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 functions to fail to have representations over finite bases.
Example 5.6.1. Let U be the unary operation on partial functions given by U(f ) = {(y, y) ∈ X2| ∃!x ∈ X : (x, y) ∈ f }.
Let F2be the algebra of partial functions, of the signature {;, ·, D, R, U} and with base N × 2, consisting
of the following five elements. • 0, the empty function,
• r, the identity function on N × {1},
• f , the function with domain d and range r sending each (n, 0) to (n, 1),
• g, a function with domain d and range r such that each (n, 1) ∈ N × {1} has precisely two g-preimages: the least two elements of N × {0} that are neither the f -preimage (n, 0) nor g- preimages of (m, 1) for m < n. See Figure 5.3.
0 0 1 1 2 2 3 3 4 4 5 5
Figure 5.3: The algebra F2. Dashed lines for f , solid lines for g
Since F2isan algebra of partial functions, it is certainly representable by partial functions. It is easy
to see that F2cannot be represented over a finite base. Indeed, R(f ) = U(f ), so in any representation f
is a bijection from its domain, the d-vertices, to its range, the r-vertices. On the other hand, R(g) 6= U(g) so g maps the d-vertices onto the r-vertices but not injectively. Hence these sets of vertices cannot have finite cardinality.
By including the operation U in less expressive signatures, it is possible to give slightly simpler examples than Example 5.6.1. However, we chose a supersignature of the signature {;, ·, D, R} in order to contrast with the other supersignatures that are the subject of this chapter, 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 mono- tonically as a function of expressivity. Indeed U is expressible in terms of domain and opposite, U(f ) = D(f−1), and so we have
{;, ·, D, R} ⊂ {;, ·, D, R, U} ⊂ {;, ·, D, R,−1, U}
Algebras of multiplace functions for signatures
containing antidomain
The work in this chapter has been published as: Brett McLean, Algebras of multiplace functions for signatures containing antidomain, Algebra Universalis 78 (2017), no. 2, 215–248.1
ABSTRACT. We define antidomain operations for algebras of multiplace partial functions. For all sig- natures 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 repres- entation property holds for the representation class. We show that for a large set of signatures, the representation classes have equational theories that are coNP-complete.
6.1
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 set-theoretically-defined operations possible on such functions, for example function com- position or set intersection. Finally, study the isomorphism class of algebras that consist of some such functions together with the specified set-theoretic operations. We have discussed the basic case—unary functions—extensively in previous chapters, particularly in Section 3.2.1.
The study of algebras of so-called multiplace functions started with Menger [76]. Here the objects in the concrete algebras are (usually partial) functions from Xn to X for some fixed X and n. Since then, representation theorems—axiomatisations of isomorphism classes via explicit representations— have been given for various cases [22, 107, 106, 23, 24].
For unary functions, the antidomain operation yields the identity function restricted to the comple- ment of a function’s domain. In the setting of partial functions, this operation seems first to have been described in [54], where it is referred to as domain complement.2 Some recent work has been direc-
1Reproduced with permission of Springer.
2Though for an earlier appearance in the setting of binary relations, see [51], where Hollenberg calls the operation dynamic
ted towards providing representation theorems in the case of unary functions for signatures including antidomain [57, 44].
In this chapter we define, for n-ary multiplace functions, n indexed antidomain operations by simul- taneous 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 6.2.
The majority of this chapter, Sections 6.3–6.8, consists of representation theorems for multiplace functions for signatures containing composition and the antidomain operations. Much of this is a straightforward translation of [57], where the same is done for unary functions.
In Sections 6.3 and 6.4 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 6.5 we use a single quasiequation to extend the axiomatisation of Section 6.3 to a finite quasiequational axiomatisation for the case of injective multiplace partial functions.
In Section 6.6 we add intersection to our signature and for both partial multiplace functions and in- jective partial multiplace functions are able to give finite equational axiomatisations of the representation class.
In Sections 6.7 and 6.8 we consider all our previous representation questions with the preferen- tial 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 6.9 we switch our focus to equational theories. We prove that for any signature contain- ing operations that we mention, the equational theory of the representation class of multiplace partial functions lies in coNP. If the signature contains the antidomain operations and either composition or intersection then the equational theory is coNP-complete.