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ALGEBRAIC LOGIC AND LOGICAL GEOMETRY. TWO IN ONE
B. Plotkin
Institute of Mathematics, Hebrew University, ISRAEL, Professor, plotkin@macs.biu.ac.il
To N. A. Vavilov, a wonderful mathematician and my young colleague, on the occasion of his 60th anniversary
1. Logic for Universal Algebraic Geometry
1.1. Getting started. For me personally, the topic of this paper originates from two main sources. The first one was my interest to mathematical models in knowledge theory, knowledge bases and databases, see, in particular, [6, 5, 7, 9].
Let us describe briefly how the bridge between algebra and knowledge theory works. We consider the following three components of knowledge:
(1) A syntactical part of knowledge, based on a language of the given logic, is the description of knowledge.
(2) The subject of knowledge is an object in the given applied field, i.e., an object for which we determine knowledge. In algebraic terms the subject of knowledge is presented by an algebra H in a variety © or by a model over this algebra.
(3) The content of knowledge (its semantics). Using some abuse of language we can consider the content of knowledge as a reply to the query to a knowledge base.
A certain category of formulas in algebraic logic is related to a knowledge description. We consider a knowledge description as a system of equations or, more generally, a system of formulas. It corresponds the knowledge content, which consists of solutions of the given system. These solutions are presented by points in the corresponding affine space. The category of knowledge content having definable sets in the affine space as objects, is defined in a natural way. Any passage from the knowledge description to the knowledge content is determined by a functor from the first category to the second. This functor depends on the subject of knowledge.
All these notions are defined with respect to some variety of algebras ©. In algebraic setting a variety © is a counter-part of the notion of a data type defined for databases. The described approach to knowledge theory motivates studies in logical geometry. Another inspiration is related to acquaintance with the works of E.Rips and Z.Sela particularly presented at the Amitsur Seminar in Jerusalem, and also with the works of V.Remeslennikov, O.Kharlampovich, A.Myasnikov and others. In these papers algebraic geometry in free groups has been developed. In parallel, within many years I was influenced by general viewpoints of A.Tarski and A.Maltsev on elementary theories of algebras and models.
© B. Plotkin, 2013
Both sources described above gave rise to the idea of universal algebraic geometry (UAG). In UAG we try to transit from classical algebraic geometry associated with the variety of commutative and associative algebras over a field to geometry and logic in an arbitrary variety © and fixed algebra H g ©. The case when © is the variety of groups and H is a free group in © remains of principal importance in UAG. We shall note that the general viewpoint from the positions of UAG to the classical variety provides some new tasks in the classical geometry as well.
1.2. Points and spaces of points. Let an algebra H g © be given. Take a finite set of variables X = {x\,.xn}. Define points as maps of the form j : X ^ H. Each point j determines the sequence (hi,.hn), where hi = j(xi). Since we are working in a given variety ©, one also can define a point j as a homomorphism
j : W (X) ^ H,
where W (X) is the free in © algebra over X.
The set Hom(W(X),H) of all homomorphisms from W(X) to H is regarded as an affine space or, what is the same, a space of points.
Every point defined in such a way has the classical kernel Ker(j) and, as we will see later, the logical kernel LKer(j).
Along with free in © algebras W (X) we consider also algebras of formulas $(X) which are also associated with the given ©. We leave the precise definition of the algebra $(X) till Subsection 1.5. Right now we can note that $(X) is an extended Boolean algebra which means that $(X) is a Boolean algebra with the operations 3x, x g X called existential quantifiers and with nullary operations of the form w = w', where w,w' g W(X), called equalities. There is a bunch of axioms regulating $(X).
We view all formulas u g $(X) as equations. In particular, the formulas of the form w = w', where w,w' g W(X) are equations, since they are the elements of $(X).
Let a point j : W(X) ^ H be given. The logical kernel LKer(^) consists of all formulas u g $(X) valid on the point ^ (see Definition1.6). This is an ultrafilter in the algebra $(X).
1.3. Algebras and categories Hale(H). Algebras from the variety Hale will be the main structures in our setting. We approach to these algebras by introducing the category Hale(H). Let us start to do that.
For each algebra H g © and every finite set X g r consider the algebra Bool(W(X), H). This is the Boolean algebra of subsets of Hom(W(X), H) equipped with quantifiers 3x, x g X and equalities.
Define, first, quantifiers 3x, x g X on Bool(W(X),H). Let A be a set from Bool(W(X), H). We define ^ g 3xA if and only if there exists v g A such that p(y) = v(y) for every y g X, y = x. It can be checked that 3x defined in such a way is, indeed, an existential quantifier.
An equality [w = w']H in Bool(W(X), H) is defined by
¡j, g [w = w']h (w, w') g Ker(p).
For some reason we denote the obtained algebra by Halq (H). This is an example of the extended Boolean algebra in the variety ©. An algebra of formulas $(X) is a structure of this kind.
We will define the algebras $(X) in Subsection 1.5. In Subsection 1.5 we define the important homomorphisms
ValX : $(X) ^ Hal%(H).
Define now the category Hal0(H). Its objects are just defined algebras HalX(H), where H is given and X g r. In order to define morphisms in Halo(H), consider first the category ©(H) of affine spaces.
The objects of ©(H) are affine spaces Hom(W(X),H). Assign to each morphism s : W(X) ^ W(Y) the map s : Hom(W(Y),H) ^ Hom(W(Y),H) defined by the rule s(^) = ¡s : W(X) ^ H, for ¡i : W(Y) ^ H. These s are morphisms in ©(H).
The correspondence W(X) ^ Hom(W(X),H) and s ^ s defines a contravariant functor ©0 ^ ©(H) which determines duality.
Theorem 1.1. The categories ©0 and ©(H) are dually isomorphic under this functor if and only if Var(H) = ©.
Morphisms s^ in Hale(H) are defined as follows. Every homomorphism s : W(X) ^ W (Y) gives rise to a Boolean homomorphism
sf : Bool(W(X), H) ^ Bool(W(Y), H),
defined by the rule: for each A C Hom(W(X), H) the point ¡i belongs to s*A if s(^) = ¡s g A.
The defined category HoJq(H) can be treated as a multi-sorted algebra Hale(H) = (HalX(H), X g r), with objects as domains and morphisms
sf : Bool(W(X), H) ^ Bool(W(Y), H),
as operations.
The algebra of formulas $ = Hal°0 = ($(X), X g r) is defined in a similar way.
1.4. Signature of algebras Hal0(H). Our next aim is to describe the signature of operations for the multi-sorted algebras Hal0(H). This signature should be also multi-sorted.
Consider an arbitrary W(X) in ©, and take the signature
LX = {V, A, -., 3x, MX}, for all x g X.
Here MX is the set of all formulas w = w', w,w' g W (X) over the algebra W (X). We treat these formulas as symbols of relations of an equality over W(X), that is there is a map =: W (X) x W (X) ^ MX which satisfies axioms of an equational predicate on W(X). These are the only symbols of relations in use. Symbols w = w' can be regarded also as symbols of nullary operations.
Signature LX is the signature of the one-sorted extended Boolean algebras. Now we define the signature L0 for the multi-sorted algebras Hal0 (H) .
Along with the set of symbols of equalities MX consider the set SX,Y of symbols of operations s* of the type t = (X; Y), where X,Y g r. Symbols s* are just symbols of
operations but we keep in mind that each homomorphism s : W (X) ^ W (Y) induces the operation s+ in Hale(H) of the type t = (X; Y).
By the same reason we assume that given s : W(X) ^ W(Y) and s' : W(Y) ^ W(Z), the axiom
(ss )* — s*st
holds. Here the operation (ss')* has the type t = (X; Z). Define the signature
Le = {lx,sx,y ; X, Y g T]
The signature Le is multi-sorted. We consider the constructed multi-sorted algebras Hale(H) in this signature with the natural realization of all operations from Le. We will take Le also as the signature of an arbitrary algebra from the variety of multi-sorted algebras Hale .
Now we define algebras which belong to the variety Hale.
Definition 1.2. We call an algebra L = (LX,X g r) in the signature Le a Halmos algebra, if
(1) Every domain lx is an extended Boolean algebra in the signature Lq .
(2) Every mapping s* : lx ^ ly is a homomorphism of Boolean algebras. Let s :
W(X) ^ W(Y), s' : W(Y) ^ W(Z), and let u g lx. Then s*(s* (u)) = (s's)*(u).
(3) The identities controlling the interaction of s* with quantifiers are as follows:
(a) si*3xa, = s2*3xa, a g L(X), if si(y) = s2(y) for every y = x, x, y g X.
(b) s*3xa, = 3(s(x))(s*a), a g L(X), if s(x) = y and y is a variable which does not belong to the support of s(x'), for every x' g X, and x' = x.
This condition means that y does not participate in the shortest expression of the element s(x') g W (Y).
(4) The identities controlling the interaction of s* with equalities are as follows:
(a) s*(w = w') = (s(w) = s(w')).
(b) (sW)*a A (w = w') < (sW)*a, where a g L(X), and sW g End(W(X)) is defined by sW (x) = w, and sW (x') = x', for x' = x.
One should not be upset with the looking complicated axioms from the items 3-4. First of all, we have already algebras Hale(H) as an example of Halmos algebras, So, one can verify in the very straightforward way that Hale(H) satisfy these axioms. Second, the purely logical explanations on the language of first order formulas of the similar axioms for the one-sorted polyadic algebras are contained in [1], see also [5].
Definition 1.3. The variety Hale consists of the multi-sorted algebras in the signature Le subject to axioms from Definition 1.2.
The conditions specified in Definition 1.2 have the form of identities and they actually define a variety.
1.5. Free algebras in Hal0. We construct the free in Hal0 algebra $ in an explicit way. Denote by M = (MX ,X g r) the multi-sorted set of equalities with the components mx .
Let us build the absolutely free algebra of formulas in the signature LQ. Each formula in this algebra has two parameters: the length and the sort. We define formulas by induction.
Each equality w = w' is a formula of the length zero, and of the sort X if w = w' g mx . Let u be a formula of the length n and the sort X. Then the formulas —u and 3xu are the formulas of the same sort X and the length (n + 1). Further, for the given s : W(X) ^ W(Y) we have the formula s*u with the length (n + 1) and the sort Y. Let now ui and U2 be formulas of the same sort X and the length ni and n2 accordingly. Then the formulas ui V u2 and ui A u2 have the length (ni + n2 + 1) and the sort X. In such a way, by induction, we define lengths and sorts of arbitrary formulas.
We construct a big set formulas L0.
Let LX be the set of all formulas of the sort X. Each LX is an algebra in the signature LX and we have the algebra
L0 = (L0X ,X g r)
in the signature Lq. By construction, the algebra L0 is the absolutely free algebra of formulas over equalities M = (MX ,X g r) concerned with the variety of algebras ©. Its elements are considered as pure formulas.
Denote by n the verbal congruence in L0 generated by the identities of Halmos algebras from Definition 1.2.
Define the Halmos algebra of formulas as
$ = L0/Sr.
It can be written as $ = ($(X), X g r), where
$(X) = LX/Sx,
where each $(X) is an extended Boolean algebra of the sort X in the signature LX.
The algebra $ is the free algebra in the variety HalQ of all multi-sorted Halmos algebras associated with the variety of algebras ©, with the set of free generators M = (mx,X g r).
This approach to the free in HalQ algebra through the factorization of the absolutely free algebra of formulas by the verbal congruence can be viewed as a syntactic approach.
We could built $ also semantically. In each LX the formulas of the sort X are collected. Recall that for s : W(X) ^ W(Y) and u g LX the formula v = s*u lies in LX. All these formulas can be treated as pure formulas of a logic which possesses some axioms and rules of inference which correlate with the definition of the variety HalQ.
One can show that if we factor out component-wisely the algebra L0 by the many-sorted Lindenbaum-Tarski congruence, then we get the same algebra $. This observation provides a bridge between syntactical and semantical description of the free multi-sorted Halmos algebras.
Using the correspondence between multi-sorted algebras and categories we are able to define the category of algebras of formulas HalQ. Its objects are algebras $(X), its
morphisms have the form s* : $(X) ^ $(Y), where s : W(X) ^ W(Y) is a morphism in ©0. According to Definition 1.2 the morphisms s* preserve the Boolean structure of $(X) and correlated with quantifiers and equalities. Hence the correspondence
W (X) ^ $(X) and s ^ s*
determines a covariant functor a : ©0 ^ Hale.
1.6. Variety Hale and algebras Hale(H). As we have seen, the algebras Hale(H) belong to Hale. Moreover,
Theorem 1.4. Algebras Hale(H) generate the variety Hale-
This means that we could define the algebra $ using algebras Hale(H). We have a unique homomorphism of the algebra L to Hale(H). The kernel of this homomorphism is the system of identities of Hale(H). Intersection of all kernels through all H g © and coincide with the verbal congruence with respect to the variety Hale
One can prove that these algebras are simple with respect to congruences and all simple algebras are exhausted by algebras Hale(H) and their subalgebras.
1.7. The Value homomorphism.. The free algebra $ = ($(X), X g r) and an arbitrary algebra Hale(H) = (HalQ(H), X g r) belong to the same variety Hale. We will define a homomorphism
ValH : $ ^ Hale(H),
which induces the homomorphism
ValX : $(X) ^ HalXX(H)
of the one-sorted extended Boolean algebras, for every X in r.
Since equalities M = (MX ,X g r) freely generate the free multi-sorted Halmos algebra $, it is enough to assign an equality in HalQX (H) to the corresponding equality w = w' in $(X).
Recall that we have defined equalities [w = w']h in HalQX (H) by
j g [w = w']h ■ (w, w') g Ker(j).
Hence, the element [w = w']H in HalQ(H) is assigned to the element w = w' in $(X). This correspondence gives rise to the homomorphism of multi-sorted Halmos algebras
ValH : $ ^ Hale(H).
Since $ = ($(X),X g r), where each component $(X) is an extended Boolean algebra, the homomorphism ValH induces homomorphisms
ValX : $(X) ^ HalQX(H),
of the one-sorted extended Boolean algebras. In particular,
ValQ(w = w') = {j | j(w) = j(w')].
Definition 1.5. A point j : W(X) ^ H satisfies the formula u g $(X) if ValX(u) contains j.
This definition has the same meaning as the standard model theoretic one. Now we are in a position to define formally the logical kernel of a point.
Definition 1.6. A formula u g $(X) belongs to the logical kernel LKer(j) of a point j if and only if j g Val'X (u).
If u g $(X) then ValX (u) is the set of points j satisfying the formula u. This means
that
u g LKer(j) ■ j g ValX (u).
Since we consider each formula u g $(X) as an "equation" and VaH (u) as the value of the formula u in the algebra Bool(W(X),H), then ValX(u) is a set of points j : W(X) ^ H satisfying the "equation" u. We call ValX(u) solutions of the equation u. We also say that the formula u holds true in the algebra H at the point j.
It can be verified that the logical kernel LKer(j) is always a Boolean ultrafilter of $(X).
Note that
Ker(j) = LKer(j) n MX.
Now let Th(H) = (ThX (H), X g r) be the multi-sorted representation of the elementary theory of H. We call its component ThX (H) the X-theory of the algebra H. Since Ker(ValH) is the set of formulas satisfied by all points of H, we have have:
Ker(ValH) = Th(H),
Ker(ValX) = ThX (H).
This means, in particular, that the algebra $(X) can be represented, modulo elementary theory, in the more transparent algebra HalX (H). We can also present the X-theory of the algebra H as:
ThX (H) = P| LKer(j).
¡j.:W (X)^H
Definition 1.7. An algebra H is called saturated if for every finite X every ultrafilter T in $(X), which contains ThX(H) coincides with a Lker(j) for some j : W(X) ^ H.
This notion stimulates a lot of problems.
The key diagram which relates logic of different sorts in multi-sorted case is as follows:
$(X) S* : $(Y)
ValX
ValY
HalX (H) s" : HalQ(H)
Here the upper arrow represent the syntactical transitions in the category Halo, the lower level does the same with the respect to semantics in Halo, and the correlation is provided by the vertical value homomorphism.
With this diagram we finish exposition of the necessary ideas from algebraic logic and switch to a logically-geometric stuff. We defined formally the multi-sorted algebra
of formulas $ and its domains $(X), where X g r. These domains could be informally treated as dynamic algebras of formulas, which means that formulas (elements) from a particular $(X) interact with formulas from other $(Y).
2. Some results and problems
2.1. Types and isotypeness. In model theory for each set X, X = {xi,...,xn] the notion of X-type (n-type) is defined (see [2]). Given an algebra H with the elementary theory Th(H), a set P of formulas u, such that all free variables in u belongs to X = {x\,..., xn] is an X-type (n-type), if P U Th(H) is satisfiable. Denote by TH(j) the X-type of the point j : W(X) ^ H, i.e., the set of all formulas u valid on j. The types of the form TH(j) will be called MT-types or model-theoretic types of points.
We consider also LG-types or logically-geometric types of the points j : W(X) ^ H. According to Definition 6.2 from [10] an LG-type of a point j is the logical kernel LKer(j) of the point j in the algebra $(X). The following result connects MT-types and LG-types, and plays a key role in all considerations:
Theorem 2.1 ([11]). Let the points j : W(X) ^ Hx and v : W(X) ^ H2 be given. The equality
TH (j)= TH2 (v)
holds if and only if we have
LKer(j) = LKer(v).
Now we will describe the idea of isotypeness of algebras.
Definition 2.2. Two algebras Hi and H2 are called isotypic if for every X and every point j : W (X) ^ Hi there exists a point v : W (X) ^ H2 such that the types of j and v coincide, and for every point v : W (X) ^ H2 there exists a point j : W (X) ^ Hi such that the types of j and v coincide.
• - In view of Theorem 2.1 two algebras Hi and H2 are isotypic with respect to MT-types (MT-isotypic) if and only if they are isotypic with respect to LG-types (LG-isotypic). Thus, in Definition 2.2 one can equally rely on MT-types and LG-types.
• - Since in the definition of isotypeness one can grounds on coincidence of logical kernels of the points, then this definition has a geometric nature and extends the notion of geometrically equivalent algebras. One can say that Theorem 2.1 visualizes the bridge between geometrical and logical ideas.
• - According to Definition 2.2 isotypeness of algebras implies their elementary equivalence. So, this notion is more strong than the notion of elementary equivalence and expresses a logical property of algebras which should in many cases be closed to isomorphism.
Indeed, it is easy to see that if algebras Hi and H2 are isotypic, then they are locally isomorphic. This means that every finitely generated subalgebra in Hi is isomorphic to a finitely generated subalgebra in H2 and vice versa. Of course, locally isomorphic groups are not necessarily isomorphic and isotypic. For example every two free groups Fm and Fn are locally isomorphic, but not isomorphic provided m = n. As we will see Fm and Fn are not also isotypic. This follows from the result of C. Perin and R. Sklinos [3] on logical homogenity of a free group.
Moreover, it easy to see that if Hi = Fn is a finitely generated free group and H2 is a finitely generated group, then their isotypeness implies isomorphism. This follows from the local isomorphism of Hi and H2, which means that H2 can be viewed as a subgroup of Hi. Since every subgroup of Fn is free, it remains to use the above mentioned result on isotypeness of free finitely generated groups. We come up with the following conjecture:
Conjecture 2.3. Let Fn be a free group of the rank n > 1 and H be a group. If Fn and H are isotypic then they are isomorphic.
Recently, R. Sklinos* gave a positive answer to this problem. His proof is also based on the logical homogenity of a free group in the form of the following theorem of Pillay:
Theorem 2.4 (Pillay). Let Fn be the free group with free generators ei,...,en. Consider the points j : W(X) ^ Fn and v : W(X) ^ Fn defined by j(xi) = ei and v(xi) = ai, respectively, where i = l,...n, and ai are arbitrary elements in Fn. Suppose that
TH (j)= TH (v).
Then ai,... ,an are the free generators of Fn.
We see that all groups isotypic to a free finitely generated group Fn should be iso-morphic to Fn. The next problems are related to the general case:
Problem 2.5. Let the groups Hi and H2 be isotypic and Hi be finitely generated. Is it true that H2 is finitely generated?
Problem 2.6. Let Hi and H2 be two finitely generated isotypic groups. Are they isomor-phic?
Since all the theory grounds on an arbitrary variety of algebras Problems 2.6 and 2.5 which are formulated for the variety of all groups make sense for an arbitrary variety of algebras ©. Their solution heavily depends on the choice of ©. So, let Hi and H2 be two algebras from a variety ©.
Problem 2.7. Let the algebras Hi and H2 be isotypic and Hi be finitely generated. Is it true that H2 is finitely generated?
Problem 2.8. Let Hi and H2 be two finitely generated isotypic algebras. Are they iso-morphic?
Let us point out one more closely related question
Problem 2.9. Let Hi and H2 be two finitely generated isotypic groups. What can be said about isotypeness of the groups algebras KHi and KH2 where K is a field.
As we have mentioned above a local isomorphism does not imply isotypeness. Here is an example when local isomorphism of algebras implies isotipicity:
Example 2.10. Any two infinite dimensional vector spaces Hi and H2 are locally isomorphic. It can checked that their local isomorphism implies isotypeness of Hi and H2. Take now two non-isomorphic infinitely dimensional vector spaces. Then, in view of above they provide an example of isotypic, locally isomorphic but not isomorphic algebras.
•"Unpublished.
References
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2. Marker D. Model Theory: An Introduction. Springer Verlag, 2002.
3. Perin C., Sklinos R. Homogenety in the free group // Duke Mathematical Journal, to appear. Preprint arXiv: 1003.4095v1 [math.GM].
4. Pillay A. On genericity and weight in the free group // Proc. Amer. Math. Soc. 2009. Vol. 137. P. 3911-3917.
5. Plotkin B. Universal algebra,algebraic logic and databases. Kruwer Acad. Publ., 1994.
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8. Plotkin B. Algebras with the same algebraic geometry // Proceedings of the International Conference on Mathematical Logic, Algebra and Set Theory, dedicated to 100 anniversary of P. S. Novikov. Proceedings of the Steklov Institute of Mathematics. 2003. Vol. 242. P. 176-207. Arxiv: math.GM/0210194.
9. Plotkin B., Plotkin T. Geometrical aspects of databases and knowledge bases // Algebra Universalis. 2001. Vol.46, issue 1-2. P. 131-161.
10. Plotkin B., Aladova E., PlotkinE. Algebraic logic and logically-geometric types in varieties of algebras // Journal of Algebra and its Applications. 2013. Vol.12, no. 2.
11. Sklinos R., Unpublished.
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