The Finite Representation Property

# The finite representation property for composition, intersection, domain and range

## Abstract.

We prove that the finite representation property holds for representation by partial functions for the signature consisting of composition, intersection, domain and range and for any expansion of this signature by the antidomain, fixset, preferential union, maximum iterate and opposite operations. The proof shows that, for all these signatures, the size of base required is bounded by a double-exponential function of the size of the algebra. This establishes that representability of finite algebras is decidable for all these signatures. We also give an example of a signature for which the finite representation property fails to hold for representation by partial functions.

The first author would like to thank his PhD supervisor Robin Hirsch for many helpful discussions.

## 1. Introduction

The investigation of the abstract algebraic properties of partial functions involves studying the isomorphism class of algebras whose elements are partial functions and whose operations are some specified set of operations on partial functions—operations such as composition or intersection, for example. We refer to an algebra isomorphic to an algebra of partial functions as representable.

One of the primary aims is to determine how simply the class of representable algebras can be axiomatised and to find such an axiomatisation. Often, the representation classes have turned out to be axiomatisable by finitely many equations or quasi-equations [8, 2, 5, 3, 4, 1].

Another question to ask is whether every finite representable algebra can be represented by partial functions on some finite set. Interest in this so-called finite representation property originates from its potential to help prove decidability of representability, which in turn can help give decidability of the equational or universal theories of the representation class.

Recently, Hirsch, Jackson and Mikulás established the finite representation property for many signatures, but they leave the case for signatures containing the intersection, domain and range operations together open [1].

In this paper we prove the finite representation property for the most significant group of outstanding signatures, which includes a signature containing all the most commonly considered operations on partial functions. From our proof we obtain a double-exponential bound on the size of base set required for a representation. It follows as a corollary that representability of finite algebras is decidable for all these signatures. As an additional observation, we give an example showing that there are signatures for which the finite representation property does not hold for representation by partial functions.

The results presented here originate with McLean [6]. The contribution of the second author is to translate the original proof of the finite representation property into a semantical setting, so that the presence of antidomain is not necessary.

## 2. Algebras of Partial Functions

In this section we give the fundamental definitions that are needed in order to state the results contained in this paper.

Given an algebra , when we write or say that is an element of , we mean that is an element of the domain of . We follow the convention that algebras are always nonempty.

###### Definition 2.1.

Let be an algebraic signature whose symbols are a subset of . An algebra of partial functions of the signature is an algebra of the signature whose elements are partial functions and with operations given by the set-theoretic operations on those partial functions described in the following.

Let be the union of the domains and ranges of all the partial functions occurring in an algebra . We call the base of . The interpretations of the operations in are given as follows:

• the binary operation is composition of partial functions:

 f;g={(x,z)∈X2∣∃y∈X:(x,y)∈f and (y,z)∈g},

that is, ,

• the binary operation is intersection:

 f∧g={(x,y)∈X2∣(x,y)∈f and (x,y)∈g},
• the unary operation is the operation of taking the diagonal of the domain of a function:

 D(f)={(x,x)∈X2∣∃y∈X:(x,y)∈f},
• the unary operation is the operation of taking the diagonal of the range of a function:

 R(f)={(y,y)∈X2∣∃x∈X:(x,y)∈f},
• the constant is the nowhere-defined empty function:

 0=∅,
• the constant is the identity function on :

 1′={(x,x)∈X2},
• the unary operation is the operation of taking the diagonal of the antidomain of a function—those points of where the function is not defined:

 A(f)={(x,x)∈X2∣∃y∈X:(x,y)∈f},
• the unary operation is fixset, the operation of taking the diagonal of the fixed points of a function:

 F(f)={(x,x)∈X2∣(x,x)∈f},
• the binary operation is preferential union:

 (f⊔g)(x)=⎧⎨⎩f(x)if f(x) definedg(x)if f(x) undefined, but g(x) definedundefinedotherwise
• the unary operation is the maximum iterate:

 f↑=⋃0≤n<ω(fn;A(f)),

where and ,

• the unary operation is an operation we call opposite:

 f−1={(y,x)∈X2∣(x,y)∈f and ((x′,y)∈f⟹x=x′)}.

The list of operations in Definition 2.1 does not exhaust those that have been considered for partial functions, but does include the most commonly appearing operations.

###### Definition 2.2.

Let be an algebra of one of the signatures permitted by Definition 2.1. A representation of by partial functions is an isomorphism from to an algebra of partial functions of the same signature. If has a representation then we say it is representable.

In [3], Jackson and Stokes give a finite equational axiomatisation of the representation class for the signature and similarly for any expansion of this signature by operations in .

In [1], Hirsch, Jackson and Mikulás give a finite equational axiomatisation of the representation class for the signature and similarly for any expansion of this signature by operations in . For expanded signatures containing the maximum iterate operation they give finite sets of axioms that, if we restrict attention to finite algebras, axiomatise the representable ones.

The operation that we call opposite is described in [7], where Menger calls the concrete operation ‘bilateral inverse’ and uses ‘opposite’ to refer to an abstract operation intended to model this bilateral inverse. The opposite operation appears again in Schweizer and Sklar’s [9] and [10], but thereafter does not appear to have received any further attention. In particular, for signatures containing opposite, axiomatisations of the representation classes remain to be found.

###### Definition 2.3.

Let be a signature. We say that has the finite representation property (for representation by partial functions) if whenever a finite algebra of the signature is representable by partial functions, it is representable on a finite base.1

In [1], Hirsch, Jackson and Mikulás establish the finite representation property for many signatures that are subsets of . Assuming composition is in the signature, they prove the finite representation property holds for any such signature not containing domain, any not containing range and almost all that do not contain intersection. This leaves one significant group of cases, which they highlight as an open problem: signatures containing .

In this paper we prove that and any expansion of by operations that we have mentioned (including opposite) all have the finite representation property.

## 3. Uniqueness of Presents and Futures

In this section we derive some results about representations of finite algebras, which we will use in the following section to prove the finite representation property holds.

Throughout this section, let be a signature with and let be a finite representable -algebra. Since is representable, we may freely make use of basic properties of algebras of partial functions in the process of our deductions.

First note that the algebra is a meet-semilattice, with meet given by . Whenever we treat as a poset, we are using the order induced by this semilattice. The set of domain elements forms a subsemilattice of .

In any representation of we have that if and only if is greater than or equal to the meet of the finite set . Hence we may identify each representation of with a particular edge-labelled directed graph (with reflexive edges). The label of an edge is the least element of . Since we take the base of a representation to be the union of the domains and ranges of all the partial functions, every vertex participates in some edge. Given that the domain and range operations are in the signature, this means that all vertices will have a reflexive edge.

The previous paragraph motivates our interest in the following type of object, of which representations of are a special case.

###### Definition 3.1.

A network over will be an edge-labelled directed graph, with labels drawn from and with a reflexive edge on every vertex.

Given a network , we will follow the usual convention of writing to mean is a vertex of and we will denote the label of an edge by . We will speak of an element holding on an edge when . We will call a vertex an -vertex if the reflexive edge at that vertex is labelled , that is if .

Note that if is an -vertex of a representation then is necessarily a domain element. Indeed if holds on it follows that holds on . Since is the least such element, we have . It follows that , by a property of partial functions.

###### Definition 3.2.

Let be a network (over ) and let be a subset of the vertices of . We define the future of to be the subnetwork induced by the vertices reachable via an edge starting in . We define the future of a vertex to be the future of the singleton set containing .

Since is finite and we are representing by partial functions, in a representation, the future of a vertex must be a finite network.

Note also that in a representation, the taking of futures is a closure operator. Indeed, each is reachable from via the reflexive edge at . If there is an edge from to , labelled , and from to , labelled , then is reachable via an edge starting in , labelled .

###### Definition 3.3.

Let be a network (over ). The present of a vertex of is the set of all vertices such that is in the future of and is in the future of .

We are interested in presents and futures because in Section 4 we will describe how to use the presents and futures extant in representations in order to construct a representation on a finite base.

###### Definition 3.4.

Let . If there exists a representation of in which labels an edge, then we will call realisable.

###### Proposition 3.5.

For any realisable domain element , any two -vertices from any two representations have isomorphic futures.

###### Proof.

Let be an -vertex: . We claim that there is an edge starting at labelled with if and only if .

Suppose first that . Then, by the definition of the domain operation, there must be an edge starting at labelled with some . Then must hold on and so . From and it follows that , by a property of partial functions. Hence there is an -labelled edge starting at .

Conversely, suppose there is an edge labelled with starting at , ending at say. Then holds on and so . Since is a domain element, this implies , by a property of partial functions. But holds on and so , since . We conclude that .

Note that as the elements of are represented by partial functions, there cannot be multiple edges starting at on which the same element holds. In particular there cannot be multiple edges with the same label. We therefore now know that for any -vertex , the edges starting at are precisely a single edge labelled for every with (the -labelled edge being the reflexive one). So we have an obvious candidate for an isomorphism between the futures of -vertices: given two -vertices and in networks and respectively, we let if and only if .

For and with , let and be the two edges starting at and labelled by and respectively. To show that the future of is uniquely determined up to isomorphism, we only need show that the set of elements of holding on is uniquely determined. We claim that

 (1) if N(x,y)=a and N(x,z)=b, then c holds on (y,z) if and only if a;c=b,

which gives us what we want.

Suppose first that . Then as is the unique edge starting at on which holds, must hold on in order that composition be represented correctly. Conversely, suppose that holds on . Then holds on and so . But is valid in all representable algebras and . From and we may conclude , by a property of partial functions. ∎

For realisable domain elements and , write if in a (or every) representation of there is a -vertex in the future of every -vertex, or equivalently if there exists an with and . Then is easily seen to be a preorder on the realisable domain elements.

###### Proposition 3.6.

For any realisable domain elements , if and are -equivalent, then any -vertex and any -vertex from any two representations have isomorphic futures.

###### Proof.

Let be an -vertex, from some representation of . As , in the same representation there is a -vertex, say, in the future of . As there is an -vertex, say, in the future of . Hence is in the future of , meaning that the future of is a subnetwork of the future of . But these are finite isomorphic objects and therefore equal. So is in the future of and therefore is in the future of . Hence the futures of the -vertex and the -vertex are equal in this representation. By Proposition 3.5, we conclude the required result. ∎

In a representation, the present of an -vertex is always the initial, strongly connected component of ’s future—the one that can ‘see’ the entire future of . So we get the following immediate corollary of Proposition 3.6.

###### Corollary 3.7.

For any realisable domain elements , if and are -equivalent, then any -vertex and any -vertex from any two representations have isomorphic presents.

Given a -equivalence class , we will speak of ’the future of ’ to mean the unique isomorphism class of futures of -vertices in representations, for any . Similarly for ’the present of ’.

## 4. The Finite Representation Property

In this section, we prove our main result: the finite representation property holds for all the signatures we are interested in. We then use our proof to calculate an upper bound on the size of base required.

To construct a representation over a finite base we will use the realisable domain elements and the preorder on them defined in Section 3. Recall that the realisable domain elements are the domain elements appearing as reflexive-edge labels in some representation.

We start with a lemma that is little more than a translation of the definition of a representation into the language of graphs, but which gives us an opportunity to state exactly what is needed in order for a network to be a representation.

###### Lemma 4.1.

Let be a finite representable algebra of a signature with and let be a network (over ). Then is a representation of if and only if the following conditions are satisfied.

1. (Relations are functions) For any vertex of and any there is at most one edge starting at on which holds.

2. (Operations represented correctly) Let be an operation in the signature (excluding ). Then (assuming for simplicity of presentation that is a binary operation) if we apply the appropriate set-theoretic operation to the set of edges where holds and the set of edges where holds then we get precisely the set of edges where holds:

 {(x,y)∣N(x,y)≤a}∗{(x′,y′)∣N(x′,y′)≤b}={(x′′,y′′)∣N(x′′,y′′)≤a∗b}.
3. (Faithful) For every with , there is an edge of on which holds, but does not.

###### Proof.

Routine. ∎

We also require the following definition.

###### Definition 4.2.

Let be a finite representable algebra of a signature with . From the relation defined in Section 3, form the partial order of -equivalence classes (of realisable domain elements of ). The depth of a -equivalence class will be the length of the longest increasing chain in this partial order, starting at . (We take the length of a chain to be one fewer than the number of elements it contains, so a maximal -equivalence class has depth zero.) Since is finite, the depth of every -equivalence class is finite and bounded by the size of . The depth of a realisable domain element will be the depth of its -equivalence class.

We are now ready to prove our main result, but note that the following theorem does not cover signatures containing opposite.

###### Theorem 4.3.

The finite representation property holds for representation by partial functions for any signature with .

###### Proof.

Let be a finite representable algebra of one of the signatures under consideration. We construct a finite network step by step, by adding copies of the present of -equivalence classes of increasing depths. Then we argue that the resulting network is a representation of . The idea of the proof is to always ’add everything we can, times’.

Assume inductively that we have carried out steps of our construction, giving us the network . We form as follows. (For the base case of this induction, we let be the empty network.) Let be a -equivalence class of depth , and a copy of the present of . A choice of edges from to labelled by elements of is allowable if adding and these edges to would make the future of in the extended network isomorphic to the future of . For every allowable choice, to we add copies of both and the edges from specified by the choice. The network is the network we have once we have done this for every -equivalence class of depth .

Note that the order that -equivalence classes of a given depth are processed is immaterial, since no allowable choice could have an edge ending at a vertex that had not been in . By induction, each is finite: assume that is finite; then as each is also finite and is finite we see that the number of allowable choices is finite, so is finite. We take to be , where is the maximum depth of any -equivalence class of .

For any vertex of , the future of during the various stages of the construction of is unaltered once has been added to the construction. So the future of in is isomorphic to the future of some vertex in a representation of , since this is true at the moment that is added to the construction, by the definition of an allowable choice.

The next lemma will ensure that allowable choices always exist. Let be the underlying network for a representation of . Fix a vertex and write for the future of in . Define to be the subnetwork of induced by vertices such that the depth of is at most .

###### Lemma 4.4.

For every , there is an embedding . Moreover, if is any future-closed subset of and is an embedding, then can be chosen so that it agrees with wherever and are both defined.

###### Proof.

We use induction on . As before, the base case for the induction is depth , so we define to be the empty map.

For , suppose we have an embedding , a future-closed subset of and an embedding such that and agree where both are defined. We can form , an extension of , as follows. First use to define an intermediate extension of to those vertices in , that is:

 f′n−1(y)={fn−1(y)if y∈Fn−1g(y)if y∈(Fn∖Fn−1)∩G.

Using the assumption that is future closed, we will show that this intermediate extension is still an embedding. Observe that for any , the future of in is isomorphic to the future of in , since the future of any vertex of is isomorphic to the future, in any representation, of any vertex with the same reflexive-edge label. Hence maps elements of to elements of , from which we see that is injective. Now for to be an embedding of , we need to show that for arbitrary and , there is an edge labelled from to if and only if there is an edge labelled from to . If then we use that is an embedding. Similarly, if then we use that is an embedding. If and then there is no edge , because there are no edges from to , nor is there an edge , because there are no edges from to . It remains to consider the case when and .

First assume there is an edge from to labelled . Then , since is future closed. Hence , by the inductive hypothesis, and the edge is labelled in , since is an embedding.

Conversely, assume there is an edge from to labelled . Then is in the future of in . Since the future of in is isomorphic to the future of in the representation , we see firstly that is the unique vertex of being the end of an -labelled edge starting at . Secondly, in there is a unique -labelled edge, say, starting at . Then , since and are future closed, and , since is an embedding. By injectivity of , we have and so , as desired. This completes the proof that is an embedding.

The remaining vertices we need to extend to are partitioned into copies of the present of various -equivalence classes of depth . Fix one copy in of one of these equivalence classes. Then and (being an embedding of into ) together specify an allowable choice of edges from to . Since every allowable choice has been replicated times during each step of the construction of , this provides not just one but possible ways to extend to and to the edges starting in . The number of vertices starting at in is bounded by the number of elements of . So , and therefore , certainly contain no more than copies of the present of any -equivalence class of depth (including any copies in ). Hence there exists a way to extend to all these copies simultaneously. ∎

It remains to show that is a representation of , so we need to show that satisfies the conditions of Lemma 4.1. It is easy to see that the relations are functions. From the fact that the future of any vertex of is isomorphic to the future of some vertex in a representation of , it follows that there is at most one edge starting at on which any given holds.

Next we need to show that the operations are represented correctly by . With the exception of range, all the operations are straightforward and similar to show. We again rely on the fact that for any vertex in , the future of is isomorphic to the future of some vertex in a representation of .

To see that composition is represented correctly, suppose first that holds on and holds on . Then as the future of matches the future of some vertex in a representation, holds on . Conversely, suppose that holds on . Then again by matching with a vertex in a representation, we know there is a such that holds on and holds on .

To see that domain is represented correctly, suppose first that holds on . Then by matching with a vertex in a representation, we know both that and that there is an edge starting at on which holds. Conversely, suppose that holds on an edge . Then by matching , we see that must hold on .

If is in the signature, then no edge in any representation of is labelled with . Hence does not hold on any edge in and so represents correctly. If is in the signature then in any representation, holds on all reflexive edges and no others. Hence the same is true of and so represents correctly.

To see that antidomain is represented correctly if it is in the signature, suppose first that holds on . Then by matching with a vertex in a representation, we know both that and that there is no edge starting at on which holds. Conversely, suppose there is no edge starting at on which holds. Then by matching , we see that must hold on .

To see that fixset is represented correctly if it is in the signature, suppose first that holds on . Then by matching with a vertex in a representation, we know both that and that holds on . Conversely, suppose holds on . Then by matching , we see that must hold on .

To see that preferential union is represented correctly if it is in the signature, suppose first that holds on . Then by matching with a vertex in a representation, we know that on either holds, or does not hold and does. Conversely, suppose that on either holds, or does not hold and does. Then by matching , we see that must hold on .

To see that maximum iterate is represented correctly if it is in the signature, suppose first that holds on . Then by matching with a vertex in a representation, we know there exist such that holds on each and there is no edge starting at on which holds. Conversely, suppose there exist such that holds on each and there is no edge starting at on which holds. Then by matching , we see that must hold on .

One direction of range being represented correctly is clear: if has an edge from to on which holds, then will hold on the reflexive edge at . For the other direction, suppose that holds on a vertex in and let be the label of . Then we know that we can find a -vertex, say, in a representation and that will hold on . So there is an edge in this representation on which holds. Since the future of is isomorphic to the future of via an isomorphism sending to , there is an embedding of the future of into sending to . Then Lemma 4.4 ensures we can embed the future of into in such a way that is mapped to and so there is an -labelled edge ending at .

For the condition that be faithful, consider any with . Then as is representable, there certainly exists some realisable having an edge in its future on which holds, but does not. Since the futures of -vertices in are isomorphic to the futures of -vertices in representations, it suffices to show that for every -equivalence class , a nonzero number of copies of the present of are added at the appropriate stage of the construction. But this is obvious, by Lemma 4.4. ∎

With a little more work we can expand the list of operations to include opposite.

###### Theorem 4.5.

The finite representation property holds for representation by partial functions for any signature with .

###### Proof.

Let be a finite representable algebra of one of the specified signatures. We may assume that has more than one element, as the one-element algebra is representable using the empty set as a base. We will argue that if the signature contains opposite, then the network described in the proof of Theorem 4.3 represents opposite correctly.

Suppose first that holds on . We want to show that is the unique edge ending at on which holds. As the future of matches the future of some vertex in a representation, we know that holds on . To show that is the unique such edge, suppose holds on . Then , and are all in the future of . So in the future of we have: holding on and holding on and . As the future of matches the future of some vertex in a representation, it follows that , as required.

Conversely, suppose that is the unique edge ending at on which holds. We want to show that holds on . Let be the label of the reflexive edge at , let be the label of the reflexive edge at and let be the label of . First note that if were in a deeper -equivalence class than , then, because of the way is constructed, there would be at least edges ending at on which holds. Hence and are in the same -equivalence class.

Now the present of is isomorphic to the present of any -vertex in any representation of . So it suffices to show that for an -vertex in a representation of , if is the -labelled edge from to a -vertex, then holds on . Being situated in a representation, we can show this by proving that is the unique edge ending at on which holds.

Suppose then that holds on and let be the label of the reflexive edge at . Suppose . We saw, in proving that range is represented correctly, that we can embed the future of into in such a way that is mapped to . So there is an edge starting at a -vertex and ending at on which holds. But this is a contradiction, as the edge , starting at an -vertex, is supposed to be the unique edge ending at on which holds. We conclude that and hence is in the present of , since and are in the same -equivalence class. We must now have , for otherwise the present of would feature two distinct edges ending at on which holds. We know this not to be the case, by comparison with , in the present of . Hence is the unique edge ending at on which holds, as required. ∎

Given some representation of an algebra , we could give an alternate definition of the realisable elements of as those appearing as edge labels in the particular representation , rather than just in any representation. Then our proofs of Theorem 4.3 and Theorem 4.5 would work equally well. However, with the definition we gave, the constructed representation is in a sense the richest possible, in that if it is possible for an element to appear as a label in a representation, then it appears as a label in the constructed representation.

It is clear that from the proof of Theorem 4.3 we can extract a bound on the size required for the base.

###### Proposition 4.6.

For any signature with every finite -algebra is representable over a base of size

 |A||A|O(|A|12).
###### Proof.

We may assume . Let and be as in the proofs of Theorem 4.3 and Theorem 4.5. Let be a -equivalence class of depth and let be a copy of the present of . An allowable choice from to is determined by the labelled edges from a single vertex of , since it follows from claim (1) in the proof of Proposition 3.5 that if is the label of an edge and is the label of an edge then is the label of the edge . There are at most labels, so at most allowable choices (unless is of depth , in which case there is a single allowable choice). When is constructed from , for each allowable choice, copies of are added, so vertices are added. The sum, over all -equivalence classes of depth , of the number of vertices in the present of each class, is at most . Hence at most vertices are added when is constructed from . We obtain

 |N0|≤|A|2,|Nn|≤|Nn−1|+|A|2|Nn−1||A| % for n≥1,

from which it is provable by induction that

 |Nn|≤|A|2(n+1)|A|n.

At least distinct elements of are required in order for there to be a -equivalence class of depth , since composition is in the signature. So the construction of is completed by a depth that is . Hence

 |N|=|A|2(O(|A|12)+1)|A|O(|A|12)=|A||A|O(|A|12).\qed

For comparison, note that in [1], whenever a signature is shown to have the finite representation property, a bound on the size required for the base is derived that has either polynomial or exponential asymptotic growth.

We mentioned in the introduction that proving the finite representation property can help show that representability of finite algebras is decidable. The most direct way this can happen is by finding a (computable) bound on the size required for a representation. Then the representability of a finite algebra can be decided by searching for an isomorph amongst the concrete algebras with bases no larger than the bound.

For most of the signatures that we have considered, decidability has already been established, because finite equational or quasiequational axiomatisations of the representation classes (or at least the finite representable algebras) are known. However this is not the case for some of our signatures. Specifically, the antidomain-free expansions of by and/or and also any of the signatures containing opposite. So it is worth stating the following corollary of Proposition 4.6.

###### Corollary 4.7.

Representability of finite algebras by partial functions is decidable for any signature with .

## 5. Entirely Algebraic Constructions

Most of the construction detailed in Section 4 can be carried out based only on direct inspection of the algebra under consideration. However we noted that the construction does depend in one respect on information contained in representations of the algebra: the representations determine which are the realisable domain elements. We also noted that our construction works equally well if our realisable elements are those appearing as edge labels in one particular representation. So if we were to give an algebraic characterisation of the elements appearing as reflexive-edge labels in a particular representation, we would have a method of constructing a representation on a finite base using only algebraic properties of the algebra. Giving such characterisations, for certain signatures, is precisely what we do in this section.

We first mention the signature and expansions of this signature by operations in . The representation that Jackson and Stokes give in [3] for these signatures uses for the base of the representation Schein’s ‘permissible sequences’, as originally described in [8]. A permissible sequence, is a sequence with and for each (and cannot participate in a sequence if it is in the signature). There is an edge on which holds, starting at such a sequence, if and only if . Hence for Schein’s representation we can identify the elements labelling reflexive edges quite easily: they are those of the form , for some (excluding if it is in the signature).

Now we examine the signature . An arbitrary representable -algebra, , has a least element, , given by for any and any representation of must represent by the empty set. We can define and in any representation this must be represented by the domain operation.

The down-set of any element forms a Boolean algebra using the meet operation of and with complementation given by . Any representation of by partial functions restricts to a representation of each as a field of sets over . From this we see that turns any finite joins in into unions.

###### Definition 5.1.

Let be a poset with a least element, . An atom of is a minimal nonzero element of . We say that is atomic if every nonzero element is greater than or equal to an atom.

A finite representable -algebra, , is necessarily atomic. Any can be expressed as a finite join of atoms of since, given , we can split as .

From the preceeding discussion, we see that in any representation of a -algebra, for any edge a finite sum of atoms holds, so at least one of the atoms holds, as finite joins are represented by unions. We know that at most one atom holds, since the meet of two distinct atoms is , which can never hold on an edge. Hence a unique atom holds on each edge and necessarily labels the edge. In every representation every atom must appear as a label, otherwise it is not separated from . We conclude that in any representation the elements appearing as edge labels are precisely the atoms and so the elements labelling reflexive edges are precisely the atomic domain elements. Hence for the signature the realisable domain elements are the atomic domain elements. This also applies to any expansion of this signature by operations we have mentioned.

The purpose of the next example is simply to illustrate that, unlike Boolean algebras for example, the set of atoms in a finite representable -algebra can be almost as large as the algebra itself. Hence applying the knowledge that the number of labels is at most the number of atoms to the calculation in Proposition 4.6, does not improve the bound.

###### Example 5.2.

Let be any finite group. We can make into an algebra of the signature by using the group operation for composition (and for all ) and defining unless , every antidomain of a nonzero element to be (and , the group identity) and every range of a nonzero element to be (and ). Then every nonzero element of is an atom. Augmenting the Cayley representation of (the representation ) by setting demonstrates that is representable.

## 6. Failure of the Finite Representation Property

Finally, one might reasonably wonder if it is possible for the finite representation property not to hold for algebras of partial functions. After all, for every signature for which it has been settled, the finite representation property has been shown to hold. We finish with a simple example showing that we can indeed force a finite representable algebra of partial factions to fail to have representations over finite bases.

###### Example 6.1.

Let be the unary operation on partial functions given by

 U(f)={(y,y)∈X2∣∃!x∈X:(x,y)∈f}.

Let be the algebra of partial functions, of the signature and with base , containing the following five elements.

• , the empty function,

• , the identity function on ,

• , the identity function on ,

• , the function with domain and range sending each to ,

• , a function with domain and range such that each has precisely two -preimages: the least two elements of that are neither the -preimage nor -preimages of for . See Figure 1.

Since is an algebra of partial functions, it is certainly representable by partial functions. It is easy to see that cannot be represented over a finite base. Indeed, , so in any representation is a bijection from its domain, the -vertices, to its range, the -vertices. On the other hand, so maps the -vertices onto the -vertices, but not injectively. Hence these sets of vertices cannot have finite cardinality.

By including the operation in less expressive signatures, it is possible to give slightly simpler examples than Example 6.1. However we chose an expansion of the signature in order to contrast with the other expansions that are the subject of this paper, for which we have seen that the finite representation property does hold.

Note that our example allows us to observe the finite representation property behaving non monotonically as a function of expressivity. Indeed is expressible in terms of domain and opposite, , and so we have

 {;,∧,D,R}⊂{;,∧,D,R,U}⊂{;,∧,D,R,−1,U}

with the finite representation property holding for the outer two signatures, but failing in the middle.

### Footnotes

1. This property has also been called the finite algebra on finite base property.

### References

1. Robin Hirsch, Marcel Jackson, and Szabolcs Mikulás, The algebra of functions with antidomain and range, preprint: arXiv:1410.3981 (2014).
2. Marcel Jackson and Tim Stokes, An invitation to C-semigroups, Semigroup Forum 62 (2001), no. 2, 279–310.
3. Marcel Jackson and Tim Stokes, Partial maps with domain and range: extending Schein’s representation, Communications in Algebra 37 (2009), no. 8, 2845–2870.
4. Marcel Jackson and Tim Stokes, Modal restriction semigroups: towards an algebra of functions, International Journal of Algebra and Computation 21 (2011), no. 7, 1053–1095.
5. Marcel Jackson and Tim Stokes, Agreeable semigroups, Journal of Algebra 266 (2003), no. 2, 393 – 417.
6. Brett McLean, The finite representation property for composition, intersection, antidomain and range, preprint: arXiv:1503.02627v1 (2015).
7. Karl Menger, An axiomatic theory of functions and fluents, The Axiomatic Method (A. Heyting L. E. J. Brouwer, E. W. Beth, ed.), North Holland, 1959, pp. 454–472.
8. Boris M. Schein, Restrictively multiplicative algebras of transformations, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika 95 (1970), no. 4, 91–102 (Russian).
9. B. Schweizer and A. Sklar, The algebra of functions, Mathematische Annalen 139 (1960), no. 5, 366–382.
10. B. Schweizer and A. Sklar, The algebra of functions. III, Mathematische Annalen 161 (1965), no. 3, 171–196.
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