every i ∈ { , . . . , n − } , we have K i ⊆ C i ⊆ Z i ∩ Y . The sets C , . . . , C n are clopen subsets of X and they are a partition of X . Furthermore, for every i ∈ { , . . . , n } , wehave K i ⊆ C i ⊆ Z i . This concludes the proof. Lemma 4.2.
Let X be a multiset and suppose that the following conditions hold. . The set X is finite and every element of X has finite denominator.2. There exists an element x ∈ X with denominator ν .Then X is regular injective in MS .Proof. Let X be a multiset satisfying items 1 and 2 above and B and A be arbitrary multisets. Let g : B ֒ → A be a regular monic arrow in MS , and let f : B → X be an arrow in MS . We shall prove that there exists anarrow h in MS such that the following diagram commutes. BA Xg fh
Let x , . . . , x n be a listing of the elements of X \ { x } . Let d , . . . , d n be the denominators of x , . . . , x n ,respectively. Fix i ∈ { , . . . , n } . Since X is a finite Stone space, the singleton { x i } is a clopen subset of X . Thus, f − [ { x i } ] is a clopen subset of B . The function g is closed by the Closed Map Lemma; therefore,the set K i := g [ f − [ { x i } ]] is closed in A . Since g is monic, the sets K , . . . , K n are disjoint. For every i ∈ { , . . . , n } , set Z i := { z ∈ A | ζ A ( z ) ≥ d i } = ζ − A [ { ν ∈ N | ν ≥ d i } ] . For every i ∈ { , . . . , n } , the set Z i is an open subset of A because ζ A is continuous, and { ν ∈ N | ν ≥ d i } is an open subset of N for d i is finite by item 1. Notice that, for all a ∈ f − [ { x i } ], it holds that ζ B ( a ) ≥ d i and, since g is regular monic, all denominators of K i are greater or equal to d i . It follows that K i ⊆ Z i .Moreover, Z = A by item 2. By Lemma 4.1, the set A can be partioned into clopen subsets C , . . . , C n such that, for every i ∈ { , . . . , n } , we have K i ⊆ C i ⊆ Z i . Define h : A → X by setting h ( c ) = x i for c ∈ C i .The function h is trivially continuous. To see that it decreases denominators, let i ∈ { , . . . , n } , and let c ∈ C i . Then, we have ζ ( h ( c )) = ζ ( x i ) = d i . Since c ∈ C i ⊆ Z i , we have ζ A ( c ) ≥ d i . Thus, the function h decreases the denominators. Finally, from g (cid:2) f − [ { x i } ] (cid:3) ⊆ C i , it is easy to derive f = h ◦ g . Remark . Let { X i } i ∈ I be a family of compact topological spaces. Each clopen subset of the topologicalproduct Q i ∈ I X i is nontrivial only on finitely many coordinates i ∈ I . Thus, if Y is a finite discrete space,every continuous function from Q i ∈ I X i to Y depends on finitely many coordinates. Moreover, it followsthat, if { Y n } n ∈ N is a countable family of finite discrete spaces, any continuous function from Q i ∈ I X i to Q n ∈ N Y n depends on at most countably many coordinates i ∈ I . Proposition 4.4.
The set { D n | n ∈ { } ∪ { p k | p ∈ P , k ∈ N + }} is an abstractly co-finite regularlyco-generating countable set of regular injective objects. Hence the category of locally finite MV-algebras isequivalent to a countably-sorted quasi-variety of finitary algebras.Proof. The set { D n | n ∈ { } ∪ { p k | p ∈ P , k ∈ N + }} is abstractly co-finite by Remark 4.3, it is regularly co-generating by Theorem 3.12, and each D n is regular injective by Lemma 4.2. In addition, MV lf is co-completeby Corollary 2.9; so, by item 3 in Theorem 2.26, MV lf is equivalent to a countably-sorted quasi-variety offinitary algebras.The countably-sorted quasi-variety in Proposition 4.4 is actually a countably-sorted variety, as we shallprove in Theorem 5.5. Lemma 4.5.
The multiset C := Y n ∈{ }∪{ p k | p ∈ P ,k ∈ N + } D n is an abstractly co-countable regular injective regular co-generator. roof. Regular injective objects are closed under products (see, e.g. [5, Lemma 5.13]), so Lemma 4.2 entailsthat C is regular injective. Moreover, C is a regular co-generator by Corollary 3.14. Finally, C is abstractlyco-countable by Remark 4.3. Proposition 4.6.
The category of locally finite MV-algebras is equivalent to a quasi-variety of algebras,with operations of at most countable arity.Proof.
By Corollary 2.9, MV lf is co-complete. By Lemma 4.5, MV lf has an abstractly countable regularprojective regular generator. Therefore, by Theorem 2.26, MV lf is equivalent to a quasi-variety of algebrasand, by Remark 2.27, the operations depend on at most countably many coordinates.The quasi-variety in Proposition 4.6 is actually a variety, as we shall prove in Theorem 5.4.
5. From quasi-varieties to varieties: effective equivalence relations
In this section we prove that all reflexive (internal) relations in MV lf are effective equivalence relations.To study equivalence relations in MV lf we look at their dual in MS . In order to do so we need to introducesome notation and tools for the dual concepts.By a co-subobject we mean the dual notion of subobject: a co-subobject on X is an equivalence class ofepic arrows with domain X . With a little abuse of notation, we take the liberty to refer to a co-subobjectjust by one of its representatives. By a (binary) co-relation on X we mean a co-subobject on X + X . Noticethat arrows with domain X + X bijectively correspond to pairs of parallel arrows with domain X . Dualisingthe respective notion for relations, we say that a co-relation q , q : X ⇒ S on X is reflexive if and only ifthere exists an arrow d : S → X such that the following two diagrams commute. X SX q X d X SX q X d Symmetric , transitive and equivalence co-relations are obtained in a similar way. Dualising Definition 2.25,we say that an equivalence co-relation q , q : X ⇒ S on the multiset X is effective provided it is the co-kernelpair of some arrow. Lemma 5.1.
If a co-relation q , q : X ⇒ S on X in MS is reflexive, then, for all i, j ∈ { , } , we have q i ( x ) = q j ( y ) implies x = y, and ζ S ( q i ( x )) = ζ X ( x ) . Proof.
By reflexivity, there exists a map d : S → X such that d ◦ q = 1 X , and d ◦ q = 1 X . Therefore, if q i ( x ) = q j ( y ), then x = d ( q i ( x )) = d ( q j ( x )) = y .Since q i is an arrow of multisets, we have ζ S ( q i ( x )) ≤ ζ X ( x ). For the opposite inequality, since d decreasesdenominators, we have ζ S ( q i ( x )) ≥ ζ X ( d ( q i ( x ))) = ζ X ( x ). Therefore, ζ S ( q i ( x )) = ζ X ( x ).It should be noted, although it will not be needed in this paper, that also the converse of the previouslemma holds, thus providing a characterisation of reflexive co-relations. For a proof, see Lemma 6.10 in theAppendix. Proposition 5.2.
Every reflexive co-relation in MS is an effective equivalence co-relation.Proof. Let q , q : X ⇒ S be a reflexive co-relation on a multiset X . Set Y := { x ∈ X | q ( x ) = q ( x ) } . ByLemma 5.1, we have q i ( x ) = q j ( y ) ⇒ x = y . Therefore, for every x, y ∈ X and every i ∈ { , } , we have1. q ( x ) = q ( y ) if and only if x ∈ Y , y ∈ Y and x = y .18. q i ( x ) = q i ( y ) if and only if x = y .Therefore, the following is a pushout in Set . Y XX S q q (8)Equip Y with the subspace topology. By Lemma 2.15, the forgetful functor from Stone to Set reflects co-limits, and therefore (8) is a pushout in
Stone , as well. Equip Y with the the denominator map ζ Y : Y →N , y ζ X ( y ). By Lemma 5.1, ζ S ( q i ( x )) = ζ X ( x ). Hence, by the construction of finite co-limits (seeLemma 2.14) in MS , (8) is a pushout in MS , as well. Corollary 5.3.
Every reflexive relation in MV lf is an effective equivalence relation. Thus in particular MV lf is a Mal’tsev category . Theorem 5.4.
The category of locally finite MV-algebras is equivalent to a variety of algebras, with opera-tions of at most countable arity.Proof.
By Proposition 4.6, MV lf is equivalent to a quasi-variety of algebras, with operations of at mostcountable arity. By Corollary 5.3, every reflexive relation in MV lf is an effective equivalence relation. Thestatement of the theorem follows by an application of Theorem 2.26. Theorem 5.5.
The category of locally finite MV-algebras is equivalent to a countably-sorted variety offinitary algebras.Proof.
By Proposition 4.6, MV lf is equivalent to a countably-sorted quasi-variety of finitary algebras. ByCorollary 5.3, every reflexive relation in MV lf is an effective equivalence relation. The statement of thetheorem follows by an application of Theorem 2.26.We conclude the main development of our work by pointing out that from our results it follows that MV lf is an exact category (in the sense of Barr), i.e., a regular category in which every equivalence relationis effective. For more details on regular and exact categories, we refer the reader to [6].
6. Appendix N In [12], the authors erroneously state that the topology on N having as an open basis the sets of theform { ν ∈ N | ν > ν n } for n ∈ N + coincides with the topology having as an open subbasis the sets of theform { ν ∈ N | ν ( p ) > k } for p ∈ P and k ∈ N . However, the error does not appear in [24, Section 8.4]. Thestatement in [12] is incorrect for two reasons. First, the sets of the form { ν ∈ N | ν > ν n } for n ∈ N + donot form a basis for any topology on N . Indeed, these sets do not cover N because, for every n ∈ N + , wehave ν / ∈ { ν ∈ N | ν > ν n } . Secondly: even if we replace “basis” by “subbasis”, the statement remainsincorrect: the topology on N having as an open subbasis the sets of the form { ν ∈ N | ν > ν n } for n ∈ N + does not coincide with the topology having as an open subbasis the sets of the form { ν ∈ N | ν ( p ) > k } for p ∈ P and k ∈ N . This is implied by the statement τ τ in the following lemma. Lemma 6.1.
Let τ , τ , τ , τ , τ , τ be the topologies on N generated by the following families of sets,respectively:1. { ν ∈ N | ν ( p ) > k } for p ∈ P , k ∈ N ; Recall that a category is called
Mal’tsev if it has finite limits and every reflexive relation is an equivalence relation. . { ν ∈ N | ν ( p ) ≥ k } for p ∈ P , k ∈ N ;3. { ν ∈ N | ν ≥ ν n } for n ∈ N + ;4. { ν ∈ N | ν > ν n } for n ∈ N + ;5. { ν ∈ N | ν ν n } for n ∈ N + ;6. { ν ∈ N | ν < ν n } for n ∈ N + .Then τ = τ = τ , and in fact they all coincide with the topology we equipped N with in Section 2. Fur-thermore, τ = τ , τ ( τ and τ ( τ , and the topologies τ and τ are not comparable. τ = τ = τ τ τ = τ Proof. τ ⊆ τ If k = 0, then { ν ∈ N | ν ( p ) ≥ k } = { ν ∈ N | ν ( p ) > k − } . If k = 0, then { ν ∈ N | ν ( p ) ≥ k } = N . τ ⊆ τ Let n = p k · · · · · p k l l be the unique prime factorisation of n . Then { ν ∈ N | ν ≥ ν n } = l \ i =1 { ν ∈ N | ν ( p i ) ≥ k i } .τ ⊆ τ We have { ν ∈ N | ν ( p ) > k } = { ν ∈ N | ν ≥ ν p k +1 } . τ ⊆ τ We have { ν ∈ N | ν > ν n } = S m ∈ N + : ν m >ν n { ν ∈ N | ν ≥ ν m } . τ ⊆ τ We have { ν ∈ N | ν ν n } = S p ∈ P { ν ∈ N | ν ( p ) > ν n ( p ) } . τ ⊆ τ For all ν ∈ N and n ∈ N + , the condition ν < ν n holds if and only if there exists m ∈ N + with ν m < ν n such that ν ≤ ν m . Contrapositively, the condition ν < ν n holds if and only if for every m ∈ N + with ν m < ν n we have ν ν m . Then we have { ν ∈ N | ν < ν n } = \ m ∈ N + : ν m <ν n { ν ∈ N | ν ν m } . Note that the intersection is finite. τ ⊆ τ Since { ν ∈ N | ν ≤ ν n } = T p ∈ P { ν ∈ N | ν < ν np } , we have { ν ∈ N | ν ν n } = [ p ∈ P { ν ∈ N | ν < ν np } .τ τ For every n ∈ N + , the set { ν ∈ N | ν ν n } = N \ { ν ∈ N | ν ≤ ν n } is co-finite. Therefore, τ isgenerated by co-finite sets; thus, every nonempty element in τ is co-finite. The topology τ containsnonempty elements which are not co-finite, such as { ν ∈ N | ν > ν } : indeed, for every m ∈ P \ { } ,the element ν m belongs to N \ { ν ∈ N | ν > ν } .20 τ We have
N \ { ν , ν } = { ν ∈ N | ν ν } ∈ τ . We claim that N \ { ν , ν } / ∈ τ . We show that thereis no finite list n , . . . , n k of elements of N + such that ν ∈ T ki =1 { ν ∈ N | ν > ν n i } ⊆ N \ { ν , ν } .Indeed, suppose, by way of contradiction, that such a list exists. Then, for every i ∈ { , . . . , k } , wehave ν ∈ { ν ∈ N | ν > ν n i } , i.e. ν > ν n i , i.e. n i = 1. Thus, k \ i =1 { ν ∈ N | ν > ν n i } = k \ i =1 { ν ∈ N | ν > ν } = k \ i =1 N \ { ν } = ( N \ { ν } if k = 0; N if k = 0 . This set contains the element ν , and hence it is not included in N \ { ν , ν } : a contradiction. τ τ Even if this statement follows from the previous ones, we provide a direct prove of this fact. Set A := { ν ∈ N | ν (2) > } . Notice that A ∈ τ . We claim that A / ∈ τ . The element ν belongs to A ; weshow that there is no finite list n , . . . , n k of elements of N + such that ν ∈ T ki =1 { ν ∈ N | ν > ν n i } ⊆ A .Indeed, suppose, by way of contradiction, that such a list exists. Then, for every i ∈ { , . . . , k } , wehave ν ∈ { ν ∈ N | ν > ν n i } , i.e. ν > ν n i , i.e. n i = 1. Thus, k \ i =1 { ν ∈ N | ν > ν n i } = k \ i =1 { ν ∈ N | ν > ν } = k \ i =1 N \ { ν } = ( N \ { ν } if k = 0; N if k = 0 . This set contains ν , and thus it is not included in A (because ν / ∈ A ): a contradiction. Remark . By Lemma 6.1, for every p ∈ P , every k ∈ N and every n ∈ N + , the following subsets of N areopen in the topology τ :1. { ν ∈ N | ν ( p ) > k } ;2. { ν ∈ N | ν ( p ) ≥ k } ;3. { ν ∈ N | ν ≥ ν n } ;4. { ν ∈ N | ν > ν n } ;5. { ν ∈ N | ν ν n } ;6. { ν ∈ N | ν < ν n } .Recall that an ideal of a lattice is a subset which is downward closed and closed under finite joins. Noticethat N can be identified with the “ideal-completion” of N + with the divisibility order. By [16, Lemma 1.1and 1.2], N can be regarded as the “sobrification” of N + endowed with the Alexandrov topology inducedby the divisibility order .The topology τ on N is not Hausdorff. So N with the identity function is not a multiset, which seemssomewhat unnatural. However, N can be made into a Stone space by taking the “patch topology”, i.e.,taking as an open subbasis the family of subsets U p,k := { ν ∈ N | ν ( p ) > k } and N \ U p,k := { ν ∈ N | ν ( p ) ≤ k } for p ∈ P , k ∈ N . We let ¯ N denote the set N with the patch topology.The multiset ¯ N is not a regular co-generator, even if it is almost so. The only reason why ¯ N is nota regular co-generator is that it does not contain two distinct points of denominator ν . However, ¯ N is“almost a regular co-generator” because, for every p ∈ P and k ∈ N , it contains an element of denominator ν p k , thus satisfying item (ii) of Lemma 3.11. Moreover, one can show ¯ N to be regular injective. The set ofmultisets { , ¯ N } is then a regularly co-generating set of regular injective objects, and the multiset ¯
N × isa regular injective regular co-generator object of MS . We are indebted to A. Moshier (Chapman University) for this remark. .2. Representation of co-subobjects Recall that, given a multiset X , a co-subobject of X is an equivalence class of epic arrows of multisetswith domain X , where two epic arrows f : X ։ Y and g : X ։ Z are equivalent if and only if there existsan iso h : Y → Z such that g = h ◦ f . We denote with Q ( X ) the set of co-subobjects of X , and we equip Q ( X ) with a partial order defined as follows: the equivalence class of f : X ։ Y is below the equivalenceclass of g : X ։ Z if and only if there exists a (necessarily unique and epic) arrow h : Y → Z such that thefollowing diagram commutes. X YZ fg h
Our next goal is to represent co-subobjects on X internally on X . We recall that this is possible forStone spaces thanks to the notion of Boolean relation . Definition 6.3.
Let R be an equivalence relation on a Stone space X . A subset P of X is said to be compatible with R if P is a union of equivalence classes of R ; in other words, if an element x is in P , thenthe entire equivalence class of x (modulo R ) is included in P . The equivalence relation R is called Boolean if for any two distinct equivalence classes of R , there is a clopen subset of X that is compatible with R andthat includes one of the equivalence classes, but not the other. Lemma 6.4 ([15, Lemma 1, Chapter 37]) . Let X be a Stone space and R be an equivalence relation on X .The quotient space X/R is Boolean if and only if the relation R is Boolean. Similarly, we shall identify a co-subobject f : X ։ Y with a pair ( ∼ f , µ f ), where ∼ f is an equivalencerelation and µ f is a function from X to N , as follows. Definition 6.5.
Given an epic (= surjective) arrow f : X → Y of multisets, we set ∼ f = { ( x , x ) ∈ X × X | f ( x ) = f ( x ) } . and we denote by µ f : X → N the composite ζ Y ◦ f .The idea is that, up to an iso, an epic arrow f can be recovered from ( ∼ f , µ f ). In order to establish aninverse for the assignment f ( ∼ f , µ f ), we identify the characterising properties of ( ∼ f , µ f ). Definition 6.6.
Given a multiset X , we call multiset relation on X a pair ( ∼ , µ ), such that1. ∼ is a Boolean relation on X ,2. µ : X → N is a continuous function such that µ ≤ ζ ,3. for all x, y ∈ X , if x ∼ y , then µ ( x ) = µ ( y ). Lemma 6.7. If f : X → Y is an epic arrow of multisets, then ( ∼ f , µ f ) is a multiset relation on X .Proof. Item 1 in Definition 6.6 follows from Lemma 6.4. Items 2 and 3 in Definition 6.6 are immediate.We shall now see how one recovers an epic arrow f from the multiset relation ( ∼ , µ ). Definition 6.8.
Let (
X, ζ X ) be a multiset and ( ∼ , µ ) be a multiset relation on X . By Lemma 6.4, X/ ∼ is aStone space. Furthermore, define ζ µ : X/ ∼ → N by setting ζ µ ([ x ]) = µ ( x ). It is immediate that this functionis well defined and continuous, and so we obtain an epic arrow of multisets π ∼ : ( X, ζ X ) ։ ( X/ ∼ , ζ µ ).For X a multiset, we let R ( X ) denote the set of multiset relations on X . We turn R ( X ) into a partiallyordered set by setting ( ∼ , µ ) ≤ ( ∼ , µ ) if and only if ∼ ⊆ ∼ and µ ≥ µ .22 heorem 6.9. The assignments G : Q ( X ) −→ R ( X ) and F : R ( X ) −→ Q ( X )( f : X ։ Y ) ( ∼ f , µ f ) ( ∼ , µ ) ( π ∼ : ( X, ζ X ) ։ ( X/ ∼ , ζ µ )) establish an isomorphism of partially ordered sets.Proof. Let f : X → Y be an arrow of multisets. We want to prove that f = F ( G ( f )) = π ∼ f in Q ( X ). It issufficient to show that there is an isomorphism ε f : X/ ∼ f → Y such that the following triangle commutes. X X/ ∼ f Y π ∼ f f ε f Define ε f by setting ε f ([ x ]) := f ( x ). Observe that, for every x, y ∈ X , we have x ∼ f y ⇐⇒ f ( x ) = f ( y ).Therefore, the function ε f is well defined and injective. By definition, f = ε f ◦ π ∼ f . Furthermore ε f issurjective, for f is. We now claim that ε f is an arrow of multisets. By definition of quotient topology, ε f is continuous if and only if the composite ε f ◦ π ∼ f is continuous; this is so, because the composite is f . Bydefinition, for every x ∈ X , µ f ( x ) = ζ Y ( f ( x )); since the former is the denominator of [ x ] and the latter is thedenominator of the image of [ x ] under ε F , the map ε f preserves denominators. Finally, since isomorphismsin MS are precisely the bijective denominator-preserving arrows (Lemma 2.17), we conclude that ε f is anisomorphism.Let us now check that ( ∼ , µ ) = G ( F ( ∼ , µ )) = ( ∼ π ∼ , µ π µ ). By definition, for every x, y ∈ X , x ∼ π ∼ y ifand only if π ∼ ( x ) = π ∼ ( y ) if and only if x ∼ y . Furthermore, again a simple inspection of the definitionsshows that, for every x ∈ X , we have µ π µ ( x ) = ( ζ µ ◦ π µ )( x ) = ζ µ ( π µ ( x )) = ζ µ ([ x ]) = µ ( x ) , so that µ π µ = µ .We conclude the proof showing that the assignments in the statement preserve the order. Suppose that f : X → Y , g : X → Z are in Q ( X ) and there exists an arrow h : Y → Z in MS such that h ◦ f = g . Then µ g = ζ Z ◦ g = ζ Z ◦ h ◦ f ≤ ζ Y ◦ f = ζ f , and if f ( x ) = f ( y ) then also g ( x ) = h ( f ( x )) = h ( f ( y )) = g ( y ). Thus µ g ≤ µ f and ∼ f ⊆ ∼ g , which givesexactly ( ∼ f , µ f ) ≤ ( ∼ g , µ g ).Vice versa, suppose ( ∼ , µ ) , ( ∼ , µ ) are two multiset relations such that ( ∼ , µ ) ≤ ( ∼ , µ ). Thus, ∼ ⊆ ∼ and µ ≤ µ . The former entails that the function h : X/ ∼ → X/ ∼ that maps [ x ] ∼ to [ x ] ∼ iswell-defined and the latter entails that h decreases denominators. Continuity follows from the obvious factthat π = h ◦ π and the definition of quotient topology. This concludes the proof.The representation of co-subobjects obtained in Theorem 6.9 is quite useful to study co-relations in MS .We illustrate this by characterising reflexive co-relations in MS . The next lemma re-proves Lemma 5.1 andshows that also the converse holds.By Theorem 6.9, we have a bijective correspondence between multisets relations on X + X (i.e., elementsof R ( X + X )) and equivalence classes of co-relations on X (i.e., elements of Q ( X + X )).Let us recall the notion of reflexivity for a co-relation ( q , q ) : X + X → S : X + XS X ( q ,q ) (1 X , X ) ∃ d emma 6.10. For any multiset X , a multiset relation ( ∼ , µ ) on X + X corresponds to a reflexive co-relationon X if and only if the following hold: ( x, i ) ∼ ( y, j ) = ⇒ x = y, and µ ( x, i ) = ζ X ( x ) . Proof.
Let ( ≈ , η ) be the relational structure associated with (1 X , X ), i.e.( x, i ) ≈ ( y, j ) ⇐⇒ x = y, and η ( x, i ) = ζ X ( x ) . By Theorem 6.9, ( ∼ , µ ) is reflexive if and only if ( ∼ , µ ) ≤ ( ≈ , η ), i.e. ∼ ⊆ ≈ and µ ≥ η , by definition of thepartial order on R ( X ). The condition ∼ ⊆ ≈ is precisely the condition ( x, i ) ∼ ( y, j ) = ⇒ x = y .Note that we have η = ζ X + X and that, by definition of multiset relation, we have µ ≤ ζ X + X ; thus, wehave µ ≤ η , and therefore the condition η ≤ µ is equivalent to η = µ , i.e. µ ( x, i ) = ζ X ( x ). Acknowledgements
The research of both authors was supported by the Italian Ministry of University and Research throughthe PRIN project n. 20173WKCM5
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