1 Introduction. - University of Colorado Boulder

Simple equations on real intervals by Walter Taylor version of August 22, 2008

Abstract. If A is an absolute retract in the class of metric spaces, and if Σ is a consistent set of simple equations, then A is compatible with Σ, i.e. there are continuous operations on A that model Σ.

1

Introduction.

1.1

Main results

Our main result is Theorem 6, in which we prove that if A is an absolute retract (see §2.2) in the class of metric spaces, and if Σ is a consistent set of simple equations (defined at the end of §1.2 and in §2.1.1), then there are continuous operations on A that model Σ — i.e. A is compatible with Σ. In particular, Theorem 6 applies when A is any convex subset of Rk (k = 1, 2, . . .), such as R itself. In other words, R is compatible with every consistent set Σ of simple equations. Thus Theorem 6 contrasts with our previous result [22] that, for an arbitrary finite set Σ (of not necessarily simple equations), compatibility with R is algorithmically undecidable.

1.2

Background

For a set A and operations Ft : An(t) −→ A (with t ranging over some set T ,1 often finite) we say that the operations Ft satisfy Σ (or that they model Σ), and write (1) (A, F t )t∈T |= Σ, if for each equation σ ≈ τ in Σ, both σ and τ evaluate to the same function when the operations Ft are substituted for the symbols Ft appearing in σ and τ . Here we are concerned with the satisfiability of equations by continuous operations. When we refer to continuity of an operation Ft , we intend it with 1

The set T and function hn(t) | t ∈ T i determine a similarity type; see §2.1.1.

1

respect to the product topology on An(t) . Given a topological space A and a set of equations Σ, we write A |= Σ, (2) and say that A and Σ are compatible, iff there exist continuous operations Ft on A satisfying Σ. Moreover, in this case we may also say that (A; F t )t∈T is a topological algebra modeling Σ. The emphasis here is on the continuity of the operations F t . (Their mere existence will generally not be in doubt.) If V is mod Σ, i.e. the class of all (non-topological) models of Σ, and if A is a space, we allow A |= V to be written in place of A |= Σ. (This makes sense because if mod Σ1 = mod Σ2 , then A |= Σ1 iff A |= Σ2 ). This extension of the notation will facilitate a remark at the end of §2.1.1. The author’s papers [20, 21, 22, 23] (1986–2006) contain most of the known examples of compatibility and of incompatibility, either directly or by reference; we will not repeat that material here. Suffice it to say that the relation (2) remains mysterious: among the few pairs (A, Σ) for which the truth or falsity of (2) is known at all, (2) holds true only sporadically, and with little discernible pattern; in the larger picture, its truth-values are almost completely unknown. In [21] we were able at least to see that some of the mysteries are lodged among the arcane facts of algebraic topology. In [22] we saw that some of the apparent chaos of |= can be ascribed to the algorithmic undecidability of (2), especially for A = R: as mentioned above, there is no algorithm that answers, for all finite Σ, whether R |= Σ. Our main result (Theorem 6 in §3.1 below) restricts Σ to the so-called simple equations—those equations that have at most one Ft appearing on each side (see §2.1.1 below). In this case, if Σ is consistent then Σ is compatible with R (and with many other spaces — see §3.1). In this limited context, one does have algorithmic decidability (albeit in a somewhat trivial way — see §3.2.1).

1.3

Acknowledgments

I should like to thank Professor Ralph McKenzie and Professor Artur Barkhudaryan, each of whom made essential comments, especially in pointing out errors in earlier versions. I should like to thank the University of Colorado for their support during my sabbatical year of 2005–06.

2

2

Preliminaries.

2.1 2.1.1

Simple equations. Definition and examples.

A formal equation is an ordered pair (σ, τ ), where σ and τ are both terms. This ordered pair is usually written σ ≈ τ , but nonetheless it is still merely an ordered pair. The symbol ≈ makes no assertion about equality, but merely presents two symbolic quantities for consideration: they might or might not turn out to be equal, depending on how the symbols are interpreted. An equational theory, sometimes called simply a theory, is a collection of formal equations. Universal quantification, over all variables appearing, is understood; with this understanding, an equational theory is a special kind of first-order theory. There is one place, however, where it is convenient to adopt a usage that differs from the usual first-order usage: a theory is consistent iff it does not have x ≈ y as a consequence. We work with a (possibly infinite) similarity type h n(t) | t ∈ T i. Correspondingly, for each t ∈ T we take Ft as an operation symbol of arity n(t). If A is an algebra appropriate to this similarity type, its operation corresponding to the symbol Ft will often be denoted F t or FtA . Our formal equations are built with a fixed set of variables, {x0 , x1 , . . . }. By a simple term in this language, we mean2 a term that contains at most one Ft , and moreover contains at most one instance of that Ft . In other words, according to the usual recursive definition of terms, a simple term is either a variable or created at the first stage beyond the inclusion of variables. A simple equation is a formal equation σ ≈ τ with σ and τ both simple. The case where σ and τ are both variables is of little interest, since such an equation is either tautological or inconsistent. Therefore we usually consider the case where one or both of σ and τ is not a variable. One may easily check that (modulo the symmetry that is usually accorded to ≈) every such simple equation has either the form Fs (xα(1) , · · · , xα(n(s)) ) ≈ Ft (xβ(1) , · · · , xβ(n(t)) ),

(3)

Fs (xα(1) , · · · , xα(n(s)) ) ≈ xk ,

(4)

or the form

2

This terminology was used, perhaps for the first time, in Garc´ıa and Taylor [7].

3

where s, t ∈ T , where k ∈ ω = { 0, 1, 2, . . . }, and α, β are functions from { 1, 2, . . . } to ω. Among the well-known simple Σ are various Mal’tsev conditions: permutability, modularity, distributivity, and so on. See e.g. Garc´ıa and Taylor [7] or Sequeira [14]. Note also the idempotent affine algebras studied by Fajtlowicz and Mycielski [6] and by Szendrei [17, 18]. By the way, the main result of this paper will imply that many varieties V are not equivalent to (or even interpretable3 in) any variety W defined by simple equations. Such is the case for any non-trivial V that fails to be compatible with some AR A. For if we had V ≤ W with W definable by simple equations, then we would have A |= W and hence A |= V. For example, group theory is not compatible with a closed interval; lattice theory is not compatible with a triode (three closed segments joined in the form of the letter Y). Thus neither the variety of groups nor the variety of lattices is interpretable in any variety defined by simple equations. 2.1.2

Closure under set-retraction.

Let A and B be sets. By a set-retraction from A onto B, we mean a pair (i, p) of functions such that p

B  A,

and p ◦ i = 1B .

(5)

i

If (i, p) is a set-retraction from A onto B, and A is any algebra (with universe A), then we define an algebra B of the same similarity type as follows. Its universe is B, and its operations are defined by the formulas FtB (b1 , · · · , bn(t) ) = p(FtA (i(b1 ), · · · , i(bn(t) ))

(6)

(one formula for each relevant t). This B is said to be determined from A by the set-retraction (i, p). In general, an algebra B (with universe B) is said to be determined by set-retraction from A iff there exist maps i and p making (5) true and making (6) true for all b1 , b2 , . . .. We emphasize that the functions i and p appearing in (5) and (6) are generally not homomorphisms between A and B. As one may check, if 3

For a detailed explanation of this concept, we must refer the reader to Garc´ıa and Taylor [7]. In brief, variety V is interpretable in variety W if there are W-terms that allow W-algebras to be interpreted as V-algebras. (Denoted V ≤ W.) If this holds, then A |= W obviously implies A |= V.

4

(i, p) is a set-retraction, then i is a homomorphism if and only if i[B] is a subuniverse of A; in this case B is isomorphic to that subalgebra. For a class K of algebras of the same type, we say that K is closed under set-retractions iff the following holds: if A ∈ K and B is determined by setretraction from A, then B ∈ K. We say “set-retraction” to emphasize that p and i are not required to be homomorphisms. Obviously every variety is closed under retractions, if we insist that i and p should be homomorphisms. One direction of Theorem 1 that follows—namely the fact that a variety defined by simple equations is closed under set-retractions—will be used for four results that follow: Corollary 3, Theorems 6 and 7, and the example of ΣH in §3.2.3. The converse direction is interesting, but not applied in this article. It is reasonable to suppose that Theorem 1 might already exist in the mathematical literature, but we have not found it there. A version of ´ Theorem 1 has recently been found4 independently by K. Kearnes and A. Szendrei. Theorem 1 Let V be a variety (class of algebras modeling a fixed set Σ of equations of a fixed similarity type hn(t) : t ∈ T i). Then V is closed under set-retractions iff V is definable by simple equations (i.e. there exists a set Σ0 of simple equations such that V is the class of algebras of type hn(t) : t ∈ T i that model Σ0 ). Proof. Let us first assume that there exists such a Σ0 , and prove that V is closed under set-retractions. So let us suppose that A ∈ V and that B is related to A through maps p and i, in the manner of (6). We need to show that B ∈ V. Our approach will be to use the fact that A |= Σ0 and prove that B |= Σ0 . We now must show that the operations FtB defined by (6) satisfy the equations Σ0 . Each equation in Σ0 is simple and hence has the form of Equation (3) or Equation (4). The two forms are similar, with (3) simpler, so we shall consider only (4). So, let us be given k ∈ ω and α : {1, · · · , n(s)} −→ ω such that Equation (4) lies in Σ0 . To verify the satisfaction of (4) in the algebra B = hB, FtB it∈T , we use the definition (6) and the fact that (4) holds in A, to calculate FsB (bα(1) , · · · , bα(n(s)) ) = p ◦ FsA (i(bα(1) ), · · · , i(bα(n(s)) )) = p(i(bk )) = bk . 4

K. Kearnes, private communication.

5

This completes the proof that B |= Σ0 , and hence that V is closed under set-retraction. Conversely, let us suppose that V is closed under set-retraction. Taking Σ0 to be the set of simple equations that hold identically on V, we clearly have V ⊆ mod Σ0 ,

(7)

where mod Σ0 denotes the class of algebras that model (i.e. satisfy) the equations Σ0 . To finish the proof, we shall show that equality holds in (7). So, let us take an algebra B that satisfies Σ0 , and prove that B ∈ V. Let A ∈ V be an algebra that is V-freely generated by B (the universe of our B). We now consider two sorts of condition that we shall impose on a function p : A −→ B: p(b) = b p(FsA (b1 , · · ·

, bn(s) )) =

FsB (b1 , · · ·

(8) , bn(s) ),

(9)

where s ranges over T , and b, b1 , b2 , · · · range independently over B. We claim that (8) and (9) in fact describe a partial function p : A −→ B. Clearly for certain A-values a, this description of p yields one (or more!) B-values p(a); hence the indicated domain and co-domain are correct. It remains to check that p is well-defined by (8) and (9), i.e. that p is single-valued. A failure of p to be well-defined could result either from a conflict between two instances of (9) or a conflict between an instance of (8) and an instance of (9). We shall look at the latter case in detail. The former case—which we omit—is similar, but relies on equation-form (3) rather than (4). For a disagreement to arise between (8) and (9), we would have FsA (b1 , · · · , bn(s) ) = b

(10)

FsB (b1 , · · · , bn(s) ) 6= b.

(11)

but

Let e1 , · · · , eb be the distinct elements of B appearing among b, b1 , · · · , bn(s) . Now there clearly exist k ∈ {1, · · · , b} and α : ω −→ {1, · · · , b} such that b = ek and bi = eα(i) for 1 ≤ i ≤ n(s). Thus Equation (10) may be rewritten as FsA (eα(1) , · · · , eα(n(s)) ) = ek . 6

The distinct free generators ej appearing here may be replaced by variables xj to obtain a V-identity. In this way, we discover that (4) is an identity of V, which implies that (4) belongs to Σ0 . We assumed that B satisfies Σ0 , and hence FsB (eα(1) , · · · , eα(n(s)) ) = ek , which is to say FsB (b1 , · · · , bn(s) ) = b, in contradiction to (11). Thus, in fact, no conflict occurs: (8) and (9) indeed define a partial function p. Let p be extended in any manner to a function p : A −→ B. Also take i : B −→ A to be the identity function of B. It is now clear from (8) that (5) holds, and from (9) that (6) holds. Therefore B is defined from A by set-retraction. Therefore, by our assumption, B ∈ V, as was to be proved. Thus, as we promised, the inclusion (7) is an equality of classes, and hence V is indeed defined by the simple equations Σ0 . Corollary 2 A set Σ of simple equations is consistent iff Σ has a model of every cardinality κ ≥ 1. Proof. Of course, if Σ has a model of every cardinality, then Σ is consistent. For the converse, we assume that Σ is consistent and consider κ ≥ 1. As is well known, there exists an algebra A modeling Σ, with |A| ≥ κ. If we take B to be an arbitrary set of cardinality κ, there exist functions i and p satisfying (5). Now if we let B be the algebra determined from A by the set-retraction (i, p) (i.e. the algebra on B with operations defined by (6)), then B |= Σ by Theorem 1, and B has the desired cardinality κ.

2.1.3

Algorithmic consistency for sets of simple equations.

Corollary 3 There exists an algorithm to determine, for each finite simple Σ, whether Σ is consistent. Proof. By Corollary 2, it is enough to check whether Σ has a two-element model. This clearly is an algorithmic problem for finite Σ.

7

2.2

Absolute retracts in topology.

Let K be a class of topological spaces. A space A is said to be an absolute retract (AR) in K iff the following conditions hold: (i) A ∈ K; (ii) If A is a closed subspace of F ∈ K, then there exists a continuous map φ : F −→ A such that φ(a) = a for a ∈ A (in other words φ  A = 1A ). In fact, for reasons outlined on page 95, Hu [10] takes the class M of all metrizable spaces as the preferred class for the rˆole of K in this definition. In other words, when he says simply that a metrizable space A is an AR, he means an AR for the class M. (We shall occasionally follow this practice.) In fact, if A is a separable and topologically complete metrizable space that is an AR in this sense, then A is an AR for the class of all normal topological spaces [10, Theorem III.4.1(d), page 87]. A metrizable space A is an AR for M iff A is homeomorphic to a retract of a convex subset of a real Banach space — and, for compact A, iff A is homeomorphic to a retract of the Hilbert cube [0, 1]ω — see Hu [10, Propositions III.6.1 and III.6.3, pages 95–96]. These results derive ultimately from Tietze’s Extension Theorem. In particular, open and closed balls and simplices—even R itself, and even a half-open interval, are AR’s. For a more general class of examples, one may consider trees, which are connected acyclic one-dimensional simplicial complexes. (The trees that we shall consider in §4.1 and §4.2 are finite complexes of this type, e.g. the shape of the letter Y.) The notion of an absolute retract in K is closely related to that of an absolute extensor (AE) in K, which is defined as follows. A space A is called an AE in K iff the following conditions hold: (iii) A ∈ K; (iv) If B is a closed subspace of F ∈ K, and if g : B −→ A is a continuous function, then there exists a continuous function φ : F −→ A such that φ  B = g. (see [10, pages 34–35]). If we specialize (iv) by taking B to be A, and g to be the identity function on A, we clearly obtain (ii); thus every AE in K is an AR in K. In fact, in 8

many contexts, the converse holds, and every AR is an AE, as may be seen in the following theorem. For a proof, see Hu [10, Theorem III.3.2, page 84]; in fact the same result is proved there for many natural classes K of spaces. Theorem 4 If K is the class M of metrizable spaces, then the AR’s and the AE’s in K are the same. This close affinity between AR and AE allows one to prove that, for many K, the absolute retracts form a class that is closed under products and retraction. For K = M, and for finite or countable products, this is proved in [10, Theorems III.7.5–7, page 97]. Our only use of the theory of AR’s and AE’s occurs in the proofs of Theorems 6 and 7 below. For the existence of φ in (14), we apply Theorem 4 to see that a certain space A is an AE for M. The desired φ will then come from clause (iv) above. (For Theorem 6, we may use clause (ii) directly, without requiring Theorem 4.) Thus Theorems 6 and 7 apply to many well-known spaces, including those mentioned above: intervals, balls, acyclic unions of arcs, a half-open interval, the letter Y , and so on. We have referred mainly to Hu [10], but one may also consult Borsuk [3] for a general reference to the theory of retracts and extensors.

2.3

Topologizing free algebras.

For V a variety of type hn(t)|t ∈ T i, and κ a cardinal, we define FV (κ), the free algebra of type hn(t)|t ∈ T i on κ generators, as in [12, Definitions 4.107 and 4.118]. If Σ is a set of equations, we let FΣ (κ) denote FV (κ), where V is the class of models of Σ, i.e. the variety defined by Σ, in the smallest type that makes sense for the given context. In other words, unless the context requires5 some further operations, the operations Ft of V should be just those that appear in Σ. ´ The following theorem was proved by S. Swierczkowski [16] in 1964. The ´ final assertion (about the metric case) was implicit in the Swierczkowski paper, and was made explicit in 1977 by Taylor [19]. In 1979 A. Bateson proved [2] that if A has the topology of a CW-complex, then FΣ (A) can also be given the topology of a CW-complex. In 1992 Coleman gave [4] a number of generalizations, improved proofs, and so on, including a general form of a 5 Other operations may be included if they are wanted for some other reason; they play no role in, and have no influence on, what happens in this article.

9

related but not identical metric that Graev [9] had introduced for free groups in 1948. Theorem 5 Let (A, T0 ) be a completely regular topological space, and Σ any consistent equational theory. There exists a topology T on F = FΣ (A), the Σ-free algebra on A, such that (a) the operations of FΣ (A) are T -continuous; (b) (F r A) ∈ T ; i.e. A is a closed subset of F with respect to T ; (c) T0 is the topology inherited by A from T ; (d) T is completely regular; (e) if T0 is Hausdorff, then T is Hausdorff. In fact if T0 is defined by a metric d0 on A, then there exists a metric d on F such that T is defined by d and d  A2 = d0 .

3 3.1

The main theorems. Statements and proofs.

Theorem 6 If Σ is a consistent set of simple equations, and if A is a completely regular space that is an AR in the class of all completely regular spaces, then A is compatible with Σ. Moreover, if A is metrizable, then we only need A to be an AR in the class M of all metrizable spaces. Theorem 7 If Σ is a consistent set of simple equations, and if A is a completely regular space of more than one element that is an AE in the class of all completely regular spaces, then (a) If Γ is a finite set of simple equations, none of which is a logical consequence of Σ, then there exists a topological algebra A, whose underlying space is A, such that A satisfies Σ but satisfies no equation in Γ. (b) In particular, if the similarity type is finite, then there exists a topological algebra A whose underlying space is A, and whose simple identities are precisely the simple consequences of Σ. 10

Moreover, if A is metrizable, we only need A to be an AR in the class M of all metrizable spaces. Proofs. We shall first prove Theorem 7 in detail, and then we shall make some comments on how that proof may be modified for Theorem 6. To begin the proof of Theorem 7, we remark that even in the metrizable case, where we have assumed only that A is an AR, Theorem 4 tells us immediately that A is an AE in M. Thus in all cases of Theorem 7, we may assume that A is an AE in the relevant class of spaces. Part (b) of Theorem 7 is a special case of Part (a), and so we shall only prove Part (a). Moreover, we shall prove Theorem 7(a) only in the case that A ∈ M and that A is an AR for M; only very small changes are required for the class of completely regular spaces, and these will be omitted. For the rest of the proof, we consider the free algebra F = FΣ (A) as topologized in Theorem 5 of §2.3. Moreover, as described in §2.3, F will have a definite similarity type hn(t) | t ∈ T i. Further references in the proof to continuity are implicitly referring to the topology of Theorem 5. From Theorem 5 we know that the operations of F are continuous, that A is a subspace of F , and that F and all its subspaces lie in M. First let us take N so large that each equation in Γ involves only the variables xi (0 ≤ i ≤ N ). For σ ≈ τ ∈ Γ we may consider each term σ and τ to involve the variables x0 , · · · , xN , even if some of these variables fail to make an explicit appearance. In this way, the term operations σ F and τ F may be treated as operations defined on F N . Finally we note that no finite metric space of more than one element is an absolute retract; hence A is infinite. We therefore have a finite sequence c0 , c1 , · · · , cN of distinct elements of A. We first consider the finite subset of F , K = {c0 , · · · , cN } ∪ {σ F (c0 , · · · , cN ) : σ ≈ τ ∈ Γ} ∪ ∪ {τ F (c0 , · · · , cN ) : σ ≈ τ ∈ Γ}. Since A is infinite, it is not hard to see the existence of a function φ : K ∪A −→ A such that: if a ∈ A, then φ(a) = a; φ is one-to-one on K.

11

(12) (13)

From the fact that A is closed and K is finite, it is not hard to see that φ is continuous on K ∪ A. Since K is finite, A ∪ K is closed in F . Since A is an AE in M we may extend φ to a continuous map φ : F −→ A

(14)

such that (12) and (13) still hold. For each t ∈ T we define an operation FtA : An(t) −→ A as follows: FtA (a1 , a2 , · · · ) = φ(FtF (a1 , a2 , · · · ))

(15)

for a1 , a2 , · · · ∈ A. In other words, FtA = φ ◦ (FtF  An(t) ). As we mentioned above, FtF is F -continuous by Theorem 5(a), and hence FtF  An(t) is Acontinuous by Theorem 5(c). Since φ is continuous, we obviously have each FtA continuous on An(t) . Thus A = hA, FtA it∈T is a topological algebra. It is apparent that A is determined by retraction from F—see §2.1.2; by (12) and (15), φ plays the role of p in (5) and (6), while the identity function plays the role of i. It is thus immediate from Theorem 1 that A |= Σ. Hence A |= Σ, i.e. A is compatible with Σ. This is one of the desired conclusions of Part (a). To conclude the proof of Part (a), we must consider a simple equation in Γ, and show that this equation does not hold identically in A. Without loss of generality, we need consider only Equations (3) and (4), where (as before) α and β are functions from {1, · · · , n(t)} to {0, 1, · · · , N }. Again, we shall only describe what happens for Equation (4), while leaving (3) to the reader. Since the ci are distinct free generators of F, and since (4) is not a consequence of Σ, it follows from the theory of free algebras that FsF (cα(1) , · · · , cα(n(s)) ) 6= ck , In other words, FsF (cα(1) , · · · , cα(n(s)) ) and ck are distinct elements of K. Therefore, by (13), we have φ◦FsF (cα(1) , · · · , cα(n(s)) ) 6= φ(ck ), Remembering that all ci ∈ A, we use (15) and (12) to translate this last equation into FsA (cα(1) , · · · , cα(n(s)) ) 6= ck , 12

which tells us that Equation (4) fails in the topological algebra A. This completes the proof of Theorem 7. For Theorem 6, we modify the proof as follows. First, we omit the last two paragraphs, concerning the equation-set Γ, which makes no appearance in Theorem 6. Those two paragraphs contain the only use of N ; so N is not wanted for Theorem 6, and we delete the paragraph where N is defined. Finally we take K to be empty. Now (13) holds vacuously, and (12) merely defines φ to be the identity function. Thus the extension (14) of φ to F requires only that A should be an AR — which is what we have assumed for Theorem 6. As before, (15) defines a topological algebra that exhibits the compatibility of A and Σ.

3.2 3.2.1

Comments on the main theorems. Algorithmic decidability.

Theorem 6 applies of course to A = R, the space of real numbers, yielding that the R-compatibility of a simple equation-set Σ is equivalent to the consistency of Σ. From Corollary 3 we then have the easy consequence that there exists an algorithm to determine, for each finite simple Σ, whether Σ is compatible with R. This contrasts with the main result of W. Taylor [22], which says that no such algorithm exists if we allow Σ to range over all finite sets of equations. 3.2.2

Earlier results

Theorem 6 overlaps with a 1977 result of the author—Theorem 7.7 (page 524) of [19]—which asserts the existence of a ternary majority operation on any complex with all higher homotopy groups zero. The proof there relied on obstruction theory of algebraic topology, and hence—like the proof here— ultimately rested on Tietze’s Extension Theorem. J. Van Mill and M. Van de Vel [24] proved in 1979 that every compact AR (among metric spaces) has a continuous ternary majority operation, which they call a mixer. They extended this result to the non-compact case in 1982 [25]. See Dranishnikov [5] for a converse to this result.

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3.2.3

H-sets

There is a theory ΣH that is not axiomatizable by simple equations, but for which we can prove a version of Theorem 6. Known as the theory of H-sets, ΣH has axioms F (x, e) ≈ x ≈ F (e, x), for a binary operation symbol F and a nullary e. We first show that ΣH is not axiomatizable by simple equations. Suppose we consider the algebra C with universe C = {0, 1, 2}, with eC = 0, and with the following multiplication table for F C : 0

1 2

0 0 1 1 2 2

1 2 1 1 1 1

.

Let D = {1, 2}, let i : D −→ C be the identity map, and retract C onto D via p(0) = p(1) = 1 and p(2) = 2. One easily checks that C is an H-set, but the algebra D determined here by set-retraction is not an H-set; by Theorem 1, ΣH is not equivalent to a set of simple equations. Before proceeding, it will be helpful to know something about topological algebras A = hA; eA , F A i modeling ΣH , where A is the Hilbert cube [0, 1]ω . We claim that for any a = hai ii∈ω ∈ A, there exists such an A with eA = a. For each i ∈ ω, we let Ai be the H-set whose universe is [0, 1], for which eAi = ai , and which has F Ai (x, y) = [(x + y − ai ) ∧ 1] ∨ 0 (where a ∧ b and a ∨ b stand, respectively, for the smaller and the larger of two real numbers a and b). It is clear that if we take A to be the product of the topological algebras Ai , then eA = a, and our claim is verified. We shall now prove that if B is any compact metrizable space that is an AR in M, then B is compatible with ΣH . By Hu [loc. cit.], B is a retract of the Hilbert cube A. In other words, there are continuous maps i : B −→ A and p : A −→ B such that p ◦ i = 1B . Let us select b ∈ B, and then let a = i(b). As we just showed, ΣH has a topological model A = hA; eA , F A i for which eA = a. We now take eB to be b, and define F B according to 14

Equation (6). We will verify the first of the equations of ΣH for B (while the second equation may be similarly verified): F B (x, eB ) = F B (x, b) = p(F A (i(x), i(b))) = p(F A (i(x), a)) = p(F A (i(x), eA )) = p(i(x)) = x.

4

A construction for finite trees

Relying as it does on Tietze’s Theorem, our Theorem 6 cannot be viewed as very constructive.6 In §4.2 we exhibit a more constructive approach which is sometimes available. The method of §4.2 is valid when A is a (finite) tree and operations modeling Σ on [0, 1] are already known. A concrete example appears in §4.2.1.

4.1

An earlier construction of Sholander.

Our results, and those of Van Mill and Van de Vel reported in §3.2, were preceded in 1954 by a simple construction of M. Sholander [15] for trees. In this context, a tree is the space A associated to a connected acyclic onedimensional finite simplicial complex. For a, b ∈ A, we let [a, b] denote the intersection of all connected sets containing {a, b}. Sholander proved the following results: [a, b] is a connected set; for arbitrary a, b, c ∈ A, the intersection [a, b] ∩ [b, c] ∩ [c, a] is a singleton; if we define m(a, b, c) to be the single member of this set, then m is a continuous operation satisfying these equations: m(x, y, y) ≈ y m(x, y, z) ≈ m(y, z, x) ≈ m(z, x, y) m(m(x, y, z), u, v) ≈ m(x, m(y, u, v), m(z, u, v)). (These equations define the variety of median algebras; see [1].) This m is thus a continuous majority function, or mixer, for the tree A. Moreover, in this case there is no reliance on the Theorem of Tietze. In fact, presuming that one has in place a scheme for approximate calculation with real 6

In view of the title of Weihrauch [26], it would seem that if A is a computable metric space, then all the constructions of this paper can ultimately be exhibited as computable functions. Nevertheless the constructions outlined in §4 yield computable functions that are much simpler. See also Gherardi [8].

15

numbers—see e.g. [13]—and one for representing trees, the Sholander definition of m is algorithmic; that is to say, m is computable.

4.2

A recursive construction for trees.

Suppose that A is a tree (as defined in §4.1), and that Σ is a consistent set of simple equations. We will describe a recursive procedure for modeling Σ on A, presuming that we have a constructive way to model Σ on I, an interval, such as the methods from [22] that will be mentioned in §4.2.1. As at the end of §4.1, the method here of course assumes that one has in place a scheme for approximate calculation with real numbers, and an algorithm for the representation of trees. For our recursive definition of the operations on A, we take a representation of A as a finite acyclic 1-complex, and proceed by recursion on the number of vertices of this complex. If A has no vertex of degree 3 or higher, then A is homeomorphic to I, and so we use the given I-operations that satisfy Σ, and we are done. In the contrary case, we have a vertex V and three 1-simplices Li (i = 1, 2, 3), with V a face of each Li . For i = 1, 2, 3, we define Ai = {x ∈ A : there is a path from x to V that contains no interior point of Li }. Evidently (i) Each Ai is connected. (ii) Hence each Ai is an AR. (iii) Each Ai has fewer vertices than A; hence recursion is applicable. (iv) Each x ∈ A lies in at least two of the three Ai ’s. By (ii), for each i there exists a retraction pi : A −→ Ai (with pi  Ai the identity on Ai ). By (iv) we have m(p1 (a), p2 (a), p3 (a)) = a

(16)

for all a ∈ A, where m is the median operation on A given by Sholander’s construction (see §4.1).

16

i

By recursion (iii), for each i we have operations F t satisfying Σ on Ai . Now, for each t ∈ T , we define 1

F t (a1 , a2 , · · · ) = m(F t (p1 (a1 ), p1 (a2 ), · · · ), 2

3

F t (p2 (a1 ), p2 (a2 ), · · · ), F t (p3 (a1 ), p3 (a2 ), · · · )). We now claim that the operations F t satisfy Σ on A. To see this, we consider an equation of Σ; we may assume it has the form (3) or (4). As before, we shall consider in detail only the form (4). Using (16) and our recursive assumptions, we have: 1

F s (aα(1) , · · · ,aα(n(s) ) = m(F s (p1 (aα(1) ), p1 (aα(2) ), · · · ), 2

s

F s (p2 (aα(1) ), p2 (aα(2) ), · · · ), F t (p3 (aα(1) ), p3 (aα(2) ), · · · )) = m(p1 (ak ), p2 (ak ), p3 (ak )) = ak . 4.2.1

Minority identities on trees.

For a specific non-trivial instance of the methods of §4.2, we mention that they apply to the well-known ternary minority identities. In [22, §9.4], Equations (68) and (71), we gave an explicit definition of a ternary G on [0, 1] satisfying the minority identities G(x0 , x0 , x1 ) ≈ G(x0 , x1 , x0 ) ≈ G(x1 , x0 , x0 ) ≈ x1 plus a certain amount of symmetry: G(x0 , x1 , x2 ) ≈ G(x1 , x2 , x0 ) ≈ G(x2 , x0 , x1 ). These identities can then be modeled on an arbitrary (finite) tree by the method of §4.2.

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[3] K. Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Pa´ nstwowe Wydawnictwo Naukowe, Warsaw, 1967, 251 pages. [4] J. P. Coleman, Topologies on free algebras, Ph.D. Thesis, University of Colorado, Boulder, 1992. (UMI Dissertation Services, Order Number 9318085.) [5] A. N. Dranishnikov, Mixers, the converse of a theorem of van Mill and van de Vel, Mathematical Notes 37 (1985), 325–328. [6] S. Fajtlowicz and J. Mycielski, On convex linear forms, Algebra Universalis 4 (1974), 244–249. [7] O. C. Garc´ıa and W. Taylor, The lattice of interpretability types of varieties, Memoirs of the American Mathematical Society, Number 305, 1984, iii+125 pages. MR 86e:08006a. [8] G. Gherardi, An analysis of the lemmas of Urysohn and Urysohn-Tietze according to effective Borel measurability, Lecture Notes in Computer Science 3988 (2006), 199-208. [9] M. Graev, Free topological groups, Izv. Akad. Nauk SSSR, Ser. Mat. 12 (1948), 279–324. English translation: Amer. Math. Soc. Translations (1) 8 (1962), 305–364. [10] S. T. Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. [11] A. I. Mal’tsev, Free topological algebras, Izv. Akad. Nauk SSSR Ser. Mat. 21 (1957), 171–198. English translation: American Mathematical Society Translations (2) 17 (1961), 173–200. [12] R. N. McKenzie, G. F. McNulty, and W. Taylor, Algebras, Lattices, Varieties, Wadworth & Brooks-Cole, Monterey, 1987. [13] M. B. Pour-El and J. I. Richards, Computability in analysis and physics. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1989. xii+206 pages. [14] L. Sequeira, Ph.D. Thesis, Universidade de Lisboa, 2002. An abstract may be viewed at http://www.lmc.fc.ul.pt/~lsequeir/math/extabstract.pdf. 18

[15] M. Sholander, Medians and betweenness, Proceedings of the American Mathematical Society 5 (1954), 801–807; Medians, lattices and trees, ibid, 808–812. ´ [16] S. Swierczkowski, Topologies in free algebras, London Math. Soc. Proceedings (3) 14 (1964), 566–576. ´ Szendrei, Identities satisfied by convex linear forms, Algebra Univer[17] A. salis 12 (1981), 103–122. [18]

Identities in idempotent affine algebras, Algebra Universalis 12 (1981), 172–199.

[19] W. Taylor, Varieties obeying homotopy laws, Canadian Journal of Mathematics 29 (1977), 498–527. [20]

The clone of a topological space, Volume 13 of Research and Exposition in Mathematics, 95 pages. Heldermann Verlag, Berlin, 1986.

[21]

Spaces and equations, Fundamenta Mathematicae 164 (2000), 193–240. Equations on real intervals, Algebra Universalis 55 (2006),

[22] 409–456. [23]

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[26] K. Weihrauch, On computable metric spaces Tietze-Urysohn extension is computable, Lecture Notes in Computer Science 2064 (2001), 357-368. Walter Taylor Mathematics Department University of Colorado Boulder, Colorado 80309–0395 USA Email: [email protected] 19