The Lattice of Alter egos

We introduce a new Galois connection for partial operations on a finite set, which induces a natural quasi-order on the collection of all partial algebras on this set. The quasi-order is compatible with the basic concepts of natural duality theory, and we use it to turn the set of all alter egos of a given finite algebra into a doubly algebraic lattice. The Galois connection provides a framework for us to develop further the theory of natural dualities for partial algebras. The development unifies several fundamental concepts from duality theory and reveals a new understanding of full dualities, particularly at the finite level. The Galois connection of clone theory (between operations and relations) provides a natural framework for comparing and studying the algebras on a fixed finite set. We start this paper by introducing a new Galois connection that we will use as a framework for comparing and studying the partial algebras on a fixed finite set. Our motivation for studying partial algebras comes from the theory of natural dualities. In natural duality theory, one typically starts with a fixed finite algebra M and considers alter egos of M, which are discretely topologised structures N∼ on the same set as M but whose type may include finitary relations and partial operations. The structure of an alter ego N∼ is required to be ‘compatible’ with the operations of M. The aim is to find an alter ego of the algebra M that can be used to build a category that is dually equivalent to the quasi-variety generated by M. A classic example is Priestley duality [23], which is based on the two-element bounded lattice M = 〈{0, 1};∨,∧, 0, 1〉 and the two-element chain with the discrete topology N∼ = 〈{0, 1};!, T〉. This pair yields a dual equivalence between bounded distributive lattices and Priestley spaces and, at the finite level, a dual equivalence between finite bounded distributive lattices and finite ordered sets. The new Galois connection induces a quasi-order on the collection of all alter egos of a finite algebra M, under which they form a doubly algebraic lattice AM. Our aim is to understand how the basic concepts of duality theory sit within this lattice. This aim leads us on a new path through the foundations of natural duality theory. In particular, we further develop the basic theory of natural dualities for partial algebras. Through this development, we hope to bring out the similarities between duality theory and clone theory. We highlight the key role of entailment in duality theory, and the link with entailment in clone theory. In order to describe our duality-theoretic results more precisely, we require some specialised terminology. These terms will be defined in Section 3, before they are needed in the main body of the paper. The way that duality and strong duality sit within the lattice of alter egos AM is essentially already known. For any finite algebra M, it is easy to use the basic theory to show that: – the alter egos that yield a finite-level duality form a principal filter of the lattice AM and, if any of these alter egos yield a duality, then they all do (4.1); 2000 Mathematics Subject Classification 08A05 (primary), 06A15, 08A55, 08C15 (secondary). The second author was supported by ARC Discovery Project Grant DP0556248 and the third author by a Discovery Grant from NSERC, Canada. 2 BRIAN A. DAVEY, JANE G. PITKETHLY AND ROSS WILLARD – the alter egos that yield a finite-level strong duality form the top element of the lattice AM (4.6), and so there is essentially only one candidate alter ego for strong duality. In contrast, our understanding of full dualities has previously been fairly sketchy. From our new perspective, we will find that full dualities become less mysterious, especially at the finite level. We can give a simple and natural intrinsic description of when an alter ego yields a finite-level full duality (4.3). However, our new perspective also highlights the differences between full duality and the better behaved concepts of duality and strong duality. In particular, the alter egos that yield a full duality do not necessarily form an increasing subset of AM. In other words, by enriching the structure of an alter ego of M, we can destroy a full duality. We can nevertheless completely characterise when enrichment will not destroy full duality (5.3), and thus show that: – the alter egos that yield a finite-level full duality form a complete sublattice FM of the lattice of alter egos AM, and those that yield a full duality form an increasing subset of FM (5.5). We do not know in general whether the alter egos that yield a full duality must form a filter of FM. We finish with two applications of our results to specific examples. First we consider the four-element quasi-primal algebra R of Clark, Davey and Willard [4], which provided the first known example of a full but not strong duality. We draw the lattice of all 17 full dualities based on R (5.7). We then give a short proof of a result due to Davey, Haviar, Niven and Perkal [8]: every finite non-boolean distributive lattice has a duality that is full but not strong at the finite level (5.10). Our development of duality theory for partial algebras comes from our desire to understand full dualities for algebras. We will find that the symmetry inherent in this approach provides additional insights. For example, we shall see the sense in which the conditions (ic) and (ftc) are dual to each other (4.5), and the sense in which the Structural Entailment and the Closure Theorems are the same (3.8, 3.9). Note. We do not want the development of our general theory to be bogged down by ‘trivial’ technicalities. So we work within a somewhat restricted setting. For example, when Priestley duality is recast in our setting, the one-element lattice is removed from the usual algebraic category and the empty Priestley space is removed from the usual topological category. While this is a very minor change, still we might prefer not to make it in practice. The setting that we use in this paper is chosen to smooth the development of the general theory, the interesting part of which is certainly at the non-trivial level. In an appendix we explain how any particular duality formulated in our setting can easily be reformulated in various alternative (but essentially equivalent) settings. 1. A Galois connection for partial operations Fix a finite non-empty set M . We shall define a partial operation on M to be a map f : r → M with an associated arity k " 1, where ∅ '= r ⊆ M. (In other words, we consider finitary, nonnullary partial operations with non-empty domains.) Similarly, we define a relation on M to be a set r with an associated arity k " 1, where ∅ '= r ⊆ M. Now let PM denote the set of all partial operations on M , and let RM denote the set of all relations on M . Note 1.1. We choose to exclude partial operations that are nullary or empty in order to smooth the development of the duality theory in later sections. See the appendix for a discussion of three alternative (but essentially equivalent) settings. THE LATTICE OF ALTER EGOS 3 ! a11 a12 · · · a1! " → g(r1) a21 a22 · · · a2! → g(r2) .. .. . . . .. .. $ ak1 ak2 · · · ak! % → g(rk) ↓ ↓ ↓ ⇓ f(c1) f(c2) · · · f(c!) ⇒ f g(r1), . . . , g(rk) = g f(c1), . . . , f(c!) Figure 1.1. Compatible partial operations f and g Definition 1.2. Let f and g be partial operations on M , with arities k, ! " 1. We say that f and g are compatible provided the following condition holds, for each k × ! matrix A of elements of M : – if each row ri of A is in dom(g) and each column cj of A is in dom(f), then ∗ ( g(r1), . . . , g(rk) ) ∈ dom(f), ∗ ( f(c1), . . . , f(c!) ) ∈ dom(g), and ∗ f ( g(r1), . . . , g(rk) ) = g ( f(c1), . . . , f(c!) ) ; see Figure 1.1. Compatibility of partial operations is symmetric. The following basic lemma gives a few alternative formulations of compatibility. Lemma 1.3. Let f and g be partial operations on a finite non-empty set M . Then the following are equivalent: (i) f and g are compatible; (ii) f preserves dom(g) and graph(g), and g preserves dom(f) and graph(f); (iii) f preserves dom(g), and g preserves graph(f); (iv) g is a homomorphism with respect to the partial algebra 〈M ; f〉. When interpreted in familiar settings, this is the familiar notion of compatibility. For example, take a total operation f : M → M and a relation r ⊆ M , with k, ! " 1. Then f preserves the relation r if and only if f is compatible with the restricted projection operation π1#r. Definition 1.4. For each set F of partial operations on M , define F := { g ∈ PM ∣∣ (∀f ∈ F ) f and g are compatible } . Then the map F -→ F is an order-reversing Galois connection from the power-set lattice ℘(PM ) to itself. We seek a general description of the closure F of a set of partial operations F ⊆ PM . This task breaks up into two parts. By Lemma 1.3, a partial operation g ∈ PM belongs to F if and only if (†) g preserves the graph of every partial operation in F, and (‡) dom(g) is closed under every partial operation in F. We will see that these two conditions lead to three constructs for building the closure F. The first two constructs are adding the projections and composing operations. 4 BRIAN A. DAVEY, JANE G. PITKETHLY AND ROSS WILLARD Notation 1.5. Let Π denote the set of all total projection operations on M . For F ⊆ PM , we let Clop(F ) denote the partial clone on M generated by F : it is the smallest subset of PM containing F ∪Π that is closed under composition (with maximum, non-empty domain). This corresponds to the usual definition of a partial clone, except that we exclude empty operations. The following lemma describes the partial operations satisfying condition (†). Within clone theory, this lemma essentially dates back to Geiger [16]. Within duality theory, it was used by Davey, Haviar and Priestley [10] to describe hom-entailment (see [3, 9.4.1]). For F ⊆ PM and R ⊆ RM , we introduce the notation F #R := { f#r ∣∣ f ∈ F and r ∈ R with r ⊆ dom(f) } . For a single relation r on M , we can view the subset Clop(F )# {r} of M r as a |r|-ary relation on M , and this relation is closed under every partial operation in F . Lemma 1.6. Let F ∪ {g} be a set of partial operations on a finite non-empty set M . Then the following are equivalent: (i) every relation on M that is closed under all f ∈ F is also closed under g; (ii) the relation Clop(F )# {r} on M is closed under g, where r := dom(g); (iii) g has an extension in Clop(F ); (iv) g preserves the graph of every partial operation in F. Proof. The three implications (i)⇒ (ii), (i)⇒ (iv) and (iii)⇒ (i) are straightforward. Now let g : r → M have arity k " 1 and define r∗ := Clop(F )# {r}. We assume (ii) to prove (iii). The projections ρ1, . . . , ρk : r → M are elements of the relation r∗ ⊆ M , with (ρ1, . . . , ρk) ∈ r r . Since we are assuming that r∗ is closed under g, we get g = g r (ρ1, . . . , ρk) ∈ r∗. Thus g has an extension in Clop(F ). Now assume (iv) to prove (ii). Choose a ∈ r. Then the projection ρa : r∗ → M is in F. So g preserves graph(ρa), and it follows that g preserves dom(ρa) = r∗. The third construct for building the closure F is restriction of domains to ‘definable’ relations. Notation 1.7. Let F ⊆ PM . We will view F as a set of concrete partial operations on M and also as a set of abstract partial operation symbols with associated arities. So we can consider the partial algebra M = 〈M ; F 〉 of type F . For a short discussion of first-order logic for partial algebras, see [3, pp. 24–25]. We shall call a conjunction of atomic formulæ Ψ($v) = [ψ1($v) & · · · & ψn($v)] a conjunct-atomic formula. We say that a k-ary relation r on M is conjunct-atomic definable from F if it is described in M by a k-variable conjunct-atomic formula Ψ($v) of type F , that is, if r = { (a1, . . . , ak) ∈ M k ∣∣ Ψ(a1, . . . , ak) is true in M } . We define Defca(F ) to be the set of all relations on M that are conjunct-atomic definable from F . Note that Defca(F ) consists of all relations on M that can be obtained as a finite intersection of equalisers of partial operations in Clop(F ). Definition 1.8. A primitive positive formula is of the form ∃$ wΨ($v, $ w), where Ψ($v, $ w) is a conjunct-atomic formula. For sets of partial operations F1 and F2 on M , we say that a k-ary relation r on M is primitive-positive definable from F1 with existence witnessed by F2 if THE LATTICE OF ALTER EGOS 5 – there is a (k + !)-variable conjunct-atomic formula Ψ($v, $ w) of type F1, for some ! " 0, such that ∃$ wΨ($v, $ w) describes the relation r on M , and – there is an !-tuple $τ of k-ary terms of type F2 such that Ψ($v,$τ($v)) describes the relation r on M . The proof of the following lemma is a straightforward modification of the original proof of the Dual Entailment Theorem 3.6 [10]. We will use the lemma in this section to describe the partial operations satisfying condition (‡). In Section 5, the lemma will be used to obtain a common generalisation of two entailment theorems from duality theory. Lemma 1.9. Let F1 and F2 be sets of partial operations on a finite non-empty set M such that F1 ⊆ F2, and let r be a relation on M . Then the following are equivalent: (i) r is closed under every partial operation in F 1 with domain closed under all partial operations in F2; (ii) r is closed under every partial operation in F 1 with domain Clop(F2)# {r}; (iii) r is primitive-positive definable from F1 with existence witnessed by F2. Proof of (ii)⇒ (iii). Assume that (ii) holds. The relation r∗ := Clop(F2)# {r} ⊆ M r contains the projections ρ1, . . . , ρk : r → M , where r has arity k " 1. Enumerate the elements of r\{ρ1, . . . , ρk} as h1, . . . , h!, where ! " 0. Each hj : r → M has an extension in Clop(F2), and so there is a k-ary term τj of type F2 with hj = τ j #r. This gives us our required !-tuple of terms $τ := (τ1, . . . , τ!). We next choose a finite subset F0 of F1. For each partial operation g : r → M that is not in F 1 , choose some fg ∈ F1 such that fg and g are not compatible. Define F0 to be the set of all these partial operations fg. Then F0 is finite, as M is finite. The relation r∗ = Clop(F2)# {r} = {ρ1, . . . , ρk, h1, . . . , h!} is closed under F0. So we can construct a (k + !)-variable conjunct-atomic formula Ψ($v, $ w) of type F1 describing, via the correspondences vi ↔ ρi and wj ↔ hj , – which of the projections ρ1, . . . , ρk in r∗ are equal, and – how each partial operation f ∈ F0 acts pointwise on r∗ ⊆ M . Now define two k-ary relations on M : – let s∃ be described by the formula ∃$ w Ψ($v, $ w), and – let sτ be described by the formula Ψ($v,$τ($v)). To see that (iii) holds, it remains only to show that r = s∃ = sτ . Since sτ ⊆ s∃, it is enough to show that r ⊆ sτ and s∃ ⊆ r. Let $a = (a1, . . . , ak) ∈ r. To check that $a ∈ sτ , we need to show that Ψ($a,$τ($a)) is true in M . By the construction of Ψ, we have Ψ(ρ1, . . . , ρk, h1, . . . , h!) in M , =⇒ Ψ ( ρ1($a), . . . , ρk($a), h1($a), . . . , h!($a) ) in M , =⇒ Ψ ( a1, . . . , ak, τ M 1 ($a), . . . , τ M ! ($a) ) in M . So $a ∈ sτ , whence r ⊆ sτ . Finally, to see that s∃ ⊆ r, let $a ∈ s∃. There exists $c ∈ M ! such that Ψ($a,$c) is true in M . Since any equalities amongst the projections ρ1, . . . , ρk are expressed in the formula Ψ, we can define the partial operation g : r∗ → M by g(ρi) := ai and g(hj) := cj. As Ψ($a,$c) holds in M , the construction of Ψ ensures that g preserves graph(f), for all f ∈ F0. Since dom(g) = r∗ is closed under F0, it follows by Lemma 1.3 that g ∈ F ♦ 0 . The choice of F0 now guarantees that g ∈ F ♦ 1 . By (ii), the partial operation g preserves r. Since (ρ1, . . . , ρk) ∈ r r , this implies that $a = ( g(ρ1), . . . , g(ρk) ) ∈ r. Hence s∃ ⊆ r. 6 BRIAN A. DAVEY, JANE G. PITKETHLY AND ROSS WILLARD Proof of (iii)⇒ (i). Assume that (iii) holds via the (k + !)-variable formula Ψ($v, $ w) and the !-tuple $τ of k-ary terms. Let g be an n-ary partial operation in F 1 with domain closed under F2. We want to show that g preserves r. So let $a1, . . . ,$ak ∈ dom(g) ⊆ M with ($a1, . . . ,$ak) ∈ r n . Fix j ∈ {1, . . . , !}. We can assume that the variable wj actually occurs in the formula Ψ($v, $ w). Since Ψ($v,$τ($v)) describes r on M , this implies that r ⊆ dom(τ j ). As dom(g) is closed under F2, we can define $bj := τ n j ($a1, . . . ,$ak) ∈ dom(g). Again since Ψ($v,$τ($v)) describes r on M , it now follows that Ψ ( $a1, . . . ,$ak,$b1, . . . ,$b! ) in M, =⇒ Ψ ( g($a1), . . . , g($ak), g($b1), . . . , g($b!) ) in M , as g ∈ F 1 (see Lemma 1.3), =⇒ ( g($a1), . . . , g($ak) ) ∈ r, as ∃$ wΨ($v, $ w) describes r on M .