Algebras of Multiplace Functions for Signatures Containing Antidomain
Abstract.
We define antidomain operations for algebras of multiplace partial functions. For all signatures containing composition, the antidomain operations and any subset of intersection, preferential union and fixset, we give finite equational or quasiequational axiomatisations for the representation class. We do the same for the question of representability by injective multiplace partial functions. For all our representation theorems, it is an immediate corollary of our proof that the finite representation property holds for the representation class. We show that for a large set of signatures, the representation classes have equational theories that are coNPcomplete.
1. Introduction
The scheme for investigating the abstract algebraic properties of functions takes the following form. First choose some sort of functions of interest, for example partial functions or injective functions. Second, specify some settheoreticallydefined operations possible on such functions, for example function composition or set intersection. Finally, study the isomorphism class of algebras that consist of some such functions together with the specified settheoretic operations.
The study of algebras of socalled multiplace functions started with Menger [7]. Here the objects in the concrete algebras are (usually partial) functions from to for some fixed and . Since then, representation theorems—axiomatisations of isomorphism classes via explicit representations—have been given for various cases [1, 10, 9, 2].
For unary functions, the antidomain operation yields the identity function restricted to the complement of a function’s domain. This operation seems first to have been described in [5], where it is referred to as domain complement. Some recent work has been directed towards providing representation theorems in the case of unary functions for signatures including antidomain [6, 4].
In this paper we define, for ary multiplace functions, indexed antidomain operations by simultaneous analogy with the indexed domain operations studied on multiplace functions and the antidomain operation studied on unary functions. This definition together with other fundamental definitions we need comprise Section 2.
The majority of this paper, Sections 3–8, consists of representation theorems for multiplace functions for signatures containing composition and the antidomain operations. Much of this is a straightforward translation of [6], where the same is done for unary functions.
In Sections 3 and 4 we work over the signature containing composition and the antidomain operations. We show that for multiplace partial functions the representation class cannot form a variety and we state and prove the correctness of a finite quasiequational axiomatisation of the class. It follows, as it does for our later representation theorems, that the representation class has the finite representation property.
In Section 5 we use a single quasiequation to extend the axiomatisation of Section 3 to a finite quasiequational axiomatisation for the case of injective multiplace partial functions.
In Section 6 we add intersection to our signature and for both partial multiplace functions and injective partial multiplace functions are able to give finite equational axiomatisations of the representation class.
In Sections 7 and 8 we consider all our previous representation questions with the preferential union and fixset operations, respectively, added to the signature. In all cases we give either finite equational or finite quasiequational axiomatisations of the representation class.
In Section 9 we switch our focus to equational theories. We prove that for any signature containing operations that we mention, the equational theory of the representation class of multiplace partial functions lies in . If the signature contains the antidomain operations and either composition or intersection then the equational theory is complete.
2. Algebras of Multiplace Functions
In this section we give the fundamental definitions of algebras of multiplace functions and of the various operations that may be included.
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. We use to denote an arbitrary nonzero natural number. A bold symbol, say, is either simply shorthand for in a term of the form or denotes an actual tuple . We may abuse notation, when convenient, by writing for the tuple . If are unary operation symbols, the notation is shorthand for . When a function acts on an tuple we omit the angle brackets and write . If is an index, then ‘for all ’ or ‘for every ’ means for all .
First we make clear what we mean by a multiplace function.
Definition 2.1.
An ary relation is a subset of a set of the form . Without loss of generality we may assume all the ’s are equal. In the context of a given value of , a multiplace partial function is an ary relation validating
(1) 
We may also use the terminology ary partial function for the same concept. We import all the usual terminology for partial functions, for instance if then we may write , say ‘ is defined’, and so on.
Henceforth, we will use the epithet ‘ary’ in favour of ‘multiplace’ in order to make the arity of the functions in question explicit.
Definition 2.2.
Let be an algebraic signature whose symbols are a subset of , where we write, for example, to indicate that for some fixed . An algebra of ary partial functions of the signature is an algebra, , of the signature whose elements are ary partial functions and that has the following properties.

There is a set , the base, and an equivalence relation on with the following property. For all and all , we have that for every . That is, every partial function in the algebra contains only tuples of equivalent members of .

The operations are given by the settheoretic operations on partial functions described in the following.
In an algebra of ary partial functions

the ary operation is composition, given by:

the binary operation is intersection:

the constant is the nowheredefined function:

for each the constant is the th projection on the set of all tuples of equivalent points:

for each the unary operation is the operation of taking the th projection restricted to the domain of a function:

for each , the unary operation is the operation of taking the th projection restricted to the antidomain of a function—those tuples of equivalent points where the function is not defined:

for each , the unary operation , the th fixset operation, is the th projection function intersected with the function itself:

for each , the binary operation , the th tie operation, is the th projection function restricted to those equivalent tuples where the two arguments do not disagree, that is, either neither is defined or they are both defined and are equal:

the binary operation is preferential union:
If the equivalence relation is the universal relation, , then we say that the algebra is square.
Definition 2.3.
Let be an algebra of one of the signatures specified by Definition 2.2. A representation of by ary partial functions is an isomorphism from to an algebra of ary partial functions of the same signature. If has a representation then we say it is representable.
As we have signified, in this paper the focus is on isomorphs of algebras of ary partial functions in general, rather than the square ones in particular. However, now is an opportune moment for a brief discussion of the merits of each of these concepts and the relationship between them.
The square algebras of ary functions have the advantage of being the simpler and more natural concept. However for certain signatures they are not as algebraically well behaved, failing to be closed under direct products. Indeed there are simple examples of pairs of algebras that are each representable as square algebras of functions but whose product is not. The presence of the antidomain operations in the signature will always cause this problem, as the example we now give demonstrates.
Example 2.4.
Assume and work over any one of the signatures specified by Definition 2.2 containing the indexed antidomain operations . Consider the twoelement algebra consisting of both of the ary partial functions on some base of size one. As is a square algebra of partial functions it is trivially representable as a square algebra of partial functions. We argue that is not representable as a square algebra of functions.
Suppose, for contradiction, that is a square representation of with base . Since , we know must contain at least two distinct points, in order that distinguishes all the elements of . Let be any tuple from not lying on the diagonal. Denote the two elements of by and . Then and for every . So , and hence the domains of the partial functions and must partition . Without loss of generality we may assume is not in the domain of . But then for every . As every is the same function, namely , all components of are equal, contradicting the assumption that is not on the diagonal. We conclude that cannot be represented as a square algebra of partial functions.
An immediate consequence of not being closed under direct products is that the class of algebras having a square representation cannot be a quasivariety. We note however that these classes always possess universal axiomatisations in firstorder logic, for any of the signatures covered by Definition 2.2. This can be seen by appealing to Schein’s fundamental theorem of relation algebra [8]. There are two conditions of Schein’s theorem that need to be checked. The first is that ary partial functions can be defined as those ary relations satisfying a recursive set of sentences in the firstorder language with equality and a countable supply of ary relation symbols, which is precisely what we did in Definition 2.1 by using (1). The second is that, using the same firstorder language, the operations we are considering can each be defined using a formula with free variables. Definitions of the operations for square algebras can be formed from the more general definitions we gave in Definition 2.2 by removing any stipulations of equivalence. The resulting definitions are of the required form.
The purpose of relativising operations to in Definition 2.2 is to ensure that the class of algebras representable by ary partial functions is closed under direct products. A direct product of representable algebras can be represented using a ‘disjoint union’ of representations of the factors.
Definition 2.5.
Let be a family of algebras all of the same signature and be a corresponding family of homomorphisms to algebras of ary partial functions, with having base and equivalence relation on .
A disjoint union of is any homomorphism out of formed by the following process. First rename the elements of the ’s in such a way that the ’s are pairwise disjoint. Then the codomain of will be an algebra consisting of all ary partial functions of the form for some element . The base of will be and the equivalence relation on will be . The operations on will be given by the concrete operations described in Definition 2.2. Define for each element of . The map is straightforwardly a homomorphism.
A disjoint union of injective homomorphisms will be injective and that is why we remarked that a product of representable algebras can be represented by a disjoint union of representations of the factors. If our definition of algebras of ary partial functions were restricted to square algebras only, then we could not guarantee that a disjoint union of representations would be a representation, since the disjoint union of two universal equivalence relations is not universal.
Our final remark about square algebras of partial functions is that it is easily seen that every algebra representable by ary partial functions is a subalgebra of a product of algebras each having a square representation. Hence the general representation class is contained in the quasivariety generated by the square representation class.
For algebras of ary functions, the first representation theorem was provided by Dicker in [1], showing that the equation that has come to be known as the superassociativity law axiomatises the representation class (for total functions, although the equation is valid for partial functions) in the signature consisting only of composition. Trokhimenko gave equational axiomatisations for the signatures of composition and intersection, in [10], and composition and domain, in [9]. In [2], Dudek and Trokhimenko gave a finite equational axiomatisation for the signature of composition, intersection and domain.
The subject of this paper is signatures containing composition and antidomain. Note that and and are all definable using composition and antidomain, using , for any , and then and using (that is, a double application of ). Further, in the presence of composition and antidomain, the tie operations and intersection are interdefinable. The tie operations are definable as , where . Intersection is definable as . This leaves , and as the only interesting additional operations among those we have mentioned. When intersection is present, the fixset operations are definable as .
We include here, for ease of reference, a summary of the results about representation classes contained in this paper. All classes have finite axiomatisations of the relevant form.
Signature  Partial functions  Injective partial functions 

proper quasivariety  quasivariety  
variety  variety  
variety  quasivariety  
variety  variety  
quasivariety  quasivariety  
quasivariety  quasivariety 
Note that whenever a representation class has a finite quasiequational axiomatisation the decision problem of representability of finite algebras is solvable in polynomial time, and if we know such an axiomatisation then we know such an algorithm. We observed earlier that, for each signature, the representation class is contained in the quasivariety generated by the algebras having square representations. Hence another point to note is that our results identify the representation classes as equal to these generated quasivarieties.
Beyond representability, we may also be interested in representability on a finite base. Our final fundamental definition can be invoked in any circumstance where there is a notion of representability.
Definition 2.6.
The finite representation property holds if any finite representable algebra is representable on a finite base.
3. Composition and Antidomain
First we examine the signature consisting of composition and the antidomain operations. After presenting some equations and one quasiequation that are valid for algebras of ary partial functions, we deduce some consequences of these (quasi)equations that we use in Section 4 to prove that our (quasi)equations axiomatise the representation class.
In [6], Jackson and Stokes give a finite quasiequational axiomatisation of the representation class of unary partial functions for the signature of composition, antidomain.^{1}^{1}1Actually, their signature also contains the constants and , but these are definable from composition and antidomain. They call algebras satisfying these laws modal restriction semigroups.
Definition 3.1.
A modal restriction semigroup [6] is an algebra of the signature satisfying the equations
(the twisted law for antidomain)
and the quasiequation
where for any (and the third equation says this is a welldefined constant), and .
Note that the definition of modal restriction semigroups given by Jackson and Stokes states they should be monoids, so should also be a right identity. But this is a consequence of the equations we gave in Definition 3.1, for
using the twisted law for the second equality.
For ary functions, working over the signature , we can try to write down valid ary versions of the (quasi)equations appearing in Definition 3.1. This is easy in every case except that of the twisted law for antidomain, which needs more care.
This is a good point at which to note that we do not need to bracket expressions like since this can only mean . When we do write the brackets, we do so only for emphasis.
Proposition 3.2.
The following equations and quasiequations are valid for the class of algebras representable by ary partial functions.
(2)  
(3)  
(4)  
(5)  
(6)  
(7)  
(the twisted laws for antidomain) 
(8) 
where for any (and (4) says this is a welldefined constant), and (a double application of ).
Proof.
We noted in the previous section that every algebra representable by ary partial functions is isomorphic to a subalgebra of a product of algebras having a square representation. As the validity of quasiequations is preserved by taking products and subalgebras, it suffices to prove validity only for algebras having square representations. Further, since representations are themselves isomorphisms, it is sufficient to prove validity for an arbitrary square algebra of ary partial functions. So suppose we have such an algebra, with base .
The validity of the superassociative law has been recognised since Menger noted it in [7]. We turn next to (4). Given an ary partial function , if is to be defined at an tuple then there should be a with for each and with defined at . Since each is a restriction of the th projection, can only be . But if is defined at then cannot be. Hence is the nowheredefined function. So is well defined, that is, the value of does not depend on the choice of , and so (4) is valid. The validity of (5) and the validity of (6) are now both clear.
Now is the th projection restricted to those tuples in where is not defined. So is, as the notation indicates, the th projection on the set of all tuples in . The validity of (3) is now clear.
For the twisted laws for antidomain, first suppose that is defined at . Then we know that are all defined at and that is not defined at . Hence is defined at for every and is not defined at . It follows that is defined at for every . It is now apparent that is defined at with value —the same value as .
If is not defined at an tuple , then this is either because is undefined at for some or all are defined at , but is not defined at . If is undefined at then it is clear that cannot be defined at . In the second case, must be defined at and so is defined at . Again it is clear that cannot be defined at .
For (8), suppose the antecedent of the implication is true. Let be an tuple in . If is defined on then is defined at for each and accordingly says that either or both and are undefined at . If is undefined at then is defined at for each and this time says that either or both and are undefined at . ∎
Note that the naive ary versions of the twisted law for antidomain, namely , for every , are not valid (except in the unary case). Indeed if at an tuple, is defined, but is undefined for some different to , then is undefined, but will be defined.
To compensate for the complication with the twisted laws, we introduce as an axiom the equation
(9) 
whose validity is clear and has been noted before; for example it appears as Equation (10) in [2].
In addition we will need one extra indexed set of equations (trivial in the unary case) namely
(10) 
whose validity we now prove.
Proposition 3.3.
The indexed equations of (10) are valid for the class of algebras representable by ary partial functions.
Proof.
As before it is sufficient to prove validity for an arbitrary square algebra of ary partial functions. So suppose we have such an algebra, with base .
Suppose that is defined on an tuple , necessarily with value . Then is not defined on . Hence is defined on . It follows that is not defined on and from there we deduce that is defined on , necessarily with value . Hence the function is a restriction of . By symmetry the reverse is true and the two functions are equal. ∎
We are going to prove that (2)–(10) axiomatise the class of algebras that are representable by ary partial functions and hence the representation class is a quasivariety. But before we do that, we show that the representation class is not a variety.
Proposition 3.4.
The class of algebras that are representable by ary partial functions is not closed under quotients and hence is not a variety.
Proof.
We adapt an example given in [6] to describe an algebra of ary partial functions having a quotient that does not validate (8) and so is not representable by partial functions.
We describe an algebra of ary partial functions, with base . The equivalence relation to which the antidomain operations are relativised partitions the base into and . The elements of are the following elements.

the empty function

the th projection on , for each

the function with domain that is constantly

the function with domain that is constantly

each of the aforementioned functions with the pair adjoined
It is clear that is closed under the antidomain operations. Checking that is closed under composition is also straightforward.
It is easy to check, directly, that identifying all the elements with domain produces a quotient of . Let be any element with domain , let be the element sending to and constantly elsewhere and let be the element sending to and constantly elsewhere. Then in the quotient
and  
but and are not equal. Hence (8) is refuted in the quotient. ∎
Next comes the work of deducing the various consequences of (2)–(10) that are needed to prove their sufficiency for representability.
We noted earlier that the equation is not valid, but we can obtain a version in the special case that is of the form for some .
Proof.
We will give (11) the full title: the restricted twisted laws for antidomain, but since these are the twisted laws we apply most frequently, when we refer simply to ‘the th twisted law’ we will mean the indexed version of (11).
In the following lemma and in later proofs an ‘s’ above an equality sign indicates an appeal to superassociativity, a ‘t’ an appeal to the twisted laws and any number an appeal to the corresponding equation.
Proof.
Before proceeding with (13)–(15), we note the following useful consequences of (2)–(10). By (10) then (4) we see that
(16) 
and by first applying superassociativity and then (16) to we obtain
(17) 
We will use (8) to prove (13). Firstly
and  
so we see that and coincide. We also have
and  
and so and coincide, completing the proof of (13).
Equation 14 is a simple, but useful, consequence of (13). We have
by superassociativity  
by (13)  
by superassociativity 
as required.
For (18) we have
by (13)  
by (10)  
and in the same way  
by (13)  
and we have  
by (17)  
by the definition of  
by (3)  
and similarly  
and so from an application of (8) we deduce the required equation.
Equation 19 can be deduced with two applications of (8), composing on the left with and and with and . One can show that any of the compositions with or evaluate to 0, for example
by (14)  
by the definition of  
by the th twisted law  
by the th twisted law  
by (17)  
by the definition of  
by (3)  
by (4)  
by (6) 
and the others are similar. The compositions with and both equal . Observe