June 1990 - Categories - categories.org.au

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by Vaughan Pratt 29 Jun '90

29 Jun '90

I haven't had a chance yet to follow up on Mike Barr's thoughts, but hopefully will when I get back from the Jonsson symposium. Meanwhile here's a message from Pepe Meseguer giving some key references, which would appear to be prerequisite reading for anyone formulating ideas in this area. -v All of this has been done for ordered algebras by Steve Bloom, who defined ordered theories and proved a Birkhoff variety theorem in "Varieties of ordered algebras" JCSS 13 (1976) 200-212, and for (chain) continuous ordered algebras by me, who used continuous theories, and proved a completeness theorem and two different Birkhoff variety theorems in "Varieties of Chain-Complete Algebras" J. Pure Appl. Algebra, 19 (1980) 347-383, and in "A Birkhoff-like theorem for algebraic clases of interpretations of program schemes" in "Formalization of Programming Concepts" J. Diaz and I. Ramos (eds.) Springer LNCS 107, (1981) 151-168.

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ANnouncement for bulletin board
by WELLS＠CWRU 29 Jun '90

29 Jun '90

I have collected into one file 57 diagrams coded in TeX that I am willing to transmit to anyone who asks for it. They require LaTeX and Michael Barr's CATMAC.STY. These diagrams are drawn from various papers and from our new book. --Charles Wells

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by INHB000 27 Jun '90

27 Jun '90

Dear Bob: After thinking about Vaughn's letter some more, I realized that that is not how I would have been tempted to make a Pos theory. In two respects. First, I would not have used contravariant operations. Second, I would not have stuck to discrete exponents. Let me take up the second point first. Surely you would like to be able to have an algebra that had operations whose domain { (x,y) | x < y }, which means you want an operation of arity 2 = .-->. . For example, you want to be able to have countably CPO's as axiomatized by an operation of arity \omega, which means that the operation is defined on countable increasing sequences. Suitable axioms will guarantee that it is a countably CPO. As for the first point, this bothered me. The way Vaughn has it, you lose the ability to apply the Yoneda lemma, one leg of the categorist's holy trinity (the other two being adjoint functors and representable functors; they are all essentially equivalent concepts, but one or the other looks more fundamental, depending on the context). In this setup, these functors will be representable and the Yoneda lemma is available. At first I thought I would have to make a poset-enriched version of Yoneda (which would certainly be no problem), but eventually I realized that that is not quite the way to go. In fact, this is the setup. Let V:Pos --> Set be the underlying functor on the category of posets. A poset enriched theory (or, more properly, sketch) consists of n-ary operations for each poset n, along with equations and poset-based Horn clauses. A model is a poset X equipped with a function V(n --o X) --> VX for each n-ary operation in the sketch. The point of the V is that I make no hypothesis that these functions either preserve or reverse or have any interaction with the order. So no mental gymnastics with the symmetric difference or other mixed operations. The point is that, as Vaughn's examples show, the operations do not naturally preserve the order. Or reverse it. I am reasonably confident that I could dream up a theory in which there was no way of decomposing the operations so that they can be made up from mixed variance operations in the way that Vaughn did it. Now let B be the category of models. We have U:B --> Pos and it has a left adjoint F. There is a downside to all this. Vaughn asked for an HSP theorem and I gave one for his theory. It crucially depended on the fact that powers and the op functor preserved those epimorphisms in Pos (surjections). Well, non-discrete exponentiation doesn't preserve exponentiation. In fact, it is more-or-less obvious that only split epis are preserved by arbitrary exponentiation. So now what? Over sets, the crucial step in showing that theories and triples are (ignoring certain size questions that I will keep under the rug for the moment; they have no effect on the kinds of theories that will interest computer scientists and I am perfectly to restrict to countably based sketches and \aleph_1 accessible triples) equivalent is that the natural transformations from U^n to U are in one-one correspondence with UFn. It is not hard to prove that the set of natural transformations from U^n to U is equivalent to VUn and from there to show that if that set of natural transformations is given the obvious order: \alpha < \beta if for all f:n\to UB, \alpha B(f) < \beta B(f), then that set of natural transformations is order isomorphic to UFn. From this, I think the equivalence between theories and triples will follow. However, there is a downside to all this. Surjections are not split and therefore are not preserved by triples. This being crucial to the argument about HSP subcategories, that would seem to go down the drain as well. There are some possbilities, which I will discuss shortly, but I don't believe any simple recovery is possible. First an example. Suppose we consider a theory with just one operation of arity 3, meaning .-->.-->. This means a value I will call <x,y,z> defined whenever x < y < z. Put in the following: z < <x,y,z> and <x,x,z> = <y,z,z> = z. Then there is an algebra whose underlying poset is 2 + 2, that is the disjoint union of two arrows. The values of this operation are forced. In fact, this is the free algebra on 2 + 2. If we now identify the head of one arrow with the tail of the other, then there are now three distinct x < y < z. Thus we get elements <x,y,z> which is now strictly greater than z, as well as <y,x,<x,y,z>> and <z,<x,y,z>,<y,z,<x,y,z>>>,..., not to mention <x,z,<x,y,z>> and many others. Thus the coequalizer of that identification is infinite. This shows that images are not always very simple. Now here is what you can do. I feel more comfortable with triples. Suppose that B is a category of algebras and C is a full subcategory and let us suppose it is reflective. If we suppose at least that C is closed under U contractible quotients, then the composite is tripleable. This means that whenever X -->> Y is an arrow, split as posets and X is in C, then so is Y. Call the triple S. Then we have T --> S a map of triples. There is (of course) a category of triples and in that category, there is a factorization T -->> R >--> S where the first is a strict epi and the second monic. The real question is to what extent can T -->> R be viewed as the introduction of equations. I don't know at this point, but I don't expect there to be any easy connection. Of course, if T -->> R is actually surjective, then it can be viewed as the introduction of equations. The trouble is that there might be new operations forced by the identifications. In fact, the example above, although it takes place at the level of models, not theories, gives an example how this happens. Still, the new operations are a result of an identification that was made (or perhaps of a new ordering introduced), so this may still give a kind of HSP theorem. But to illustrate the kind of thing that will have to be taken into account, consider that T --->> R is an epimorphism in the category of triples. Unless it is an epimorphism in the functor category, one cannot even infer that a T algebra that is a subobject of an S algebra is an S algebra. This is the S of the HSP theorem. I don't see any obvious way around this. Regards, Mike

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by INHB000 25 Jun '90

25 Jun '90

Dear Bob: One thing I didn't say anything about in my message last night is what I meant by HSP. The P is standard. As for S it is pretty clear from what I said that S stands for strict (=extremal) subobjects. It follows that H stands for surjective images. This probably contradicts expectation, so a few words of explanation are necessary. First place, H and S must determine each other as a factorization system. Also H has to be closed under products. I'll come back to that. Let us look at S. If S just stands for subobjects, then even such a simple relation as x < x+y isn't inherited by subobjects and so doesn't define an HSP class. This is clearly not what is wanted. Thus only strict subojects are wanted. This forces H to include all surjective images and then it is obvious that H/S give a factorization system. Also H is closed under products. (In fact, it is closed under pullbacks, which implies there is a regular epi/mono factorization system, but not every morphism in H is regular.) H is also invariant under the endofunctor * on the category of posets that takes each poset to its opposite. Now let Th be one of Vaughn's theories. Then there are operations of arity (m,n) for which a model is a poset X equipped with a monotone map X*^m x X^n ---> X. Then these will be subject ot some constraints (equations and inequalities) and we get a category of models. Suppose that X ---> Y is a morphism of models and we factor that in the category of posets as X --->> Z >---> Y where the first map is surjective and the second is strong monic in Pos. We have the following rectangle: X*^m x X^n --->> Z*^m x Z^n ---> Y*^m x Y^n ! ! ! ! ! ! ! ! ! ! ! ! ! v v v X -------------> Z >-----------> Y where the outer vertical arrows are the given sturcture maps and the middle one exists by virtue of the factorization system. This shows that in the category of algebras there is a factorization into surjections/S-monomorphisms, the latter being strict in the underlying category Pos. They are not necessarily strict in the category of models. For example in the category of po-monoids, the inclusion of N into Z (both with usual ordering) is strict in Pos but not in the category since it is epic and a strict-monic/epic is an isomorphism. The fact that Pos is not a regular category makes me think it unlikely that anything could be done with a regular-epic/monic factorization, even if that were desirable. I guess that regular epics are closed under finite (but probably not infinite) products, so the first part of the analysis could be carried out for finitary theories. But even if it were possible, it would mean that only equations, not inequalities could be specified and that is certainly not what is wanted. I haven't checked out any of the details in this case. Regards, Mike

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by Colin B Jay 25 Jun '90

25 Jun '90

Ordered algebras are a special case of algebras for a 2-dimensional theory, i.e. a 2-category, in which the 2-cells express the ordering of operations. I considered the case of such theories with a tensor product (2-props) in "Languages for Monoidal Categories" (JPAA 59 (1989) 61-85). Barry Jay

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by INHB000 25 Jun '90

25 Jun '90

A couple of preliminary thoughts on Vaughn's communication. First, the whole business about symmetric difference was not well considered. Symmetric difference is a binary operation and calling it a quaternary operation doesn't make it one (cf. A. Lincoln). Of course, there is a quaternary, or more properly a (2,2)-ary operation that takes (x,y,z,w) to z.x- + w.y- and that can be specialized to give symmetric difference, but it isn't symmetric difference. If you want symmetric difference, it can also be defined from the operations of +, ., and -. In either case it is a derived operation, not a primitive one. This is not an important point, just to set the record straight. I have no knowledge on question I. As for II, I have some thoughts, at least on II i. First off the answer cannot be that an HSP subcategory is described by equations. This is already clear from the HSP subcategory of the theory of one binary (that's (0,2)-ary to be precise) operation +. Don't suppose it is associative or commutative. The Horn clause x < y => x+y = y defines an HSP subcategory. I don't think that, lacking associativity and commutativity, you can get that subcategory equationally. Even with them, I'm not sure you can, although you could get x+y<y from y+y=y. No, I haven't proved this, it just seems likely to me that these claims are true. Does that mean that HSP corresponds to Horn clauses. Well yes, but only Horn clauses that are stateable in the theory of posets. An equation is a Horn clause in the theory of sets. Of course the theory of sets has only one predicate, that of equality. (Membership doesn't count here because we don't use over subsets.) So an equation like x.x*=1 is the same as the Horn clause y=x* => xy=1. In a similar way, the theory of posets has only the predicates of = and <. (By the way, it is clear that Vaughn used < for less than or equal and I follow him in that usage.) I claim (or conjecture, if you prefer, but it is pretty clear) that HSP subcategories correspond to the Horn clauses that can be stated using only the predicates = and <. The reason is that if Pos is the theory of posets and if B is the category of models of one of these poset theories, then the underlying functor U:B--->Pos has an adjoint F (and in fact is tripleable). If C is an HSP (full) subcategory of B, then it is immediate that if C includes all the free algebras (that is F(Pos)), then B=C. Thus if B isn't C, there is some poset X such that F(X) isn't in C. Now C is reflective (trivial) so there is a free C algebra generated by X and it is a quotient of F(X). I am pretending that all these theories are finitary, although the infinitary generalization would probably offer no real difficulty. Suppose that X consists of elements x_1,...,x_n and some order relations such as x_i < x_j. Now there will be two terms t(x_1,...,x_n) and s(x_1,...,x_n) which are distinct in F(X), but not in its reflection in C. Or perhaps they are unrelated in F(X) but s < t in the C-reflection. This will give you Horn clauses of the sort x_i < x_j, ..., => s = t or x_i < x_j, ..., => s < t in the predicates available in Pos. The left hand sides are simply a listing of all the order relations in X. Or at least a generating family of such. In order to prove this, you must investigate the nature of quotients. However, unlike in Cat, they are surjective and are of the two types alluded to in the argument above. I don't fully understand II ii, but I think the above bears on that question too. There are obvious generalizations of this to bases other than Pos; I guess that is evident. It is a matter of taste, but I would prefer to leave the predicate of equality in rather than derive it from <. Cheers, Mike

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by pratt＠cs.stanford.edu 23 Jun '90

23 Jun '90

Guess you didn't get the revision I mailed at 7:50 PDT on Jun 22. I no longer see how to make the coding I had in mind for semigroup word problems in conjecture 9 go through. Oh well, at least it's at the end of the list of conjectures for which I had expressed the least confidence. Please forward. -v

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hat -> backquote
by Vaughan Pratt 21 Jun '90

21 Jun '90

Oops, I forgot about hat's being tilde'd by bitnet or whoever. The following replaces hat by backquote, in case you haven't sent it off yet. Since quote is only used on 0 and 1, which are their own converse, no confusion should result. There are three tilde's at the end of the message that will turn into c's, but since those tilde's denote a congruence the c can be read as c for congruence (cuch cerendipity!) ORDERED ALGEBRAS Vaughan Pratt Stanford Univ. June 20, 1990 I'd like to ask the group what it knows about the following notion, which out of ignorance of the standard term if any I will here call an ordered algebra. Which is more fundamental, = or < ? Frequency of usage gives little clue, both are in heavy use. When x+y (disjunction, sup, max) or xy (conjunction, inf, min) is in the language we can take = as basic and view x<y as merely an abbreviation for x+y = y, or x = xy. However what if x+y or xy is in the language solely so that we *can* so express x<y as an equation? This is the situation we find in the theory of residuated posets, where x\y and x/y can be defined inequationally via x;(x\y) < y < x\(x;y) (y/x);x < y < (y;x)/x without reference to x+y or xy other than that implicit in the expansion of x<y to x+y=y. Having so defined residuation we may, based on the axioms in Judith Ng's thesis (UCB Math, 1984), proceed to define reflexive transitive closure x* as the monotone operation satisfying 1' < x* (x* is reflexive) x*;x* < x* (x* is transitive) x < x* (x\x)* = x\x (pure induction, or "iteration preserves invariance") all in terms of equations that continue to make no other reference to x+y or xy beyond that implied by reading x<y as x+y=y. (Incidentally it may be verified, see e.g. the section on star in my article in LNCS 425, that x* is implicitly defined, i.e. uniquely determined as an operation on a residuated poset, by these inequations.) These considerations indicate a need for a notion along the following lines. An ordered algebra is a poset (A,<) equipped with (m,n)-ary operations f: (A,>)?m x (A,<)?n -> (A,<), where (A,>) denotes the order dual of A. We say that such a function f is of arity (m,n), having m antimonotone arguments and n monotone arguments. The application of such an f to its m+n arguments is written f(x1,...,xm;y1,...,yn). For example the signature of a relation algebra (R, +, ., -, ->, #, ;, `, 0, 1, 0', 1') would be (0,2),(0,2),(1,0),(1,1),(2,2),(0,2),(0,1),(0,0),(0,0),(0,0),(0,0). That is, disjunction x+y and conjunction xy are of arity (0,2) and hence monotone in both arguments, negation x- of arity (1,0) would be antimonotone in its one argument, material (Boolean) implication x->y of arity (1,1) would be antimonotone in one argument and monotone in the other, exclusive-or x#y would have to be viewed as a quaternary (2,2)-ary operation rather than a binary operation, relative product x;y of arity (0,2) would be monotone in both arguments, converse x` of arity (0,1) would be monotone in its one argument, and the constants 0,1,0',1' are of course all of arity (0,0). Specifying the signature takes care of the otherwise awkward question of how to axiomatize monotonicity itself without using Horn clauses or x+y; it simply becomes part of the syntax. The notion of equational theory is now replaced by that of an inequational theory, defined in the obvious way, which the following examples should make more obvious. Example 1. Semilattices. A semilattice (X,+) consists of just one operation x+y, of arity (0,2), satisfying the following inequations. x < x + y y < x + y x + x < x The first two inequations say that x+y is an upper bound on x and y, and the third says that it is the least such (easy exercise). The signature (0,2) and these three inequations constitute the entire definition of "semilattice." We may further equip it with constants 0 and 1, each of arity (0,0), defined via: 0 < x x < 1 Example 2. Ordered monoids. An ordered monoid (X,;,1') consists of an operation x;y of arity (0,2) and a constant 1' of arity (0,0) satisfying x;(y;z) = (x;y);z x;1' = x = 1';x We read x=y as abbreviating the pair of inequations x<y<x. (But we could shorten the four inequations of x;1' = x = 1';x to three via x < x;1' < 1';x < x.) We may expand an ordered monoid to a residuated poset by adding right and/or left residuation, both of arity (1,1). The official syntax calls for \(x;y) and /(x;y) in order to indicate that it is antimonotone in x and monotone in y, but we write these standardly as x\y and y/x respectively, remembering the relationship to the official syntax in order to keep track of the arity. The defining inequations are: x;(x\y) < y < x\(x;y) (y/x);x < y < (y;x)/x Had we thought earlier to introduce converse x`, of arity (0,1), satisfying x`` = x (x;y)` = y`;x` we could have dispensed with one of the two equations for 1', and done away with y/x altogether by treating it as just an abbreviation for (x`\y`)`. And we could weaken x;(y;z) = (x;y);z to x;(y;z) < (x;y);z. We may further extend this algebra to include star (ancestral, reflexive transitive closure) exactly as defined in the introduction. Exercise. Define a residuated unital semilattice with converse and star to be the combination of examples 1 and 2. Show in such: x;(y+z) = x;y + x;z (x+y);z = x;z + y;z x;0 = 0 = 0;x (x+y)` = x`+y` x;(y;x)* = (x;y)*;x (x+y)* = (x*;y*)* Now for my main questions. 1. Does this notion of ordered algebra occur in the equational logic or universal algebra literature? 2. What does that literature or any other literature (e.g. the categorical literature) have to say about the following? (i) Semantic completeness. What is the analog for ordered varieties of Birkhoff's HSP theorem for varieties? (ii) Syntactic completeness. If an equational theory Th(V) is exactly a substitutive congruence on an algebra of terms over a set V of variables, what exactly is an inequational theory on an ordered algebra of terms? (Call a congruence ^ *substitutive* when u^v implies s(u)^s(v) for any substitution s, i.e. any endomorphism s:T(V)->T(V).) (iii),(iv) Ditto for ordered quasivarieties. Vaughan Pratt

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by LICS＠B.GP.CS.CMU.EDU 09 Jun '90

09 Jun '90

MAILING LIST FOR LOGIC IN COMPUTER SCIENCE As of Fall 1990 only people on a new LICS database will receive directly announcements about the IEEE Logic in Computer Science Conferences. If you are interested in receiving such information, please fill out the form below and return it by email (or physical mail), by July 8 if possible. If you are NOT INTERESTED, please tell us by return email, so as to spare us the cost of approaching you by physical mail after that date. PLEASE FORWARD this message to potentially interested persons who you suspect might not be on our present mailing list. We apologize for multiple copies of this message. Our use of several mailing lists makes this inevitable. Thank you. lics(a)cs.cmu.edu Logic in Computer Science School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 =================================== NAME (first, middle initial, last): POSTAL ADDRESS: EMAIL ADDRESS(ES): ******* Would you be satisfied with receiving LICS CALLS FOR PAPERS (and similar announcements) via EMAIL ONLY? (yes/no) Would you be satisfied with receiving FINAL PROGRAMS and REGISTRATION FORMS via email only? (yes/no) ?Note: you will need to print registration forms and send them along with a check by PHYSICAL mail?. ******* Optional information (might be used in relation to paper submittals): -------------------- PHONE: FAX/TELEX:

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