Dear category theorists, Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct. For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws. To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F(X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws". You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory. What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory? Thanks, Tom
Hi Tom, a lot is known about this. I will leave it to more qualified others to give the category-theoretic account. In set-like language, the answer to your question is provided by universal algebra. Denote by Th(G) the theory associated to a particular algebra G (over a given signature). More generally, to a class of algebras S (all over the same, from now on fixed, signature), associate Th(S), the theory of all those equations satisfied by all the algebras in S. Also, to a given theory T, let V(T) be the class of all algebras satisfying the equations in T (also called a variety of algebras). Birkhoff's HSP theorem states that a class C of algebras is of the form V(T), for some T, if and only if C is closed under isomorphism, and under the operations of taking quotient algebras, subalgebras, and cartesian products. (HSP stands for "homomorphic image, subalgebra, product"). As a direct consequence, let C=V(Th(G)), the class of all groups satisfying those equations that a particular group G satisfies. Then C is precisely the class of groups that can be obtained, up to isomorphism, from G by repeatedly taking quotients, subalgebras, and cartesian products. [Proof: certainly, the right-hand side is contained in C. Conversely, by the HSP theorem, the right-hand side class is of the form V(T), for some T. Since G is in the class, T can only contain equations that hold in G, thus T is a subset of Th(G). By contravariance of the "V" operation, it follows that C=V(Th(G)) is a subset of V(T)]. Moreover, since a subalgebra of a quotient is a quotient of a subalgebra, and a cartesian product of quotients [subalgebras] is a quotient [subalgebra] of a cartesian product, the three HSP operations can be taken in this particular order: Thus, a group satisfies all the equations that G satisfies, if and only if it is isomorphic to a quotient of a subalgebra of some (possibly infinite) product G x ... x G. There are generalizations to properties other than equational ones, but I don't remember them as well. A "Horn clause" is an implication between equations, or more precisely a property of the form (forall x1...xn)(P1 and ... and Pn => Q), where P1,...,Pn,Q are equations. Of course, every equation is trivially a Horn clause (for n=0), but not the other way around. A typical example of a Horn clause is cancellability, the property xz=yz => x=y. (This holds in groups, but not in monoids, and cannot be expressed equationally in monoids, because it is not preserved under quotients). If you want to consider the class of algebras (in general smaller than V(Th(G))) that satisfy all the Horn clauses that G satisfies, then you have to drop the homomorphic images. I believe that the algebras in question will be precisely the subalgebras of products of G, but someone might correct me if I remember this wrongly. -- Peter Tom Leinster wrote:
Dear category theorists,
Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if
whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct.
For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws.
To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F(X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws".
You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory.
What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory?
Thanks, Tom
Hi,
If you want to consider the class of algebras (in general smaller than V(Th(G))) that satisfy all the Horn clauses that G satisfies, then you have to drop the homomorphic images. I believe that the algebras in question will be precisely the subalgebras of products of G, but someone might correct me if I remember this wrongly.
Isomorphic images of subalgebras of products and ultraproducts, I believe - the standard notation for this seems to be $ISP_U$. Further, there is the interesting result that TH(G) = Th(H) for any free nonabelian groups G and H. The following paper gives a summary of this result and a discussion of equations in free groups: http://www.math.mcgill.ca/olga/V00228H7.pdf best, Jon -- http://rsise.anu.edu.au/~jon
Hi Peter, Thanks a lot. Of all the helpful replies that I received, this was the most helpful. Previously I hadn't even thought about the most basic case of them all, viz. the theory of sets. If G is a set with more than one element than it obeys no equational laws, so Th(G) is again the theory of sets. If G has 0 or 1 elements then it obeys the law "x = y" (and essentially no other laws), so Th(G) is the theory of sets with at most one element. These results square with the HSP theorem (as they must!). Actually, I was aiming at something more general than what I indicated in my original question. I don't think it's the same generalization as passing from equational to Horn properties, but it might be something similar. I'll write a mail to the list about it. All the best, Tom
Hi Tom,
a lot is known about this. I will leave it to more qualified others to give the category-theoretic account. In set-like language, the answer to your question is provided by universal algebra.
Denote by Th(G) the theory associated to a particular algebra G (over a given signature). More generally, to a class of algebras S (all over the same, from now on fixed, signature), associate Th(S), the theory of all those equations satisfied by all the algebras in S. Also, to a given theory T, let V(T) be the class of all algebras satisfying the equations in T (also called a variety of algebras).
Birkhoff's HSP theorem states that a class C of algebras is of the form V(T), for some T, if and only if C is closed under isomorphism, and under the operations of taking quotient algebras, subalgebras, and cartesian products. (HSP stands for "homomorphic image, subalgebra, product").
As a direct consequence, let C=V(Th(G)), the class of all groups satisfying those equations that a particular group G satisfies. Then C is precisely the class of groups that can be obtained, up to isomorphism, from G by repeatedly taking quotients, subalgebras, and cartesian products. [Proof: certainly, the right-hand side is contained in C. Conversely, by the HSP theorem, the right-hand side class is of the form V(T), for some T. Since G is in the class, T can only contain equations that hold in G, thus T is a subset of Th(G). By contravariance of the "V" operation, it follows that C=V(Th(G)) is a subset of V(T)].
Moreover, since a subalgebra of a quotient is a quotient of a subalgebra, and a cartesian product of quotients [subalgebras] is a quotient [subalgebra] of a cartesian product, the three HSP operations can be taken in this particular order: Thus, a group satisfies all the equations that G satisfies, if and only if it is isomorphic to a quotient of a subalgebra of some (possibly infinite) product G x ... x G.
There are generalizations to properties other than equational ones, but I don't remember them as well. A "Horn clause" is an implication between equations, or more precisely a property of the form (forall x1...xn)(P1 and ... and Pn => Q), where P1,...,Pn,Q are equations. Of course, every equation is trivially a Horn clause (for n=0), but not the other way around. A typical example of a Horn clause is cancellability, the property xz=yz => x=y. (This holds in groups, but not in monoids, and cannot be expressed equationally in monoids, because it is not preserved under quotients).
If you want to consider the class of algebras (in general smaller than V(Th(G))) that satisfy all the Horn clauses that G satisfies, then you have to drop the homomorphic images. I believe that the algebras in question will be precisely the subalgebras of products of G, but someone might correct me if I remember this wrongly.
-- Peter
Tom Leinster wrote:
Dear category theorists,
Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if
whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct.
For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws.
To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F(X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws".
You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory.
What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory?
Thanks, Tom
One aspect of the categorisation of "equation" is the approach of Banaschewski and Herrlich. This replaces an equation as a pair (s,t) of terms that belong to some free algebra F by the least quotient (coequaliser) e: F -->> F/E identifying s and t. An algebra A satisfies equation (s,t) precisely when every homomorphism F --> A lifts across e to a homomorphism F/E --> A. Thus the notion of an equation becomes that of a regular epi e with free domain, and an object satisfies such an e when it is injective for e. To obtain a notion intrinsic to a given category, the free domains were replaced by domains that are regular-projective, i.e. projective for all regular epis, this being a property enjoyed by free algebras in categories of universal algebras. The approach has been dualised in the coalgebra literature, with a "coequation" being defined as a regular mono with regular-injective codomain, and a "covariety" as the class of coalgebras that are projective for some given class of coequations. Some (not all) relevant references are below. cheers, Rob @Article{ bana:subc76, author = "B. Banaschewski and H. Herrlich", title = "Subcategories Defined by Implications", journal = "Houston Journal of Mathematics", year = "1976", volume = "2", number = "2", pages = "149--171" } @Article{ adam:vari03, author = {Ji{\v{r}}{\'\i} Ad{\'a}mek and Hans-E. Porst}, title = "On Varieties and Covarieties in a Category", journal = "Mathematical Structures in Computer Science", year = "2003", volume = {13}, pages = "201-232" } @Techreport{ awod:coal00, author = "Steve Awodey and Jesse Hughes", title = "The Coalgebraic Dual of {B}irkhoff's Variety Theorem", institution = "Department of Philosophy, Carnegie Mellon University", year = "2000", number = "CMU-PHIL-109", note = "\url{http://phiwumbda.org/~jesse/papers/ index.html}" } On 8/08/2006, at 1:36 AM, Tom Leinster wrote:
Dear category theorists,
Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if
whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct.
For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws.
To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F (X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws".
You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory.
What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory?
Thanks, Tom
Dear All, What Tom says now, and what Bill calls a simple answer referring to Michel Hebert, also suggests to mention [H. Andréka and I. Németi, Los lemma holds in every category, Stud. Sci. Math. Hung. 13, 1978, 361-376] (although a previous paper of the same authors would be needed, I think). And I am sure many other people also considered many other candidates for the concept of a "law" producing a suitable Galois connection. And - no doubt - many such constructions would produce interesting Galois closed classes. However, I think "the Universal Algebra of 75 years ago" gave a beautiful and fundamental example, were the Galois closed classes are fully and beautifully described (that Galois connection deals with subvarieties of a fixed variety, with fixed "basic operators" and so there is no problem with "What is a variety?" of course). This does not mean that I am trying to argue with Bill: Of course it is true that Bill's thesis was a great further enlightenment, and of course it is true that TODAY seeing only those Galois connections and not seeing adjunctions containing them and much more (also in Galois theory itself!) is too bad! George Janelidze
A simple answer to Tom Leinster's question involves the Galois connection well-analyzed by Michel Hebert at Whitepoint (2006): in a fixed category an object A can "satisfy" a morphism q: F->Q iff q*: (Q,A) -> (F,A) is a bijection. Then for any class of objects A there is the class of "laws" q satisfied by all of them, and reciprocally. If the category itself is mildly exact, one could instead of morphisms q consider their kernels as reflexive pairs. For example, if there is a free notion, a reflexive pair F' =>F has a coequalizer which could be taken as a law q.
However, the "categorical story" that Tom was missing is not told well by the "Universal Algebra" of 75 years ago. Unfortunately, Galois connections in the sense of Ore are not "universal" enough to explicate
Thanks to Bill Lawvere for pointing out the close connection between the questions I was asking and the recent work of Jiri Adamek, Michel Hebert and Lurdes Sousa, presented at both CT06 and the Glasgow PSSL. It must have lodged itself in my mind in some subliminal way; apologies for not mentioning it earlier. Perhaps the following is rather basic, but I'm failing to understand one of the points in Bill's message. As I read it, he's saying that the Galois connection of traditional universal algebra (connecting sets of laws and varieties of algebras) is contained within the structure-semantics adjunction of his thesis. I don't see how this works. I *do* understand the following points: 1. Traditional universal algebra - given a signature S, one has the set E of equations between S-terms and the class V of S-algebras, and the relation "satisfaction" gives a Galois connection between the power-sets/classes P(E) and P(V). A Galois connection is, of course, a contravariant adjunction on the right between posets. 2. Categorical algebra - structure and semantics form a contravariant adjunction on the right between the category Th of Lawvere theories and (roughly speaking) the category K of categories over Set. One is a section of the other: if T is a theory then Struc(Sem(T)) = T. 3. If T is a theory, any equation between the operations in T can be construed as a pair of parallel arrows in T, and so induces a map from T to the quotient theory T' obtained by imposing this equation. Such a map T ---> T' is an epimorphism, although not every epi in the category of theories arises in this way. 4. Given a contravariant adjunction on the right, both functors turn colimits into limits, hence epis into monos. In particular, any set of equations between the operations of a theory T induces an epi T ---> T', hence a mono Sem(T') ---> Sem(T) between the categories of models, which may perhaps be the inclusion of a full subcategory. This makes it look as if there's going to be a Galois connection between the poset of quotient objects of T (i.e. epis out of T) and subobjects of Sem(T), for every theory T. But there seem to be two problems: (i) the functors in an adjunction on the right don't in general turn monos into epis, so I don't see why the structure-semantics adjunction is going to turn subcategories of Sem(T) into quotient theories of T; (ii) even if this did work, epis out of T are more general than equations, and monos into Sem(T) are more general than full subcategories, so it wouldn't exactly recover the classical Galois connection of universal algebra. I guess I've made a wrong turn somewhere; can someone put me right? Thanks, Tom the related universal phenomena in algebra, algebraic geometry, and functional analysis. The mere order-reversing maps between posets of classes are usually restrictions of adjoint functors between categories, and noting this explicitly gives further information. For example, Birkhoff's theorem does not apply well to the question:
"Do groups form a variety of monoids?"
Indeed, does the word "variety" mean a kind of category or a kind of
course the category of groups does become a variety if we adjoin an additional operation to the theory of monoids.
In my thesis (1963) (now available on-line as a TAC Reprint, and extensively elaborated on by Linton and others in SLNM 80) I isolated an adjoint
"Structure/Semantics" strictly analogous to the basic "Function algebra/Spectrum) pairs occurring in algebraic geometry and in functional analysis. In that context, note that the epimorphisms in the category of
given by equations, dual semantically to Birkhoff subvarieties) as well as localizations (laws given by adjoining inverses to previously given operations, semantically corresponding to "open" algebraic subcategories). Can these "open" inclusions between algebraic categories be characterized semantically?
The technical notion "Structure of" was motivated by the example of cohomology operations: in general, the totality of natural operations on
than those of the codomain category. The example illustrates that such adjoints are of much broader interest than the mere perfect duality that one might obtain by restricting both sides (one does not expect to recover a space from its cohomology, and the category of spaces studied is not even an algebraic category).
As an important further example of a large adjoint which specializes both to Galois connections in each space as well as to a perfect duality on suitable subcategories, consider Stone's study of the relation between spaces and real commutative algebras; for computational purposes, the spaces of the form C(X) need to receive morphisms from algebras A (like
inclusion functor? In algebraic geometry, an analogous question concerns whether an algebraic space that is a subspace of another one is closed (i.e. definable by equations) or not. Often instead it is defined by inverting some global functions, giving an open subscheme, not a subvariety, but still a good subspace. The analogy goes still further; a typical open subspace of X is actually a closed subspace of X x R, and of pair theories (categories with finite products) include both surjections (laws the values of a given functor involves both more operations and more laws polynomial algebras) that are not of that form; such homomorphisms are by adjointness equivalent to continuous maps X --> Spec(A), where Spec(A) would map further to R^n if n were a chosen parameterizer for generators of A in a presentation.
Best wishes to all.
Bill
************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************
The thoughts being developed by Tom Leinster give renewed hope that results of 40 years ago are being further developed, beyond mere icons, into tools for actual analysis of algebraic problems *. I should perhaps have mentioned Lourdes Sousa's very interesting talk at White Point. She and Michel Hebert had divided the presentation of the work into existential (or injective) logic, and uniquely existential logic, and the latter is more directly relevant to the present discussion. The fact that a contravariant adjoint pair gives rise to Galois connections at each object is important in many different situations, for example in the classical study of rings of continuous functions on topological spaces. The information inherent in this remark is less visible if one arbitrarily restricts consideration to general epis and general monos (it was here that Tom made a "wrong turn" in his items 4.(i) and 4.(ii) ). To recover the classical Galois connection of universal algebra one must apply adjointness, and then take surjective (or regular epimorphic) images. The above general remark depends on the availability of operations like image in the categories that are being confronted in the adjointness. Again, I emphasize that in the classical case of continuous functions, it is important to consider algebras A which are not of the form C(Y). A possibly useful remark is that, under suitable restrictions on the category of spaces, the Stone-Weierstrass theorem can be interpreted as the statement that A --> C(X) is an epimorphism of rings iff its mate X --> Spec(A) is a monomorphism. But then by restricting the kind of epimorphisms and monomorphisms considered one obtains Galois connections which were originally considered by Stone in the 1930s before the more general functorial formulation which many of us learned from J.L. Kelley's General Topology. If one considers a tractable functor X --> Sem(T) from a very general category X into a very special kind of category of the classical universal algebra type, (i.e. where T is from the special doctrine of categories with finite products, etc.) then the structure functor Str assigns to the composite functor X --> Sets a definite algebraic theory Str(X) with a morphism of theories T -->Str(X). The surjective image of the latter morphism, of course, is the embodiment of the classical construction in the special case where the original tractable functor was a full inclusion. However, there are many full inclusions for which one has X = Sem(Str(X)), i.e. X is an algebraic category even though it is not a variety in Sem(T). Remark: The difference between an object (in this case an algebraic category) and an inclusion map (in this discussion a full inclusion functor) is of course not eliminated by restricting to the case where T is a free theory. It seems reasonable to use the classical term "variety" to refer to those inclusion functors fixed under Birkhoff's Galois correspondence, i.e. to those for which not only is X equal to the semantics of its structure, but for which moreover the morphism of theories T --> Str(X) is already surjective (which of course it is not in the example mentioned of groups in monoids). There are several local studies possible within the context of a given global adjoint. It seems to be an open problem to describe those full inclusions X --> Sem(T) for which T --> Str(X) is a localization (in which case the inclusion, if fixed, might be called "open" in contrast to the Birkhoff subvarieties which are clearly analogous to "closed" subspaces). A further problem: "locally closed" is a kind of inclusion of interest in geometry, so why should it not be also here? * see also the paper "Some Algebraic Problems..." following the Thesis in the TAC Reprints. On Sat, 12 Aug 2006, Tom Leinster wrote:
Thanks to Bill Lawvere for pointing out the close connection between the questions I was asking and the recent work of Jiri Adamek, Michel Hebert and Lurdes Sousa, presented at both CT06 and the Glasgow PSSL. It must have lodged itself in my mind in some subliminal way; apologies for not mentioning it earlier.
Perhaps the following is rather basic, but I'm failing to understand one of the points in Bill's message. As I read it, he's saying that the Galois connection of traditional universal algebra (connecting sets of laws and varieties of algebras) is contained within the structure-semantics adjunction of his thesis. I don't see how this works.
I *do* understand the following points:
1. Traditional universal algebra - given a signature S, one has the set E of equations between S-terms and the class V of S-algebras, and the relation "satisfaction" gives a Galois connection between the power-sets/classes P(E) and P(V). A Galois connection is, of course, a contravariant adjunction on the right between posets.
2. Categorical algebra - structure and semantics form a contravariant adjunction on the right between the category Th of Lawvere theories and (roughly speaking) the category K of categories over Set. One is a section of the other: if T is a theory then Struc(Sem(T)) = T.
3. If T is a theory, any equation between the operations in T can be construed as a pair of parallel arrows in T, and so induces a map from T to the quotient theory T' obtained by imposing this equation. Such a map T ---> T' is an epimorphism, although not every epi in the category of theories arises in this way.
4. Given a contravariant adjunction on the right, both functors turn colimits into limits, hence epis into monos. In particular, any set of equations between the operations of a theory T induces an epi T ---> T', hence a mono Sem(T') ---> Sem(T) between the categories of models, which may perhaps be the inclusion of a full subcategory.
This makes it look as if there's going to be a Galois connection between the poset of quotient objects of T (i.e. epis out of T) and subobjects of Sem(T), for every theory T. But there seem to be two problems:
(i) the functors in an adjunction on the right don't in general turn monos into epis, so I don't see why the structure-semantics adjunction is going to turn subcategories of Sem(T) into quotient theories of T;
(ii) even if this did work, epis out of T are more general than equations, and monos into Sem(T) are more general than full subcategories, so it wouldn't exactly recover the classical Galois connection of universal algebra.
I guess I've made a wrong turn somewhere; can someone put me right?
Thanks, Tom
A simple answer to Tom Leinster's question involves the Galois connection well-analyzed by Michel Hebert at Whitepoint (2006): in a fixed category an object A can "satisfy" a morphism q: F->Q iff q*: (Q,A) -> (F,A) is a bijection. Then for any class of objects A there is the class of "laws" q satisfied by all of them, and reciprocally. If the category itself is mildly exact, one could instead of morphisms q consider their kernels as reflexive pairs. For example, if there is a free notion, a reflexive pair F' =>F has a coequalizer which could be taken as a law q.
However, the "categorical story" that Tom was missing is not told well by the "Universal Algebra" of 75 years ago. Unfortunately, Galois connections in the sense of Ore are not "universal" enough to explicate the related universal phenomena in algebra, algebraic geometry, and functional analysis. The mere order-reversing maps between posets of classes are usually restrictions of adjoint functors between categories, and noting this explicitly gives further information. For example, Birkhoff's theorem does not apply well to the question:
"Do groups form a variety of monoids?"
Indeed, does the word "variety" mean a kind of category or a kind of inclusion functor? In algebraic geometry, an analogous question concerns whether an algebraic space that is a subspace of another one is closed (i.e. definable by equations) or not. Often instead it is defined by inverting some global functions, giving an open subscheme, not a subvariety, but still a good subspace. The analogy goes still further; a typical open subspace of X is actually a closed subspace of X x R, and of course the category of groups does become a variety if we adjoin an additional operation to the theory of monoids.
In my thesis (1963) (now available on-line as a TAC Reprint, and extensively elaborated on by Linton and others in SLNM 80) I isolated an adjoint pair "Structure/Semantics" strictly analogous to the basic "Function algebra/Spectrum) pairs occurring in algebraic geometry and in functional analysis. In that context, note that the epimorphisms in the category of theories (categories with finite products) include both surjections (laws given by equations, dual semantically to Birkhoff subvarieties) as well as localizations (laws given by adjoining inverses to previously given operations, semantically corresponding to "open" algebraic subcategories). Can these "open" inclusions between algebraic categories be characterized semantically?
The technical notion "Structure of" was motivated by the example of cohomology operations: in general, the totality of natural operations on the values of a given functor involves both more operations and more laws than those of the codomain category. The example illustrates that such adjoints are of much broader interest than the mere perfect duality that one might obtain by restricting both sides (one does not expect to recover a space from its cohomology, and the category of spaces studied is not even an algebraic category).
As an important further example of a large adjoint which specializes both to Galois connections in each space as well as to a perfect duality on suitable subcategories, consider Stone's study of the relation between spaces and real commutative algebras; for computational purposes, the spaces of the form C(X) need to receive morphisms from algebras A (like polynomial algebras) that are not of that form; such homomorphisms are by adjointness equivalent to continuous maps X --> Spec(A), where Spec(A) would map further to R^n if n were a chosen parameterizer for generators of A in a presentation.
Best wishes to all.
Bill
************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************
Back from holidays I am slowly working through the various interesting e-mails of Tom Leinster. The examples he presents in his e-mail on August 9 seem to lead to the following question: given on object A characterize the full subcategory L_A of all objects satisfying all "laws" that A satisfies. The algebraic case ("law" meaning equation) has two obvious generalizations: orthogonality and injectivity. For both of them the answer to the above question is nice and easy. INJECTIVITY: let H be the class of morphisms generated by {A} in the Galois connection "to be injective to". (That is, H consists of all morphisms to which A is injective.) The opposite class L_A = Inj H of all objects injective w.r.t. H consists of precisely all split subobjects of powers of A. This holds in every category with powers. Proof: injectivity classes are clearly closed under product and split subobject. Conversely, if B lies in Inj H, then it is a split subobject of the power of A to hom(B,A). In fact, the canonical morphism m from B to the power of A to hom(B,A) lies in H: given a morphism f: B -> A, then f factorizes through m via the projection of A^hom(B,A) corresponding to f. Consequently, B is injective w.r.t. m, and since B is the domain of m, this implies trivially that m is a split mono. ORTHOGONALITY: let K be the class of morphisms generated by {A} in the Galois connection "to be orthogonal to". The opposite class L_A = Ort K of all objects orthogonal to K is the closure of {A} under limits. This holds in every complete and cowellpowered category. Proof: orthogonality classes are clearly closed under limit, thus, Ort K contains the limit closure L of {A}. To prove the opposite inclusion observe that L is a reflective subcategory due to Freyd's SAFT: A is easily seen to be a cogenerator of L. For every object B in Ort K a reflection r: B -> B' in L lies in K (since A lies in L). Thus, B is orthogonal to r. This implies that r is a split mono. Now L contains B' and is closed under split subobjects, thus B lies in L. FINITARY LAWS The algebraic case has another feature: every equation, when translated as injectivity or orthogonality w.r.t. a morphism e:A-> B, has the property that both A and B are finitely presentable. We can thus decide to restrict our attention to finitary morphisms, i.e., morhisms with finitely presentable domains and codomains, as our "laws". If H is the class of all finitary morphisms to which A is injective, then the injectivity class Inj H is the closure of {A} under product, filtered colimit and pure subobject. This was proved by J. Rosicky, F. Borceux and myself in TAC 10 (2002), 148-161. If K is the class of all finitary morphisms to which A is orthogonal, then the orthogonality class Ort K is the closure of {A} under product, filtered colimit and A-pure subobject as proved by L. Sousa and myself in JPAA 276 (2004), 685-705. (The concept of A-pure subobject is a bit artificial, but unfortunately the above result is false if one substitutes it with pure morphism. Surprisingly, when generalizing finitary morphisms to k-ary morphisms for uncountable cardinals k, the corresponding result does hold with pure subobjects: see M. Hebert and J. Rosicky, Bull. London Math. Soc 33 (2001) 685-688.) xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Hi Tom, a lot is known about this. I will leave it to more qualified others to give the category-theoretic account. In set-like language, the answer to your question is provided by universal algebra. Denote by Th(G) the theory associated to a particular algebra G (over a given signature). More generally, to a class of algebras S (all over the same, from now on fixed, signature), associate Th(S), the theory of all those equations satisfied by all the algebras in S. Also, to a given theory T, let V(T) be the class of all algebras satisfying the equations in T (also called a variety of algebras). Birkhoff's HSP theorem states that a class C of algebras is of the form V(T), for some T, if and only if C is closed under isomorphism, and under the operations of taking quotient algebras, subalgebras, and cartesian products. (HSP stands for "homomorphic image, subalgebra, product"). As a direct consequence, let C=V(Th(G)), the class of all groups satisfying those equations that a particular group G satisfies. Then C is precisely the class of groups that can be obtained, up to isomorphism, from G by repeatedly taking quotients, subalgebras, and cartesian products. [Proof: certainly, the right-hand side is contained in C. Conversely, by the HSP theorem, the right-hand side class is of the form V(T), for some T. Since G is in the class, T can only contain equations that hold in G, thus T is a subset of Th(G). By contravariance of the "V" operation, it follows that C=V(Th(G)) is a subset of V(T)]. Moreover, since a subalgebra of a quotient is a quotient of a subalgebra, and a cartesian product of quotients [subalgebras] is a quotient [subalgebra] of a cartesian product, the three HSP operations can be taken in this particular order: Thus, a group satisfies all the equations that G satisfies, if and only if it is isomorphic to a quotient of a subalgebra of some (possibly infinite) product G x ... x G. There are generalizations to properties other than equational ones, but I don't remember them as well. A "Horn clause" is an implication between equations, or more precisely a property of the form (forall x1...xn)(P1 and ... and Pn => Q), where P1,...,Pn,Q are equations. Of course, every equation is trivially a Horn clause (for n=0), but not the other way around. A typical example of a Horn clause is cancellability, the property xz=yz => x=y. (This holds in groups, but not in monoids, and cannot be expressed equationally in monoids, because it is not preserved under quotients). If you want to consider the class of algebras (in general smaller than V(Th(G))) that satisfy all the Horn clauses that G satisfies, then you have to drop the homomorphic images. I believe that the algebras in question will be precisely the subalgebras of products of G, but someone might correct me if I remember this wrongly. -- Peter Tom Leinster wrote:
Dear category theorists,
Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if
whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct.
For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws.
To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F(X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws".
You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory.
What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory?
Thanks, Tom
To respond to Leinster's inquiry, "Laws" (or "equations"), as the set-based universal algebraists understand them, are ordered pairs of members of free algebras (i.e., pairs e = (e_1, e_2) in F x F, for F an algebra free on some set of "free generators." Actually, far more often than not, the variety of algebras these F are free in is presented by means of operations only, and the F are then called "absolutely free." A given equation e "holds" in an algebra A with the given operations iff under each homomorphism from F to A the elements e_1 and e_2 of F are shipped to some same value in A.
From this perspective the Abelianness equation xy=yx is the pair (xy, yx) in F2 x F2 (F2 denoting the absolutely free algebra on the two free generators x & y based on, say, three operations, one binary (multiplication), one unary (inversion), one nullary (choice of base point).
The associativity equation x(yz) = (xy)z is another equation in this sense. One need not, of course, insist dogmatically on taking as equations ONLY pairs in absolutely free algebras: no harm in considering pairs in free algebras of any variety. Thus, for example, (xy, yx) is still a reasonable equation for groups. But (x(yz), (xy)z) doesn't do what you think: the RHS and LHS are ALREADY equal in every group, and the pair is simply the diagonal entry (xyz, xyz) (the INTENDED associativity is already a FACT for groups, not, like commutativity, a condition that, capable of failing, may meaningfully be imposed). If these comments don't fully address the concerns raised, please let me know. In any event, the laws most UAers speak of refer to equations in absolutely free algebras coming from the "lawless" variety whose algebras use the same operations as another variety one is more interested in, but are subject to the imposition of no equations at all. -- Fred Tom Leinster had written:
Dear category theorists,
Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if
whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct.
For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws.
To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F(X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws".
You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory.
What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory?
Thanks, Tom
The following seems so obvious that I suspect it's not what Tom is really asking for; but it seems to me to be an answer to his question. A law in Tom's sense is just a parallel pair of arrows F(X) \rightrightarrows F(1) in the algebraic theory T under consideration (thinking of T as the dual of the category of finitely-generated free algebras). To get the theory of algebras satisfying a given set S of laws, you just need to construct the product-respecting congruence on T generated by S (i.e., the usual closure conditions for a congruence, plus the condition that f ~ f' and g ~ g' imply f x g ~ f' x g'), and factor out by it. Now any T-algebra A (in a category C, say) corresponds to a product- preserving functor F: T --> C; and the set of laws satisfied by A is just the (necessarily product-respecting) congruence generated by F, i.e. the set of parallel pairs in T having the same image under F. Is there anything more to it than that? Peter Johnstone ------------ On Mon, 7 Aug 2006, Tom Leinster wrote:
Dear category theorists,
Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if
whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct.
For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws.
To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F(X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws".
You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory.
What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory?
Thanks, Tom
Dear Tom, "Any theory"?... If it is about Lawvere theories, we go back to classical universal algebra: Let V be a variety of universal algebras, X be a fixed infinite set and F(X) the free algebra on X. A pair (w,w') holds in an algebra A in V if, for every map f : X ---> A, the induced homomorphism f* : F(X) ---> A makes f*(w) = f*(w'); and in this case we write A |= (w,w'). Thus |= becomes a relation between V and F(X)xF(X) (where x is used as the cartesian product symbol). As every relation does, |= determines a Galois connection between the subsets in V and the subsets in F(X)xF(X). Galois closed subsets in V are exactly subvarieties (by definition), and Galois closed subsets in F(X)xF(X) are called algebraic theories. Now, as every universal-algebraist knows, every algebra A in V has its theory T(A) - the one corresponding to the subvariety <A> in V generated by A. By a classical theorem, due to Garrett Birkhoff, <A> is the smallest subclass in V containing A and closed under products, subalgebras, and quotients. Moreover, there is also a well-known completeness theorem for algebraic logic, according to which T(A) can be described directly (i.e. without using any algebras other then A and F(X); in the language of universal algebra it is the fully invariant congruence on F(X) generated by the intersection of all congruences determined by homomorphisms F(X) ---> A). If we now move from classical universal algebra to the more elegant language of Lawvere theories, and begin with such a theory T, then it is better not to fix X and instead of the pairs (w,w') above talk about pairs of parallel morphisms in T - and the story above can be easily modified accordingly. And in the new story T(A) is in fact not set-based anymore: Indeed, if C is a category with finite products, A an internal T-algebra in C, and (t,t') a pair of parallel morphisms in T, then A |= (t,t') should be understood as A(t) = A(t') (elegant indeed!). And then T(A) can be defined as "the largest quotient theory" of T obtained by making t = t' whenever A |= (t,t'). The only thing to have in mind is that not every C is "good enough" to get the "C-completeness" theorem. Moving further from Lawvere theories to other kinds of theories, we will only need to know if "the largest quotient theory" does exist. On the other hand, moving back to, say, classical (non-categorical) first order logic, we are in the well known situation again: if T is a first order theory and A a model of T, everybody knows what is the elementary theory of A. What I do not know is if anyone ever considered any kind of logic (categorical or not) where one cannot do this. I think Michael Makkai is the right person to be asked. Best regards, George ----- Original Message ----- From: "Tom Leinster" <t.leinster@maths.gla.ac.uk> To: <categories@mta.ca> Sent: Monday, August 07, 2006 3:36 PM Subject: categories: Laws
Dear category theorists,
Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if
whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct.
For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws.
To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F(X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws".
You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory.
What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory?
Thanks, Tom
Filtered as Spam Please contact my secretary Alex Judd ajudd[at]ed ac uk [join with dots] with your identity and your email address so I can ensure future mail from you reaches me. On 8 Aug 2006, at 06:08, selinger@mathstat.dal.ca (Peter Selinger) wrote:
Hi Tom,
a lot is known about this. I will leave it to more qualified others to give the category-theoretic account. In set-like language, the answer to your question is provided by universal algebra.
Denote by Th(G) the theory associated to a particular algebra G (over a given signature). More generally, to a class of algebras S (all over the same, from now on fixed, signature), associate Th(S), the theory of all those equations satisfied by all the algebras in S. Also, to a given theory T, let V(T) be the class of all algebras satisfying the equations in T (also called a variety of algebras).
Birkhoff's HSP theorem states that a class C of algebras is of the form V(T), for some T, if and only if C is closed under isomorphism, and under the operations of taking quotient algebras, subalgebras, and cartesian products. (HSP stands for "homomorphic image, subalgebra, product").
As a direct consequence, let C=V(Th(G)), the class of all groups satisfying those equations that a particular group G satisfies. Then C is precisely the class of groups that can be obtained, up to isomorphism, from G by repeatedly taking quotients, subalgebras, and cartesian products. [Proof: certainly, the right-hand side is contained in C. Conversely, by the HSP theorem, the right-hand side class is of the form V(T), for some T. Since G is in the class, T can only contain equations that hold in G, thus T is a subset of Th(G). By contravariance of the "V" operation, it follows that C=V(Th(G)) is a subset of V(T)].
Moreover, since a subalgebra of a quotient is a quotient of a subalgebra, and a cartesian product of quotients [subalgebras] is a quotient [subalgebra] of a cartesian product, the three HSP operations can be taken in this particular order: Thus, a group satisfies all the equations that G satisfies, if and only if it is isomorphic to a quotient of a subalgebra of some (possibly infinite) product G x ... x G.
There are generalizations to properties other than equational ones, but I don't remember them as well. A "Horn clause" is an implication between equations, or more precisely a property of the form (forall x1...xn)(P1 and ... and Pn => Q), where P1,...,Pn,Q are equations. Of course, every equation is trivially a Horn clause (for n=0), but not the other way around. A typical example of a Horn clause is cancellability, the property xz=yz => x=y. (This holds in groups, but not in monoids, and cannot be expressed equationally in monoids, because it is not preserved under quotients).
If you want to consider the class of algebras (in general smaller than V(Th(G))) that satisfy all the Horn clauses that G satisfies, then you have to drop the homomorphic images. I believe that the algebras in question will be precisely the subalgebras of products of G, but someone might correct me if I remember this wrongly.
-- Peter
Tom Leinster wrote:
Dear category theorists,
Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if
whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct.
For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws.
To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F(X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws".
You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory.
What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory?
Thanks, Tom
_______________________________________________ categories mailing list categories@inf.ed.ac.uk http://lists.inf.ed.ac.uk/mailman/listinfo/categories
A simple answer to Tom Leinster's question involves the Galois connection well-analyzed by Michel Hebert at Whitepoint (2006): in a fixed category an object A can "satisfy" a morphism q: F->Q iff q*: (Q,A) -> (F,A) is a bijection. Then for any class of objects A there is the class of "laws" q satisfied by all of them, and reciprocally. If the category itself is mildly exact, one could instead of morphisms q consider their kernels as reflexive pairs. For example, if there is a free notion, a reflexive pair F' =>F has a coequalizer which could be taken as a law q. However, the "categorical story" that Tom was missing is not told well by the "Universal Algebra" of 75 years ago. Unfortunately, Galois connections in the sense of Ore are not "universal" enough to explicate the related universal phenomena in algebra, algebraic geometry, and functional analysis. The mere order-reversing maps between posets of classes are usually restrictions of adjoint functors between categories, and noting this explicitly gives further information. For example, Birkhoff's theorem does not apply well to the question: "Do groups form a variety of monoids?" Indeed, does the word "variety" mean a kind of category or a kind of inclusion functor? In algebraic geometry, an analogous question concerns whether an algebraic space that is a subspace of another one is closed (i.e. definable by equations) or not. Often instead it is defined by inverting some global functions, giving an open subscheme, not a subvariety, but still a good subspace. The analogy goes still further; a typical open subspace of X is actually a closed subspace of X x R, and of course the category of groups does become a variety if we adjoin an additional operation to the theory of monoids. In my thesis (1963) (now available on-line as a TAC Reprint, and extensively elaborated on by Linton and others in SLNM 80) I isolated an adjoint pair "Structure/Semantics" strictly analogous to the basic "Function algebra/Spectrum) pairs occurring in algebraic geometry and in functional analysis. In that context, note that the epimorphisms in the category of theories (categories with finite products) include both surjections (laws given by equations, dual semantically to Birkhoff subvarieties) as well as localizations (laws given by adjoining inverses to previously given operations, semantically corresponding to "open" algebraic subcategories). Can these "open" inclusions between algebraic categories be characterized semantically? The technical notion "Structure of" was motivated by the example of cohomology operations: in general, the totality of natural operations on the values of a given functor involves both more operations and more laws than those of the codomain category. The example illustrates that such adjoints are of much broader interest than the mere perfect duality that one might obtain by restricting both sides (one does not expect to recover a space from its cohomology, and the category of spaces studied is not even an algebraic category). As an important further example of a large adjoint which specializes both to Galois connections in each space as well as to a perfect duality on suitable subcategories, consider Stone's study of the relation between spaces and real commutative algebras; for computational purposes, the spaces of the form C(X) need to receive morphisms from algebras A (like polynomial algebras) that are not of that form; such homomorphisms are by adjointness equivalent to continuous maps X --> Spec(A), where Spec(A) would map further to R^n if n were a chosen parameterizer for generators of A in a presentation. Best wishes to all. Bill ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************
Hi,
If you want to consider the class of algebras (in general smaller than V(Th(G))) that satisfy all the Horn clauses that G satisfies, then you have to drop the homomorphic images. I believe that the algebras in question will be precisely the subalgebras of products of G, but someone might correct me if I remember this wrongly.
Isomorphic images of subalgebras of products and ultraproducts, I believe - the standard notation for this seems to be $ISP_U$. Further, there is the interesting result that TH(G) = Th(H) for any free nonabelian groups G and H. The following paper gives a summary of this result and a discussion of equations in free groups: http://www.math.mcgill.ca/olga/V00228H7.pdf best, Jon -- http://rsise.anu.edu.au/~jon
participants (11)
-
F W Lawvere -
flinton＠wesleyan.edu -
George Janelidze -
Jiri Adamek -
Jon Cohen -
Michael Fourman -
Prof. Peter Johnstone -
Rob Goldblatt -
selinger -
selinger＠mathstat.dal.ca -
Tom Leinster