A Single-Sorted Theory of Multisets

An axiomatic account of multiset theory is given, where multiplicities are of the same sort as sets (with their own internal structures). Various theories are proposed covering diﬀerent existing multiset systems, as well as a stronger theory which is equiconsistent with Zermelo–Fraenkel set theory and with antifoundation. The inclusion relation receives a recursive deﬁnition in terms of membership and is shown to be not always antisymmetric.


Introduction
Multisets are sets with possibly repeated elements, somewhat natural objects that arise in various situations in both mathematics and computing. However despite numerous accounts of multisets, some by quite well-known mathematicians, there has been no consensus on how to axiomatize them. The best survey of these accounts is Wayne Blizard's "The development of multiset theory" [4], and the two most comprehensive proposals seem to be Blizard's own in [2] and [5], the latter even allowing for infinite multiplicities. However, like other multiset theories, they are both twosorted theories where the multiplicities are different types of objects from the multisets they support. This means having separate axioms for multiplicity arithmetic, and in the infinite case it assumes a predefined model of cardinal arithmetic; for example, [5] uses cardinals in a model of Zermelo-Fraenkel (ZF) set theory.
Here we will propose a one-sorted account of multisets, where multiplicities and sets come from the same domain and follow the same axioms. In this system, multiplicities are no longer predefined cardinal numbers but are also multisets with their own internal structures. The natural ordering of multiplicities will be identified with the subset relation; that is, intuitively we consider x to be less than y as multiplicities if x is a proper subset of y. The axioms we propose will mirror Zermelo-Fraenkel set theory, mutatis mutandis, the only real complication coming from the subset relation for multisets (which becomes naturally recursive upon being identified with the ordering on multiplicities). Using a graph-based model closely related to the construction in D'Agostino and Visser [6], we prove the consistency of our multiset theory with antifoundation and show that unlike in set theory, the subset relation on multisets under our definition need not be antisymmetric; hence an extra axiom is necessary if one wants to enforce the antisymmetry of the inclusion relation.

Some Notes on Notation
For clarity we keep the number of brackets to a minimum and assign descending priorities to the following symbols: universal and existential quantifiers and their variants, conjunction and disjunction, implications. If R is a binary relation or binary predicate symbol, write .8xRy/'.x/ , df .8x/ xRy ) '.x/ ; .9xRy/'.x/ , df .9x/ xRy^'.x/ : Similarly, write .8hx 1 x n iRy/'.x 1 x n / and .9hx 1 x n iRy/'.x 1 x n / with the obvious meanings.
Relations and functions mean sets of ordered pairs (or tuples in the case of higher arity) unless specified otherwise. Write f 2 Function to mean that f is a function and f 1 for the inverse ¹hy; xi W hx; yi 2 f º (which may not be a function). Write dom f for the domain of the function f and ran f for the range of f .
Write R 2 Relation to mean that R is a relation, and write Dom R for the relational domain (or field) of R, that is, the set ¹x W .9y/.hx; yi 2 R _ hy; xi 2 R/º.

Remark 2.1
For any function f , we have Dom f D dom f [ ran f .

The Theory
This is a one-sorted theory where the same variables will be used for multisets and multiplicities; we may call them sets where there is no danger of confusion with traditional sets. The membership predicate x 2 a y means that x belongs to y with multiplicity a.

Definition 3.1
The language of multisets L M has one sort of variable and two predicate symbols: the identity D and the ternary symbol 2. (In practice we write x 2 a y.)

Remark 3.2
The membership and subset relations we will define on multisets will be denoted by overlined symbols to differentiate them from their set-theoretic counterparts. This distinction is especially important when we build a model for our multiset theory from a model of set theory.

Definition 3.3
We write x 2 y for .9a/x 2 a y, that is, x belongs to y. We write y x for the unique multiplicity of x in y.

Axiom 3.0.2 (Axiom of extensionality)
We have .8x; y/ x D y , .8a; b/.a 2 b x , a 2 b y/ : If '.x; y/ has two free variables and possibly parameters such that .8x/.9Šy/'.x; y/, we say that ' defines a function class on multisets.

Axiom 3.0.3 (Axiom of empty set)
We have .9x/.8y/y 6 2 x: Extensionality ensures that the empty multiset, which we denote by ;, is unique.

Axiom Schema 3.0.4 (Axiom schema of comprehension)
We have .8x/.9y/.8z; b/ z 2 b y , z 2 b x^'.z; b/ for all formulas ' with two free variables and possibly parameters.
Note that the set given by comprehension inherits the multiplicities from the original set. By introducing axioms that deal with multiplicities separately, we can extend the basic theory to incorporate different systems of multisets, for example, those with only finite multiplicities or cardinal multiplicities.

Definition 3.6
Let ' define a function class on multisets. Write ¹x˝y W '.x; y/º for the multiset a satisfying .8x; y/.x 2 y a , '.x; y// (i.e., a contains x with multiplicity y if and only if '.x; y/ holds) if such a multiset exists.
As a special case, for any concrete natural number n we write ¹x 1˝y1 ; : : : ; x n˝yn º for the multiset a satisfying the following if it exists (though later with all of our axioms, such multisets do indeed exist): .8x; y/ x 2 y a , .x D x 1^y D y 1 / _ _ .x D x n^y D y n / : The multisets specified in the definition above are unique by extensionality. We have an ordered pair hx; yi WD ® ¹x˝;º˝;; ¹x˝;; y˝;º˝;¯:

The subset relation
Intuitively in our theory there should be an ordering on multiplicities (namely, there are more copies of something than of another). With this in mind, we regard a multiset x as a subset of y if and only if every member of x appears in y with greater or equal multiplicity. As mentioned before, identifying the subset relation with the ordering on multiplicities gives us a natural recursive definition of subset.

Axiom Schema 3.1.1
The relation is a partial order and the largest class relation such that .8x; y/ x y , .8a 2 x/ a 2 y^x a y a ÁÁ : In other words, if '.x; y/ is a formula (possibly with parameters) such that .8x; y/ '.x; y/ , .8a 2 x/ a 2 y^' x a ; y a ÁÁÁ ; then .8x; y/.'.x; y/ ) x y/.
We will prove this schema from the other axioms by formally defining x y as shorthand for the formula below. It trivially follows from that definition that if x y, then every member of x is a member of y. Furthermore, ; is a subset of everything, while nothing else is a subset of ;.

Proof
In one direction ¹x˝;º y since ; is a subset of everything; the other direction is trivial.

Axiom 3.1.2 (Axiom of subset)
We have .8x; y/.x y^y x ) x D y/:

Remark 3.10
At this point one may question the necessity of this axiom, as the subset relation in set theory is trivially antisymmetric. However, in the last section we will show that the axiom of subset is independent from the remaining multiset axioms by means of a syntactic model.
We define the union of x to be the minimal superset of all members of x. In the context of two-sorted multiset theory, this definition corresponds to taking the supremum of multiplicities of the same object, as opposed to what [2] calls the additive union where multiplicities are added.

Axiom 3.1.3 (Axiom of union)
We have .8x/.9b/.8a/ b a , .8y 2 x/y a : Following set theory convention, we denote the union of x by S x and write x [ y for S ¹x˝;; y˝;º. Since is antisymmetric, S x is unique for every x. We follow the same approach in defining replacement. Let the formula ' define a function on x, and let a be in the image of x. If a is the image of more than one y 2 x, let the multiplicity of a be the -least upper bound (i.e., the union) of x y for all preimages y of a. If y is unique, it easily follows that the multiplicity of a is x y since is both reflexive and antisymmetric.

Axiom Schema 3.1.4 (Axiom schema of replacement)
We have for all formulas ' with two free variables and possibly with parameters.

Definition 3.11
It is clear that the multiset given by replacement is unique for each x and each function class ', and we will denote it by Rep ' x.

Proof
If .8y 2 x/a 6 2 y, by comprehension let z WD ¹v˝w W v 2 w b^v ¤ aº. Let y 2 x, and let R be a witness to y b. Let A WD S ¹R˝;; ¹hy; zi˝;º˝;º, and define by comprehension from A the multiset As a binary relation, S is obtained by adding hy; zi to R. Since a 6 2 y, Hence S is a witness to y z, and thus b z by definition of union, so a 6 2 b. Conversely, let y 2 x and a 2 y; then a 2 b since y b by definition.
We will prove later that the multiplicity of a 2 S x is the union of multiplicities of a in all b 2 x, using the schema of multiplicity replacement.

Definition 3.13
The canonical power set of x is P x WD ¹y˝; W y xº.

Axiom 3.1.5 (Axiom of power set)
We have .8x/.9y/y D P x:

Lemma 3.14
For any x; y there exists the product of x and y, namely,

Proof
This is proved by comprehension from P 3 .x [ y/.

Lemma 3.15
The relation is transitive and thus a partial order.

Proof
Suppose a b and b c. If a D b or b D c the proof is trivial, so suppose that they are distinct. Let R 1 witness a b, and let R 2 witness b c; then the following relation witnesses a c: R WD ® hx; zi˝; W .9y/ hx; yi 2 R 1^h y; zi 2 R 2 ¯:

Proof
Let '.x/ hold for some x. By comprehension from P x and union we have If '.y/ holds, then v y for any v in the multiset whose union is b; hence b y by the definition of union. Therefore .8a b/a y by transitivity of . Conversely, if a y for all y such that '.y/ holds, then a x. Hence a is in the union, so a b.
For convenience we denote the intersection as defined in Lemma 3.16 by T '.
x/ x. The intersection is unique if it exists since is antisymmetric. If '.x/ is the formula

Remark 3.17
Normally one would expect x 2 ; y to denote nonmembership, since it fits with the intuition of x belonging to y zero times. However, that would give rise to odd phenomena, for example, nonempty multisets with the same members but empty intersection. Suppose that x and y are nonempty and disjoint, which always exist if the model has more than one nonempty object. Then ¹;˝xº and ¹;˝yº have the same member, namely ;, but the multiplicity of ; in the intersection must be empty since it is a subset of both x and y.

Remark 3.18
There is no negative membership in our theory since ; is already the bottom multiplicity. (Recall that ; is a subset of everything, and we already chose to identify the ordering of multiplicities with the subset relation on multisets.) For a quick overview of negative multiplicities and a theory of multisets with integer multiplicities (including the negative integers), see Blizard [3].

Definition 3.19
A multiset R is a (binary) relation if all its members are ordered pairs. Define the canonical domain (or field) of R by comprehension as We define the canonical domain and range of a function f as Note that if f is a function, then Dom f D dom f [ ran f . In general, f can be regarded as a function on any multiset with the same members as dom f . Intuitively when viewed from outside the model of our theory, a function is just a map which sends multisets to multisets; the multiplicities in the domain and in the graph of the function itself are thus of no importance. For convenience we also extend our notation with ¹f .x/˝g.y/ W '.x; y/º where f; g are either functions or formulas defining function classes on multisets, with the obvious meaning.

Definition 3.21
Say that X is a closed multiset, and write X 2 Closed if .8v 2 X /.8a; b/ a 2 b v ) .a 2 X^b 2 X / :

Definition 3.22
A relation R is a well-order if it is a total order and any nonempty multiset A Dom R has an R-minimal member.
For the axiom of infinity we define an analogue of the von Neumann !.

Definition 3.23
Write˛2 ON for the formula saying both of the following: .8x 2˛/.x D ;^.8y 2 x/.y 2˛^x y D ;//, the relation 2 restricted to˛is a well-order. Write˛C for˛[¹˛˝;º, and write˛<ˇfor˛2ˇwhen˛andˇare both ordinals.

Proof
For˛to be closed it suffices to show ; 2 a, but the 2-minimal member of has to be empty. Supposeˇ2˛and x 2ˇ; thenx D ; since˛2 ON . But x 2˛, so for any y 2 x we have y 2˛and x y D ;. Now 2 well-orders˛, andˇ2 y would violate well-foundedness of 2 on˛, so y 2ˇ. To prove that 2 well-ordersˇ, note that every member ofˇis a member of˛and 2 well-orders˛.

Remark 3.25
It follows directly from the definition of˛C that˛2 ;˛C (note that cannot be a member of itself due to the well-ordering condition in the definition of ordinals), and˛C 2 ON whenever˛2 ON .

Axiom 3.3.1 (Axiom of infinity)
We have

Definition 3.26
We call this multiset ! and use the usual numerals to stand for the appropriate finite ordinals; that is, n C 1 stands for n C .

Remark 3.27
The usual schema of induction on the ordinals also works here since multiset ordinals are defined to be closed.

Lemma 3.28
For any x, there exists a closed multiset with x as a member.

Proof
Define .v; w/ , df w D v [ ¹b˝b W .9a/a 2 b vº. By union and replacement .8v/.9w/ .v; w/. Define a function class '.x; y/ from ! to the universe of multisets as follows: '.x; y/ , df .9f 2 Function/.f .x/ D y^f .0/ D ¹x˝;º^dom f ! .8n 2 dom f /.8m < n/m 2 dom f^.8n 2 dom f /f .n C 1/ D S Rep f .n//. The formula ' defines a function on all of !: if there is n 2 ! such that .Ày/'.n; y/, without loss of generality assume that n is 2-minimal. Then n ¤ 0, so n D m C 1 for some m 2 !, and some function f witnesses '.m; z/. By replacement, union, and comprehension we can extend f to include the ordered pair hn; S Rep f .m/i; then it witnesses '.n; S Rep f .m//, a contradiction. Hence for any n 2 ! we can write F .n/ for the unique y such that '.n; y/. By replacement, let X WD Rep ' ! (see Definition 3.11). Thus .8y/ y 2 X , .9a 2 !/F .a/ D y : For any v 2 S X, there exists n 2 ! such that v 2 F .n/. Let w be such that .v; w/ holds. If a 2 b v, then a; b 2 w, so Thus S X has the desired closure property, and x 2 S X since x 2 F .0/.

Corollary 3.29
For any x there is a -minimal closed multiset containing x.

Proof
Let '.y/ state that y is closed and x 2 y; then there exists y such that '.y/ holds. Hence take the intersection of all multisets satisfying '.

Lemma 3.30
We have .8x; y/ x y , .8a 2 x/ a 2 y^x a y a ÁÁ : Moreover, .8x; y/.'.x; y/ ) x y/ for any formula '.x; y/ such that .8x; y/ '.x; y/ ) .8a 2 x/ a 2 y^' x a ; y a ÁÁÁ : Proof If x D y the first claim is trivial, so we assume otherwise. If R witnesses x y, then all members of x are members of y and R itself witnesses Let X, Y be closed, and let x 2 X, y 2 Y . Define a relation R as follows: By the proven direction of the claim, we have From this and the definition of R, we have But hx; yi 2 R by the hypothesis, so R witnesses x y and the converse holds. Now suppose that '.x; y/ is a formula such that .8x; y/ '.x; y/ ) .8a 2 x/ a 2 y^' x a ; y a ÁÁÁ and that '.x; y/ holds for some particular pair x, y. Again let X, Y be closed multisets containing x, y, respectively, and define a relation R by ÁÁ^h v; wi 2 a X˝Y ± : Then the same argument as above shows that R witnesses x y.

Transitive closures
There are two obvious candidates for the definition of transitive multisets: .8x 2 a/.8y 2 x/y 2 a or .8x 2 a/x a: The second trivially implies the first, but the converse is false: consider X WD ¹;˝¹;˝;ºº and Y WD ¹X˝;; ¹;˝;º˝;; ;˝;º; then the second condition fails for Y . Here we adopt the stronger condition.

Definition 3.31
A multiset a is transitive if .8x 2 a/x a.

Remark 3.32
For X and Y defined in the previous paragraph, X is transitive but not closed while Y is closed but not transitive.

Lemma 3.33
Let '.x/ be a formula with one free variable such that .9x/'.x/, and '.x/ only holds for transitive multisets. Then T ' x is transitive.

Proof
Let a 2 T ' x, and suppose that '.x/ holds; then a 2 x since T ' x x. By transitivity a x for all such x, so a T ' x by definition.

Remark 3.34
Ordinals are transitive: if˛2 ON and x 2˛, then for any y 2 x we have y 2˛and x y D ; y ; hence x ˛.

Lemma 3.35
For any multiset x, there exists a -minimal transitive multiset T C.x/ such that x T C.x/.
Proof Let x be any multiset. We can define a function class ' on ! (see Thus there is a transitive w such that x w, so let T C.x/ be the intersection of all such w. By Lemma 3.3, T C.x/ is transitive, and by definition of intersection x T C.x/ and T C.x/ is -minimal.
Conversely, suppose x 6 2 y and .8a 2 y/x 6 2 T C.a/. By replacement, comprehension, and union let v WD y [ [ ® T C.a/˝b W a 2 b y¯: If w 2 v, then w ¤ x. Either w 2 y so w T C.w/ v, or w 2 T C.a/ for some a 2 y so w T C.a/ v. Hence v is transitive and T C.y/ v, so x 6 2 T C.y/.
We propose the following basic system, leaving foundation for later discussion.

Definition 3.37
The theory MS consists of the following axioms and schemas: unique multiplicity, extensionality, comprehension, pairing, subset, union, replacement, power set, and infinity.

The collection of sets
In this subsection we work in a model of MS.

Definition 3.38
For any multiset x, if there exists c such that we say that c D core.x/. Write x 2 Core for .8y 2 x/ x y D ;.

Remark 3.39
The axioms of MS might not guarantee that core.x/ exists for all x, but core.x/ is unique by extensionality if it exists.
It trivially follows that the core of core.x/ is itself, and two multisets have the same core if and only if they have the same members. Note that if˛2 ON , then core.˛/ D˛by definition.
Consider the following interpretation of the language of set theory: D is the identity relation, 2 is interpreted as 2, whereas 9 and 8 are relative to Set. The settheoretic inclusion relation under this interpretation coincides with , that is, .8x; y 2 Set/ x y , .8a 2 x/a 2 y :

Theorem 3.41
If MS is consistent, so is ZF.

Proof
The given interpretation turns Set into a model of ZF minus foundation. For infinity we have ! 2 Set, while for any other axiom the multiset given by its multiset counterpart is in Set as long as the parameters are in Set.

Remark 3.43
Up to this point, since we have avoided manipulation of multiplicities, note that any model of ZF set theory provides a model for MS by interpreting x 2 a y , df x 2 y^a D ;. Hence the consistency strength of MS is trivially the same as ZF.

Multiplicity replacement
By adding different axioms to handle multiplicities, we can extend MS to implement different systems of multisets of various strengths. For example, consider the following theory.

Definition 3.44
The theory MS ! consists of all MS axioms, plus the axiom of finite multiplicities.
The theory MS ! describes a system of multisets with finite multiplicities, essentially equivalent to the theory MST in [2]. The one major difference is that in MST the multiplicity 0 means nonmembership, that is, the multiplicity n in MS ! corresponds to the multiplicity n C 1 in MST.
To define a theory where multiplicities are ZF cardinals like MSTC in [5], note that Set is a model of ZF. Thus we would only need to add the following axioms to MS (where Card is the class of set-theoretic cardinals defined in Set).

Axiom Schema 3.7.2
For any ' with two free variables and possibly parameters We have .8x; y; a/.x 2 a y ) a 2 Card/.

Remark 3.45
As long as our axiom is strong enough to replace any multiplicity by ;, two multiset ordinals will have a bijection in the universe of multisets if and only if they have a bijection in the class Set; thus the alephs in the multiset model will be the same as the set-theoretic alephs in the ZF interpretation of Set.
The strongest axiom to manipulate multiplicities is given below.

Axiom Schema 3.7.4 (Axiom schema of multiplicity replacement)
We have .8x/ .8a 2 x/.9Šb/'.a; b/ ) .9y/y D ® a˝b W a 2 x^'.a; b/¯ for any formula ' with two free variables and possibly parameters.
With this axiom, our multiset theory can no longer accept models of set theory since set-theoretic extensionality is provably false: for example, ¹;˝;º and ¹;˝¹;˝;ºº are distinct multisets with exactly the same members. In the same way, it cannot accept models of any traditional multiset theory where multiplicities are integers or cardinals, since multiplicity replacement can create multiplicities not allowed by those theories.

Definition 3.46
The theory MSR consists of MS plus multiplicity replacement.

Lemma 3.47
We have

By multiplicity replacement let
a , and the claim follows by antisymmetry of .

Definition 3.48
Say that x is well founded if every submultiset A of theminimal multiset containing x (see Corollary 3.29) has a minimal member y such that .8z 2 y/ z 6 2 A^y z 6 2 A Á : Alternatively we can define the class W F of well-founded multisets by an analogue of the von Neumann hierarchy, where V ; WD ;, V˛C 1 WD ® x˝; W .8y; a/ y 2 a x ) .y 2 V˛^a 2 V˛/ ¯; and V WD S ¹V˛˝; W˛< º for limit . A simple induction shows that the V˛are all closed and nested and that W F is precisely the class of well-founded multisets. We can then define the rank of x 2 W F as the minimal˛such that x 2 V˛, and thus

Proof
Let˛WD max¹sup y2x rank y C 1; sup y2x rank x y C 1º; then x 2 V˛C 1 .

Lemma 3.50
We have .8x 2 W F /.8y x/y 2 W F .

Proof
If there exist x and y contradicting the claim, let x be of minimal rank.

Proof
Since ordinals are closed, it is easy to show that˛C is the -minimal closed multiset that has˛as a member. Suppose A ˛C, and let y 2 A be the 2-minimal; then y satisfies Definition 3.48 straightforwardly.

Theorem 3.52
W F is a model of MSR plus foundation.

Proof
Extensionality holds since W F is closed downwards. Comprehension, pairing, power set, and multiplicity replacement all hold by Lemma 3.49. Union holds by an induction on rank, using the recursive relation in Lemma 3.47. Replacement follows from union and Lemma 3.49. Infinity holds since ! is well founded by Lemma 3.51. Foundation holds since the cumulative hierarchy defined relative to W F is exactly the same as the cumulative hierarchy that forms W F .

Lemma 3.53 (MSR)
For any multiset x there is a transitive closed multiset y such that x 2 y (see Corollary 3.29 and Lemma 3.35).

Proof
Let a be closed such that ¹x˝;º 2 a, and let b D S a; then Let c be the smallest closed multiset such that b 2 c. Then an easy induction on the minimal property of c shows that for all w 2 c, By multiplicity replacement let y WD ¹v˝b W v 2 cº; then y is also closed and obviously x 2 y. Let w 2 y; then w 2 c, so .8v 2 w/ w v b D y v . Hence w y, that is, y is transitive, as required.

Remark 3.54
Any model of Zermelo (not ZF) set theory that refutes transitive containment (such as that given in Mathias [10]) readily provides a model of MS minus replacement in which some multiset is not contained in any transitive multiset.

A Model for the Theory
We prove the consistency of MSR relative to ZF, starting with a model V of ZF.
It is possible to construct a well-founded model for our multiset theory from V by interpreting x 2 a y in the language of multisets as hx; ai 2 y within a recursively defined subclass of V . However, here we will construct a graph-based model, which will give us the consistency of MSR with antifoundation. Furthermore, in any wellfounded model of multiset theory the inclusion relation is necessarily antisymmetric (by induction on the membership relation), whereas our graph-based model will also prove the independence of the axiom of subset from the rest of MSR.

Accessible pointed hypergraphs
We will only consider directed 3-uniform hypergraphs; our model will be a definable class of these hypergraphs. A hypergraph H is thus a set of ordered triples hx; y; zi (i.e., its edges).

Remark 4.1
We sometimes write H.x; y; z/ as shorthand for hx; y; zi 2 H .

Definition 4.2
Let H be a hypergraph, and let x be any set.

Definition 4.5
A finite directed path from x 1 to x n in OEH; h is a finite sequence

Definition 4.6
We have x H contains x and all vertices of H accessible from x by a finite directed path.

Definition 4.7
Let H x be the restriction of H to x H . It is the smallest subgraph of H containing x and closed under outward edges.

Remark 4.9
For any hypergraph H and any x, OEH x ; x is accessible.

Bisimulations and extensionality
To enforce extensionality, we borrow the concept of a bisimulation from computer science. Our definition of multiset bisimulation was adapted from Aczel [1] and is very close to what D'Agostino and Visser use in [6] and [7], except that they have a fixed implementation of multiplicities (which are positive cardinals). In fact our model, when restricted to cardinal multiplicities, describes essentially the same multiset universe as in [6], though the particular constructions are different: while the latter followed a parallel course to Aczel's approach in [1], ours is generalized from the graph-based models of set theory in Hinnion [8] and Holmes [9].

Definition 4.10
Call a relation DomOEG; g DomOEH; h a bisimulation between OEG; g and OEH; h if for any a x we have .8b; c/ G.a; b; c/ ) .9y; z/ H.x; y; z/^b y^c z ^.8y; z/ H.x; y; z/ ) .9b; c/ G.a; b; c/^b y^c z : If OEG; g D OEH; h, we say that is a bisimulation on OEH; h.

Lemma 4.11
Let be a bisimulation between OEG; g and OEH; h; then we have the following. 1 where y 1 D h, y n D y, and y iC1 2 H 1 y i [ H 2 y i for all i. We show by induction on n that there is a sequence x 1 x n in DomOEG; g such that We proceed similarly for the other direction.

Definition 4.12
Say that OEH; h is extensional if the only nonempty bisimulation on OEH; h is (contained in) the identity.

Lemma 4.13
Let OEH; h be extensional; then OEH x ; x is extensional for any x 2 DomOEH; h. Furthermore, if OEH x ; x Š OEH y ; y for x; y 2 DomOEH; h, then x D y.

Proof
The first claim follows trivially from Lemma 4.11 (3) and (4), whereas the second claim follows from (4) of the same lemma.

The domain of our model
We will define a canonical member for each equivalence class. The collection of those forms a definable class of hypergraphs, which we will denote by M.

Lemma 4.14 (Quotient lemma)
For any pointed hypergraph OEH; h, there exists an extensional pointed hypergraph OEQ; q and a surjective quotient map from DomOEH; h to DomOEQ; q such that q D .h/ and .8a; b; c/ Q.a; b; c/ , .9x; y; z/ a D .x/^b D .y/^c D .z/ and such that the relation .x/ D y is a bisimulation between OEH; h and OEQ; q.
We call OEQ; q the extensional quotient of OEH; h. Furthermore, we have the following. 1 x y , df .9 /.
C^x y/: The identity on DomOEH; h is a bisimulation, so is reflexive. By Lemma 4.11 the relations ¹hy; xi W x yº and ¹hx; zi W .9y/x y zº are also bisimulations, so is symmetric and transitive. The set of relations on DomOEH; h ordered by forms a complete lattice, and the operation 7 ! C is monotonic. Hence by the Knaster-Tarski theorem (see, e.g., Tarski [11]) is the same as C .
Let DomOEQ; q be the set of equivalence classes of , let q be the equivalence class of h, and let W DomOEH; h ! DomOEQ; q be the corresponding quotient map.
Define the relation Q on DomOEQ; q by Q.a; b; c/ , df .9x; y; z/ H.x; y; z/^ .x/ D a^ .y/ D b^ .z/ D c : We show that any bisimulation Q on OEQ; q must be the identity. Define x H y , df .x/ Q .y/; then it is straightforward to check that H is a bisimulation. This means that H , so .x/ D .y/ for any x H y, and thus Q is the identity. We have shown that OEQ; q is extensional.

Definition 4.15
The hypergraph OEQ; q constructed in the quotient lemma provides a canonical example of an extensional quotient of OEH; h. From now on we simply refer to it as the extensional quotient of OEH; h.

Definition 4.16
Say that OEG; g and OEH; h are similar if their extensional quotients are isomorphic, and write OEG; g Á OEH; h.

Remark 4.17
It is immediate from the definition above and the quotient lemma that if OEG; g and OEH; h are extensional, then OEG; g Á OEH; h , OEG; g Š OEH; h.

Lemma 4.18
Let OEG; g and OEH; h be accessible. Then OEG; g Á OEH; h if and only if there exists a bisimulation between OEG; g and OEH; h such that g h.

Proof
Let G W OEG; g ! OEP; p and H W OEH; h ! OEQ; q be the quotient maps. If Â is an isomorphism between OEP; p and OEQ; q, then the relation Â.x/ D y is clearly a bisimulation, so x y , df Â G .x/ D H .y/ is a bisimulation by Lemma 4.11 (2). Furthermore, Conversely, let be a bisimulation between OEG; g and OEH; h such that g h. Define relations between DomOEP; p and DomOEQ; q and ' on DomOEP; p by x a , df .9y; b/ G .y/ D x^ H .b/ D a^y b ; x ' y , df 9a 2 DomOEQ; q .x a^y a/: By Lemma 4.11(2) again both are bisimulations. Since OEP; p is extensional, ' is the identity, so is a partial function from DomOEP; p to DomOEQ; q.
The same reasoning with OEQ; q shows that is injective. Moreover, p q and the hypergraphs are accessible, so by Lemma 4.11 we know that is defined on the whole of DomOEP; p and surjective on DomOEQ; q.
Thus is a bijection between DomOEP; p and DomOEQ; q, but it is also a bisimulation between OEP; p and OEQ; q, and furthermore p q.

Corollary 4.19
For any x; y 2 DomOEH; h, OEH x ; x Á OEH y ; y if and only if there is a bisimulation on OEH; h such that x y.

Proof
The proof is immediate from the Lemma 4.18 and Lemma 4.11.
We are now ready to define the class of canonical representations for multisets.

Definition 4.20
We call OEH; h a multigraph if it is accessible, extensional, and

Remark 4.21
Note that our definition of a multigraph is quite different from that of [6] and [7] despite the similarity of the eventual construction. The reason is both our different axiomatization of multiplicities and that our multigraphs are defined to be already extensional (or "collapsed" in other words).

Lemma 4.22
Let OEH; h be accessible, and for all a; b; c; d; e 2 DomOEH; h, let H.a; b; c/^H.a; d; e/^OEH b ; b Á OEH d ; d ) OEH c ; c Á OEH e ; e: Then the extensional quotient OEQ; q of OEH; h is a multigraph.

Proof
Let be the quotient map from OEH; h to OEQ; q. By construction OEQ; q is extensional, and its accessibility comes from OEH; h.
The relation .

Remark 4.25
The definitions above overload the notation of the language of multisets in an obvious manner and will be used in the following interpretation.
Given any formula ' in the language of multiset (see Definition 3.1), form the formula ' M in the language of set theory by restricting all universal and existential quantifiers to the class M; replacing the identity relation with the bisimilarity relation Á; replacing the membership relation x 2 a y with x 2 y^a D y x .

Definition 4.26
Write Mˆ' to say that ' M is true in the starting ZF model.
We proceed to prove the axioms of MSR under the given interpretation.

Lemma 4.27 (Axiom of extensionality)
Two pointed hypergraphs OEG; g; OEH; h 2 M are isomorphic if the following both hold: 8hg; x; yi 2 G 9hh; a; bi 2 H OEG x ; x Š OEH a ; a^OEG y ; y Š OEH b ; b ; 8hh; a; bi 2 H 9hg; x; yi 2 G OEG x ; x Š OEH a ; a^OEG y ; y Š OEH b ; b : Proof Define x y , df .x D g^y D h/ _ OEG x ; x Š OEH y ; y. One can readily check from the hypothesis that is a bisimulation between OEG; g and OEH; h. Since OEG; g, OEH; h are accessible and g h, by Lemma 4.18 OEG; g Á OEH; h, but they are extensional so OEG; g Š OEH; h.

Definition 4.28
For any OEH; h 2 M and y 2 H 1 x, write H.x; y/ for the unique z such that H.x; y; z/.

Definition 4.29
Let OEG; g; OEH; h 2 M. Say that OEG; g OEH; h if there is a relation C DomOEG; g DomOEH; h in the ZF model V , such that g C h and x C y ) .8a 2 G 1 x/.9b 2 H 1 y/ OEG a ; a Š OEH b ; b^G.x; a/ C H.y; b/ :

Proof
If OEQ; q Š OEG; g by the isomorphism ' and OEG; g OEH; g as witnessed by the relation C, then the relation '.x/ C y witnesses OEQ; q OEH; h.
The second claim is proved similarly.
The following lemma shows that is precisely the internal inclusion relation of M as defined in Definition 3.8 and satisfies Axiom 3.1.1.

Lemma 4.31
Let '.x; y/ be a formula in the language of multisets with two free variables and possibly parameters, such that that is, is antisymmetric if we interpret Š as the identity relation.
For clarity we divide the proof into several parts.

Proof
Suppose Mˆ'.OEQ; q; OEH; h/. Define a set relation C by x C y , df Mˆ' OEQ x ; x; OEH y ; y : Then C satisfies the condition in Definition 4.29, so OEQ; q OEH; h.

Remark 4.34
The operation taking C to C C depends on OEG; g and OEH; h, but we omit the associated multigraphs where there is no danger of confusion. Then C 1 C 2 ) C C 1 C C 2 given the same associated multigraphs.

Definition 4.35
Define the greatest subset relation by x y , df 9 C DomOEG; g DomOEH; h .C C C^x C y/:

Remark 4.36
Clearly OEG; g OEH; h if and only if g h.

Proof
If C C C and x C y, then by definition of C C we have G.x; a/ H.y; b/ trivially for any a 2 G 1 x and b 2 H 1 y, so x C y. This shows that C , so C CC and thus D C by maximality.

Remark 4.38
If x 2 DomOEG; g, y 2 DomOEH; h and C DomOEG x ; x DomOEH y ; y, clearly C C is the same relation whether defined relative to OEG; g and OEH; h or OEG x ; x and OEH y ; y. This means the restriction of to DomOEG x ; x DomOEH y ; y is the greatest subset relation between OEG x ; x and OEH y ; y; hence OEG x ; x OEH y ; y , x y:

Lemma 4.40
The relation is reflexive, transitive, and OEG; g OEH; h^OEH; h OEG; g ) OEG; g Š OEH; hI that is, is antisymmetric if we interpret Š as the identity relation.

Proof
If C is the identity, then trivially C C C , so is reflexive. Let C 1 witness OEQ; q OEG; g, and let C 2 witness OEG; g OEH; h. Define x C y , df .9d /.x C 1 d^d C 2 y/: It is straightforward to verify that C C C . Furthermore, q C 1 g C 2 h, so q C h and thus C witnesses OEQ; q OEH; h. This means that is transitive.
Let 1 be the greatest subset relation between OEG; g and OEH; h, and let 2 be the greatest subset relation between OEH; h and OEG; g. Define DomOEG; g DomOEH; h by Similarly if x y and H.y; c; d /, then there are a c, b d such that G.x; a; b/. Hence is a bisimulation and OEG; g Š OEH; h.
Thus all claims in Lemma 4.31 have been proved.

The axiom of union
For ease of reference we give the statement of the axiom first, whereas the proof will be divided into several lemmas.

Remark 4.42
We will define a recursive relation between vertices of OEG; g and sets of vertices of OEH; h. Intuitively the vertex a is related to the set X if the subgraph OEG a ; a is the -least upper bound of ¹OEH x ; x W x 2 Xº. Hence OEG; g is the union of OEH; h if and only if g is related to H 1 h.
Let OEG; g and OEH; h be fixed, and define (see Lemma 3.47) the following.

Definition 4.44
For any a 2 DomOEG; g and X DomOEH; h let a X , df 9 C DomOEG; g P DomOEH; h .C C C^a C X /:

Remark 4.45
It is straightforward to see that C 1 C 2 ) C C 1 C C 2 . As before, by the Knaster-Tarski theorem we have D C .

Lemma 4.46
For each X DomOEH; h there is no more than one a X.

Proof
Define a relation on DomOEG; g by a b , df 9X DomOEH; h .a X^b X /: If a b, then .8OEQ; q 2 M/.OEQ; q 2 OEG a ; a , OEQ; q 2 OEG b ; b/. But OEG; g is extensional, so G 1 a D G 1 b. Now let c 2 G 1 a;  If x C y, then by the defining property of , 8OEQ; q 2 OEH x ; x OEQ; q 2 OEG y ; y; so for any a 2 H 1 x there exists b 2 G 1 y such that OEH a ; a Š OEG b ; b.
Let X witness x C y; then H.x; a/ C G.y; b/ since the following both hold: Trivially d C g, so C witnesses OEH d ; d OEG; g and thus OEQ; q OEG; g. Conversely, suppose OEP; p is such that OEQ; q OEP; p for any OEQ; q 2 OEH; h. Define C DomOEG; g DomOEP; p to witness OEG; g OEP; p by First note that g C p as witnessed by H 1 h. Let c 2 G 1 a, and let X witness a C b. Since a X, OEG c ; c 2 OEH x ; x for some Therefore G.a; c/ C P .b; v/, and so C witnesses OEG; g OEP; p, as required.
Thus it suffices to construct a multigraph OEG; g such that g H 1 h, where is defined as in Definition 4.44. Intuitively we add a new vertex .X / for each X DomOEH; h and build a new graph D recursively so that the extensional quotient of OED .X/ ; .X / is the -least upper bound of ¹OEH x ; x W x 2 Xº. Then we let d WD .H 1 h/, and the extensional quotient of OED; d will be the required OEG; g.

Definition 4.48
Let be a fixed bijection from P .DomOEH; h/ to some set A disjoint from DomOEH; h, and let d WD .H 1 h/.

Remark 4.49
Since our underlying set-theoretic model is a model of ZF, it is trivial to obtain a uniform definition for given any OEH; h.
The range A of will be the set of new vertices corresponding to subsets of DomOEH; h, and we will build a graph C in which each vertex in A represents the union of its preimage under .

Definition 4.50
Let C be the smallest set such that Note that the unions in this definition are set-theoretic.
Informally C is built by following the recursive property of a multiset union on multiplicities (see Lemma 3.47) from the top vertex d down. In the definition of C above, the vertex b represents members of the union, their members, members of their members, and so on; we simply copy the corresponding subgraphs of OEH; h over to form the final hypergraph, that is, so that

Definition 4.51
Let Let OEG; g be the extensional quotient of OED; d , and let be the quotient map.

Lemma 4.52
We have OEG; g 2 M.

Proof
Note that the subgraph

Lemma 4.53
We have g H 1 h where is as defined in Definition 4.44.

Proof
Define a C X , df a D .X /. We first show that C C C . Let a D .X /. If v 2 H 1 x for some x 2 X, then .X /; v; ® w W .9x 2 X /H.x; v; w/¯˛2 D: We have shown that for any Q 2 M, OEQ; q 2 OEG a ; a , .9x 2 X /OEQ; q 2 OEH x ; x: Hence aC C X by definition. But g C H 1 h, so g H 1 h, as required.
This concludes the proof of Lemma 4.41.

Supertransitivity
The following lemma shows that M is in a sense supertransitive; that is, for any set X M there is a multigraph whose members in the sense of M are precisely members of X in V .

Lemma 4.54
Let ' be a function in V such that dom '; ran ' M and But also OED a ; a D OEA; a and OEQ .a/; .a/ Š OEA; a, so OEQ;q OEH;h Š 'OEH; h.

The axiom of power set
We aim to obtain the M-power set of OEH; h by supertransitivity. For any OEH; h 2 M we will define a subset of the class M which contains an isomorphic copy of every OEQ; q OEH; h.

Definition 4.55
An edge chain in OEH; h is a sequence of linked edges of H , that is, hh 2i 1 ; h 2i ; h 2i C1 i W 1 Ä i Ä n, where h 1 D h and n 1. We can denote such an edge chain by the vertex sequence bh 1 h 2nC1 c without ambiguity.
The expansion expOEH; h is the set of all edge chains in OEH; h.

Lemma 4.56
Two edge chains bg 1 g 2nC1 c and bh 1 h 2nC1 c in OEH; h are identical if OEH g 2i ; g 2i Š OEH h 2i ; h 2i for all 1 Ä i Ä n.

Proof
Since OEH; h is extensional, the hypothesis means that g 2i D h 2i for all 1 Ä i Ä n. But g 1 D h D h 1 , so the result follows by induction on n and uniqueness of multiplicity.

Proof
This follows by induction on n using Lemma 4.31 in the induction step.

Definition 4.58
Let A expOEQ; q and B expOEH; h. A projection ' W A ! B is an injection which preserves chain length, that is, and is such that the following conditions hold.
The next result follows trivially from the definition.

Corollary 4.59
Let A expOEQ; q, B expOEH; h, and C expOEG; g. If ' W A ! B and W B ! C are projections, then the composite ' W A ! C is a projection.

Lemma 4.60
Let A expOEQ; q, B expOEH; h, and let ' W A ! B be a projection.
If 'bq 1 q 2nC1 c D bh 1 h 2nC1 c, then for any 1 Ä i Ä n,

Proof
We prove the claim for i D n 1 from the case i D n (which is true by definition); then an induction on i with the same argument in the inductive step will complete the proof. Let 'bq 1 q 2n 1 c D bg 1 g 2n 1 c; then by the first condition of Definition 4.58, 'bq 1 q 2nC1 c D bg 1 g 2nC1 c for some g 2n ; g 2nC1 . Hence g m D h m for all 1 Ä m Ä 2n C 1. By the second condition of Definition 4.58 we then have OEQ q 2n ; q 2n Š OEH h 2n 2 ; h 2n 2 .

Lemma 4.61
Let A expOEQ; q, B expOEH; h, and let ' W A ! B be a projection.
Then the inverse ' 1 W ran ' ! A is also a projection.

Proof
Clearly ' 1 preserves chain length since ' does, and the second condition in Definition 4.58 is also trivially satisfied.

Lemma 4.62
We have OEQ; q OEH; h if and only if there is a projection ' W expOEQ; q ! expOEH; h.

Proof
For the first direction suppose OEQ; q OEH; h.
OEH h 2iC1 ; h 2iC1 and OEQq 2i ; q 2i Š OEH h 2i ; h 2i for all 1 Ä i Ä n. By Lemmas 4.56 and 4.57, ' is a welldefined function on all of expOEQ; q; and the second condition in Definition 4.58 holds trivially.
If 'bq 1 q 2nC1 c D 'bb 1 b 2mC1 c, then m D n since ' preserves chain length by definition. Furthermore, OEQ q 2i ; q 2i Š OEQ b 2i ; b 2i for all 1 Ä i Ä n, so by Lemma 4.56, bq 1 q 2nC1 c D bb 1 b 2mC1 c, and thus ' is injective.
If 'bq 1 q 2n 1 c D bh 1 h 2n 1 c and hq 2n 1 ; q 2n ; q 2nC1 i 2 Q, let bb 1 b 2nC1 c WD 'bq 1 q 2nC1 c: Then bb 1 b 2n 1 c D 'bq 1 q 2n 1 c by checking the definition of ', so bb 1 b 2n 1 c D bh 1 h 2n 1 c. Thus the first condition of Definition 4.58 holds, and ' is a projection.
Conversely, suppose that there exists a projection ' W expOEQ; q ! expOEH; h. Define x C y if there exists 'bq 1 q 2nC1 c D bh 1 h 2nC1 c such that x D q 2nC1 and y D h 2nC1 , or x D q and y D h.
Suppose n > 1 and 'bq 1 q 2n 1 c D bh 1 h 2n 1 c and hq 2n 1 ; q 2n ; q 2nC1 i 2 Q. By the first condition of Definition 4.58 we can extend bh 1 h 2n 1 c to bh 1 h 2nC1 c D 'bq 1 q 2nC1 c.
If n D 1, just take bh 1 h 2nC1 c WD 'bq 1 q 2nC1 c since there is nothing to extend.
Hence C witnesses OEQ; q OEH; h as in Definition 4.29.

Definition 4.63
Say that B expOEH; h is a subimage of OEH; h if it is closed under initial subchains, that is, if bh 1 h 2nC1 c 2 B where n > 1, then We will establish a correspondence between the subimages of OEH; h and the submultisets of OEH; h in our interpretation of multiset theory on M. To this end we first define a unique pointed hypergraph for each subimage.

Definition 4.64
Write dx 1 x n e for an arbitrary finite sequence, not necessarily in any expansion. Intuitively B is the graph created by first making a tree where the edge chains in B are glued together at their common initial subchains, then attaching to each even-indexed vertex dh 1 h 2n e the subgraph H dh 1 h 2n e.

Lemma 4.67
The pointed hypergraph OEB; dhe is accessible.
Proof Let x be a vertex of OEB; dhe; then it falls into one of two cases. If x is in the edge h dh 1 h 2n 1 e; dh 1 h 2n e; dh 1 h 2nC1 ei where bh 1 h 2nC1 c 2 B, the following finite directed path goes from dhe to x: there is a finite directed path from dh 1 h 2n e to x since H dh 1 h 2n e Š H h 2n (see Definition 4.65). Hence there is a path from dhe to x.

Proof
On the other hand, Dom H dh 1 h 2n e is disjoint from Dom H dg 1 g 2n e for any other sequence dg 1 g 2n e, and except for the top vertex dh 1 h 2n e it is also disjoint from the tree but OEH dh 1 h 2n e; dh 1 h 2n e Š OEH h 2n ; h 2n , so the result follows.

Lemma 4.69
The pointed hypergraph OEB; dhe satisfies the hypothesis of Lemma 4.22; hence its extensional quotient is a multigraph.

Proof
We have shown that OEB; dhe is accessible. But OEB a ; a Á OEB b ; b, so a D b since OEB x ; x is extensional. Thus v D w by the same argument as in the previous case.
Hence the extensional quotient of OEB; dhe is a multigraph.

Definition 4.70
If B is a subimage of OEH; h, define OEQ B ; q B 2 M to be the extensional quotient of OEB; dhe.

Definition 4.71
If B is a subimage of OEH; h, denote the quotient map by Then the primary projection of B is ®˝b q 1 q 2nC1 c; bh 1 h 2nC1 c2

Lemma 4.72
For any bq 1 q 2nC1 c 2 expOEQ B ; q B there exists bh 1 h 2nC1 c such that hbq 1 q 2nC1 c; bh 1 h 2nC1 ci is in the primary projection.

Proof
Suppose q 2i 1 D dh 1 h 2i 1 e for some bh 1 h 2i 1 c 2 B, or suppose i D 1 and q 2i 1 D dhe. Now hq 2i 1 ; q 2i ; q 2iC1 i 2 Q B , and by Definition 4.66 the only edges of B to come out of dh 1 h 2i 1 e are ®˝ dh 1 h 2i 1 e; dh 1 h 2i e; dh 1 h 2iC1 e˛W bh 1 h 2iC1 c 2 B¯: So there are h 2i , h 2iC1 such that q 2i D dh 1 h 2i e, q 2i C1 D dh 1 h 2iC1 e, and bh 1 h 2iC1 c 2 B.
Since q 1 D dhe, we can build the required edge chain bh 1 h 2nC1 c by induction on i .

Lemma 4.73
The primary projection P of B is a projection to expOEH; h, and its range is precisely B.

Proof
Let hbq 1 q 2nC1 c; bh 1 h 2nC1 ci, hbq 1 q 2nC1 c; bb 1 b 2nC1 ci 2 P . For any 1 Ä i Ä n, by definition of the quotient map and Lemma 4.68, h is extensional. By Lemma 4.56, bh 1 h 2nC1 c D bb 1 b 2nC1 c, so P is the graph of a partial function. By Lemma 4.72, this function is defined on all of expOEQ B ; q B , so we will denote it by ' W expOEQ B ; q B ! B.
Thus ' is injective, and we have already verified the second condition of Definition 4.58 above.
Suppose 'bq 1 q 2n 1 c D bh 1 h 2n 1 c and 'bq 1 q 2nC1 c D bb 1 b 2nC1 c. Then bb 1 b 2n 1 c 2 B and OEH h 2i ; h 2i Š OEQ B q 2i ; q 2i Š OEH b 2i ; b 2i for all 1 Ä i < n, so by Lemma 4.56, bh 1 h 2n 1 c D bb 1 b 2n 1 c and the first condition of Definition 4.58 holds.
By definition, ' preserves chain length, so it is a projection. Finally, for any bh 1 h 2nC1 c 2 B, let q i D dh 1 h i e for 1 Ä i Ä 2n C 1. By Definition 4.66 we have the following edge sequence in B: Hence bq 1 q 2nC1 c 2 expOEQ B ; q B so 'bq 1 q 2nC1 c D bh 1 h 2nC1 c by definition. Therefore ran ' D B.

Corollary 4.74
If B is a subimage of OEH; h, then OEQ B ; q B OEH; h.

Proof
The corollary is proved by Lemmas 4.73 and 4.62.
For completeness we include the next result, though we will not use it.

Lemma 4.75
If A, B are subimages of OEH; h such that OEQ A ; q A Š OEQ B ; q B , then A D B.

Proof
Let W DomOEQ A ; q A ! DomOEQ B ; q B be an isomorphism. Let ' A W expOEQ A ; q A ! A and ' B W expOEQ B ; q B ! B be the primary projections.
Suppose ba 1 a 2nC1 c 2 A; then ba 1 a 2nC1 c D ' A bq 1 q 2nC1 c for some   This means the set ¹OEQ B ; q B W B is a subimage of OEH; hº contains one representative from each isomorphism class of multigraph OEQ; q OEH; h. Define a constant function ı on this set by ıOEQ B ; q B D OE;; 0. Note that OE;; 0 corresponds to the empty multiset in our interpretation of the language, which will give us the canonical power set of OEH; h by supertransitivity.  With the presence of multiplicity replacement, the axiom of collection below implies the axiom of replacement as formulated in the theory MS.
We have showed that M interprets MSR, whence we have the following.

Theorem 4.82
If ZF is consistent, then MSR is consistent.

Proof
Suppose that M believes OEH; h is a hypergraph satisfying the condition of multiset AFA. Define a hypergraph G as 8x; y; z 2 DomOEH; h hx; y; zi 2 G , Mˆ˝OEH x ; x; OEH y ; y; OEH z ; z˛2 OEH; h : Then for any x 2 Dom G, OEG x ; x satisfies the conditions of Lemma 4.22, so its extensional quotient is a multigraph. Moreover, for all x; y; z 2 Dom G, MˆOEG y ; y 2 OEG z ;z OEG x ; x , Mˆ˝OEH x ; x; OEH y ; y; OEH z ; z˛2 OEH; h : For each x 2 Dom G we can define a canonical M x 2 M such that MˆM x D˝OEH x ; x; OEG x ; x˛: Define a constant function W ¹M x W x 2 Dom Gº ! M by '.M x / D OE;; 0. Then the supertransitivity lemma (Lemma 4.54) gives the required object in M.

A Model of Multiset Theory Where the Inclusion Relation is not Antisymmetric
In any well-founded model of MS an induction on the recursive definition of will show it to be antisymmetric; hence the axiom of foundation implies the axiom of subset in that case. However, as an application of the antifoundation property of our model M, we will modify it slightly to obtain a model of MSR where is not antisymmetric.

Modified multigraphs
Let M be the class of multigraphs as defined in Definition 4.20. In this section we will redefine multigraphs by strengthening the notions of a bisimulation as follows.

Definition 5.1
Let A WD OEh1; 0; 1i; 1, that is, the pointed hypergraph with vertices 0 and 1, where 1 is the point and 0 belongs to 1 with multiplicity 1.

A relation
DomOEG; g DomOEH; h is a bisimulation between OEG; g and OEH; h if all of the following hold.
It is a bisimulation in the old definition, that is, Definition 4.10.
Informally speaking, under this new definition A is no longer equivalent to its other isomorphic images. Nevertheless we can easily adapt the proofs of earlier results about bisimilarity to this new definition. From this point on we construct the model in exactly the same way as before except for a minor subtlety. First, we redefine the class M of multigraphs using the new definition of bisimulation, and we define relations on M to stand in for the identity and membership relations.

Definition 5.3
Say that OEH; h is extensional if any nonempty bisimulation on OEH; h is the identity.

Definition 5.4
Say that OEG; g Á OEH; h, that is, they are bisimilar, if there is a bisimulation between them such that g h.

Definition 5.5
Say that OEG; g 2 OEH; h if there exists x 2 DomOEH; h such that OEG; g Á OEH x ; x.
If OEG; g Á OEH; h, then it is easy to see that their extensional quotients are isomorphic. However, unlike in the old model, even isomorphic multigraphs need not be bisimilar. For example, let OEG; g be a distinct but isomorphic copy of A; then the extensional quotients of OEG; g and A are just themselves, but there is no bisimulation between them that relates the top vertices to each other (since by Definition 5.2 we would then have g D 1 and OEG; g D A). Nevertheless we can easily check that Á is an equivalence relation that respects 2; thus when proving previous results in the new model we need to replace all instances of isomorphism with the bisimilarity relation.

Remark 5.6
The proof of extensionality requires only trivial modification to check the case when A is involved.
Let us now address a small inconvenience. In the old model, given any collection of multigraphs we can easily create a copy in which all the multigraphs have disjoint domains. This was crucial in the proof of many axioms such as collection, where a large multigraph needs to be created. However, the special status of A in our new model means that we are longer allowed to replace it with isomorphic copies. Thus we have to slightly weaken the conditions of disjoint domains.
Say that B is an almost disjoint copy of A if B is almost disjoint and there is a bijection W A $ B such that OEG; g Á OEG; g for all OEG; g 2 A.
Note that this definition also works for classes if we allow the bijection to be a function class, but for our current purposes it suffices to consider only sets.

Lemma 5.8
For any collection A M there is an almost disjoint copy.
For any a 2 DomOEG; g let O a WD ha; OEG; gi, then trivially a ¤ 0 and a ¤ 1.
ci W ha; b; ci 2 Gº and OEG; g WD OE O G; O g. Then OEG; g Š OEG; g and the isomorphism is a bisimulation between them. If 1 2 DomOEG; g and OEG 1 ; 1 D A, define O a for a 2 DomOEG; g as above except that O 0 D 0 and O 1 D 1, and define OEG; g as above. Then OEG; g Š OEG; g is a bisimulation since both 0 and 1 are fixed. It is straightforward to check that the range of is almost disjoint.

Remark 5.9
Let A be an almost disjoint set of multigraphs, and let H be the hypergraph obtained by taking the union of all hypergraphs in A. Then for any OEG; g 2 A, OEG; g D OEH g ; g. Hence we can repeat the proof of supertransitivity, using almost disjoint multigraphs where disjoint multigraphs were needed.
The proofs of comprehension, multiplicity replacement, collection, and infinity thus carry over trivially. Now we give the updated definition of .

Definition 5.10
Say that OEG; g OEH; h if there is a relation C DomOEG; g DomOEH; h such that g C h and for all x 2 DomOEG; g, y 2 DomOEH; h, x C y ) .8a 2 G 1 x/.9b 2 H 1 y/ OEG a ; a Á OEH b ; b^G.x; a/ C H.y; b/ :

Proof
For the first part let 1 ; 2 be the bisimulations involved, and let C witness OEG; g OEP; p. Then the following relation witnesses OEH; h OEQ; q: x y , df 9a 2 DomOEG; g 9b 2 DomOEP; p .a 1 x^b 2 y^a C b/: The rest of the proof is mostly the same as for Lemma 4.31.
Note that in Lemma 4.31 we constructed a bisimulation explicitly to show that is antisymmetric, but that construction fails to be a bisimulation in the new definition. In fact is no longer antisymmetric.