In this final section, we use the conditions for complete representability that we have uncovered to obtain a finite first-order axiomatisation of the complete-representation class.
Definition 4.6.1. A poset P is atomistic if its atoms are join dense in P. That is to say that every element of P is the join of the atoms less than or equal to it.
Clearly any atomistic poset is atomic. For {;, ·, A}-algebras representable by partial functions, the converse is also true.
Lemma 4.6.2. Let A be an algebra of the signature {;, ·, A} that is representable by partial functions. If A is atomic, then it is atomistic.
Proof. Suppose A is atomic and let a ∈ A. By Lemma 4.2.6, the algebra ↓ a is a Boolean algebra and clearly it is atomic. It is well known that atomic Boolean algebras are atomistic. So we have
a =X ↓ a {x ∈ At(↓ a) | x ≤ a} =X A {x ∈ At(↓ a) | x ≤ a} =X A {x ∈ At(A) | x ≤ a}.
The second equality holds because any upper bound c ∈ A for {x ∈ At(↓ a) | x ≤ a} is above an upper bound in ↓ a, for example c · a. Hence the least upper bound in ↓ a is least in A also.
Lemma 4.6.3. Let A be an algebra of the signature {;, ·, A} that is representable by partial functions and atomic. Letϕ be the first-order sentence asserting that for any a, b, c, if c ≥ a ; x for all atoms x less than or equal tob, then c ≥ a ; b. Then composition is completely left-distributive over joins if and only if A|= ϕ.
Proof. Suppose first that composition is completely left-distributive over joins. As A is atomic it is atomistic. So for any a, b ∈ A we have
a ; b = a ;X{x ∈ At(A) | x ≤ b} =X({a} ; {x ∈ At(A) | x ≤ b}) and so ϕ holds.
Now suppose that A |= ϕ. Let a ∈ A and let S be a subset of A such thatP S exists. Then certainly a ;P S is an upper bound for {a} ; S. To show it is the least upper bound, let c be an arbitrary upper bound for {a} ; S. Then
for all s ∈ S c ≥ a ; s
=⇒ for all s ∈ S and x ∈ At(↓XS) with x ≤ s c ≥ a ; x
=⇒ for all x ∈ At(↓XS) c ≥ a ; x
=⇒ for all x ∈ At(A) with x ≤XS c ≥ a ; x
=⇒ c ≥ a ;XS.
The third line follows from the second because x ∈ At(↓P S) implies x ≤ s for some s ∈ S. To see this, consider the Boolean algebra ↓P S. When x is an atom, x s if and only if x · s = 0, which is equivalent to x ≥ s. So if x s for all s ∈ S then x ≥P S, forcing x to be zero—a contradiction. The fifth line can be seen to follow from the fourth by first writingP S as the join of the atoms below it and then using ϕ.
We now have everything we need to prove our main result.
Theorem 4.6.4. The class of {;, ·, A}-algebras that are completely representable by partial functions is a basic elementary class.
Proof. By Corollary 4.3.5, Lemma 4.4.5 and Proposition 4.5.1, an algebra of the signature {;, ·, A} is completely representable by partial functions if and only if it is representable by partial functions, atomic, and composition is completely left-distributive over joins. By Theorem 4.2.3, the property of being representable by partial functions is characterised by a finite set of first-order sentences. The property of being atomic is easily written as a first-order sentence. By Lemma 4.6.3, in the presence of the axioms for the first two properties, the property that composition is completely left-distributive over joins can be written as a first-order sentence.
We immediately obtain the following corollary (by Theorem 2.4.9).
Corollary 4.6.5. The problem of determining whether an algebra of the signature {;, ·, A} is completely representable by partial functions is decidable in polynomial time (as a function of|A|).
Any attempt at writing down our axioms will readily reveal that each can be expressed in a universal- existential-universal form. We know from Proposition 4.3.7 that no existential-universal-existential axio- matisation is possible, hence we have determined the precise amount of quantifier alternation necessary to axiomatise the class.
Note that if range had been included in our signature then the function θ in Proposition 4.5.1 would not be a representation, as it would not represent range correctly. Figure 4.2 shows how this can happen. The atom f satisfies f ; R(g) = f and so (f, f ) ∈ θ(R(g)), but there is no h such that h ; g = f and so (f, f ) 6∈ R(θ(g)). Hence questions about the axiomatisability of the complete representation class for the signature {;, ·, A, R} remain open. Equally for the less expressive signature {;, ·, D}, where the meet-complete and join-complete representations do not coincide.
f
g
The finite representation property for
composition, intersection, domain, and range
The work in this chapter has been published as: Brett McLean and Szabolcs Mikul´as, The finite repres- entation property for composition, intersection, domain and range, International Journal of Algebra and Computation 26 (2016), no. 5, 1199–1216.1
ABSTRACT. We prove that the finite representation property holds for representation by partial func- tions for the signature consisting of composition, intersection, domain, and range and for any expan- sion 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.
5.1
Introduction
The investigation of the abstract algebraic properties of partial functions involves studying the isomorph- ism 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.
As we have indicated in previous chapters, 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 [92, 52, 53, 55, 57, 44]; we detailed this earlier, in Section 3.2.1.
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.
1DOI: 10.1142/S0218196716500508. Copyright World Scientific Publishing Company.
Recently, Hirsch, Jackson, and Mikul´as established the finite representation property for many sig- natures, but they leave the case for signatures containing the intersection, domain, and range operations together open [44].
In this chapter we prove the finite representation property for the most significant group of outstand- ing 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 [71]. 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.
5.2
Algebras of partial functions
In this section we give the fundamental definitions that are needed in order to state the results contained in this chapter.
Given an algebra A, when we write a ∈ A or say that a is an element of A, we mean that a is an element of the domain of A. We follow the convention that algebras are always nonempty.
Definition 5.2.1. Let σ be an algebraic signature whose symbols are a subset of {;, ·, D, R, 0, 1’, A, F, t,
↑,−1}. An algebra of partial functions of the signature σ is an algebra of the signature σ whose
elements are partial functions and with operations given by the set-theoretic operations on those partial functions described in the following.
Let X be the union of the domains and ranges of all the partial functions occurring in an algebra A. We call X the base of A. The interpretations of the operations in σ 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, (f ; g)(x) = g(f (x)),
• the binary operation · is intersection:
f · g = {(x, y) ∈ X2| (x, y) ∈ f and (x, y) ∈ g},
• the unary operation D 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 R 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 0 is the nowhere-defined empty function: 0 = ∅, • the constant 1’ is the identity function on X:
1’ = {(x, x) ∈ X2},
• the unary operation A is the operation of taking the diagonal of the antidomain of a func- tion—those points of X where the function is not defined:
A(f ) = {(x, x) ∈ X2|
∃y ∈ X : (x, y) ∈ f },
• the unary operation F 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 t is preferential union:
(f t g)(x) = f (x) if f (x) defined
g(x) if f (x) undefined, but g(x) defined undefined otherwise
• the unary operation↑is the maximum iterate:
f↑= [
n∈N
(fn; A(f )),
where f0:= 1’ and fn+1:= f ; fn,
• the unary operation−1is an operation we call opposite:
f−1= {(y, x) ∈ X2| (x, y) ∈ f and ((x0, y) ∈ f =⇒ x = x0)}.
The list of operations in Definition 5.2.1 does not exhaust those that have been considered for partial functions but does include the most commonly appearing operations.
Definition 5.2.2. Let A be an algebra of one of the signatures permitted by Definition 5.2.1. A repres- entation of A by partial functions is an isomorphism from A to an algebra of partial functions of the same signature. If A has a representation then we say it is representable.
In [55], Jackson and Stokes give a finite equational axiomatisation of the representation class for the signature {;, ·, D, R} and similarly for any expansion of this signature by operations in {0, 1’, F}.
In [44], Hirsch, Jackson, and Mikul´as give a finite equational axiomatisation of the representation class for the signature {;, ·, A, R} and similarly for any expansion of this signature by operations in {0, 1’, D, F, t}. 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 [77], 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 [95] and [96] 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 5.2.3. Let σ be a signature. We say that σ has the finite representation property (for representation by partial functions) if whenever a finite algebra of the signature σ is representable by partial functions, it is representable on a finite base.2
In [44], Hirsch, Jackson, and Mikul´as establish the finite representation property for many signa- tures that are subsets of {;, ·, D, R, 0, 1’, A, F, t,↑}. Assuming composition is in the signature, they prove the finite representation property holds for any such signature that cannot express 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 {;, ·, D, R}.
In this chapter we prove that {;, ·, D, R} and any expansion of {;, ·, D, R} by operations that we have mentioned (including opposite) all have the finite representation property. The following example may give some intuition about this problem and its solution.
Example 5.2.4. Let F1 be the algebra of partial functions, of the signature {;, ·, D, R} and with base
Z × 2, consisting of the following five elements. • 0, the empty function,
• d, the identity function on Z × {0}, • r, the identity function on Z × {1},
• f , the function with domain d and range r sending each (n, 0) to (n, 1), • g, the function with domain d and range r sending each (n, 0) to (n + 1, 1). See Figure 5.1. −2 −2 −1 −1 0 0 1 1 2 2 3
Figure 5.1: The algebra F1. Dashed lines for f , solid lines for g
The algebra F1, being an actual algebra of partial functions, is trivially representable. However, this
representation uses an infinite base. To try to reduce the infinite representation to a finite one, we may
observe that there are two types of base points: those mapped to themself by d (the points in Z × {0}), and those mapped to themself by r (the points in Z × {1}). We may then attempt to identify all base points of the same type. But doing this produces the structure pictured on the left of Figure 5.2. This does not yield a representation of F1, for the ‘representations’ of f and g are not disjoint, conflicting
with the fact that f · g = 0 in F1.
We cannot then necessarily construct a representation using only one copy of each type of base point. However, the structure pictured on the right of Figure 5.2 uses two copies of each type, and does yield a representation of F1. Hence, in this case, with enough copies, a finite representation can be
given. The proof of the main theorem of this chapter, Theorem 5.4.3, shows it is always possible to gather ‘enough’ copies of parts of infinite representations to be able to construct a finite representation, and describes that representation.
Figure 5.2: Left: a non-representation of F1. Right: a representation of F1