Multi-sorted logic and logical geometry: some problems

The paper has a form of a survey and consists of three parts. It is focused on the relationship between the many-sorted theory, which leads to logical geometry and one-sorted theory, which is based on the important model-theoretic concepts. Our aim is to show that both approaches go in parallel and there are bridges which allow to transfer results, notions and problems back and forth. Thus, an additional freedom in choosing an approach appear.


Introduction
The paper has a form of a survey talk on the given topic. This second paper continues the first one [29]. It consists of three parts, ordered in a way different from that of [29]. The accents are also different. This paper is focused on the relationship between the many-sorted theory, which leads to logical geometry and one-sorted theory, which is based on the important model-theoretic concepts. Our aim is to show that both approaches go in parallel and there are bridges which allow to transfer results, notions and problems back and forth. Thus, an additional freedom in choosing an approach appear.
On our opinion, some simple proofs make the paper more vital. The first part of the paper contains main notions, the second one is devoted to logical geometry, the third part describes types and isotypeness. The problems are distributed in the corresponding parts. The whole material is oriented towards universal algebraic geometry (UAG), i.e., geometry in an arbitrary variety of algebras Θ. We will distinguish between the equational algebraic geometry and the logical geometry. In the equational geometry equations have the form w ≡ w ′ , where w and w ′ are elements of the free in Θ algebra W (X). In the logical geometry the elements of the multi-sorted first-order logic play the role of equations. We consider logical geometry (LG) as a part of UAG. This theory is strongly influenced by model theory and ideas of A.Tarski and A.I.Malcev.
I remember that A.I. Malcev, founding the journal "Algebra and logic" in Novosibirsk, had in mind a natural interrelation of these topics.
We fix a variety of algebras Θ. Let W = W (X) be the free in Θ algebra over a set of variables X. The set X is assumed to be finite, if the opposite is not stated explicitly. In the latter case we use the notation X 0 . All algebras under consideration are algebras in Θ. Logic is also related to the variety Θ. As usual, the signature of Θ may contain constants.

Main notions
In this section we consider a system of notions, we are dealing with. Some of them are not formally defined in this paper. For the precise definitions and references use [21], [7], [22], [27], [31], [17].
The general picture of relations between these notions brings forward a lot of new problems, formulated in the following two sections. These problems are the main objective of the paper. Some results are also presented.
2.1. Equations, points, spaces of points and algebra of formulas Φ(X). Consider a system T of equations of the form w = w ′ , w, w ′ ∈ W (X).
Each system T determines an algebraic set of points in the corresponding affine space over the algebra H ∈ Θ for every H and every finite X.
Let X = {x 1 , . . . , x n }. We have an affine space H X of points µ : X → H. For every µ we have also the n-tuple (a 1 , . . . , a n ) =ā with a i = µ(x i ). For the given Θ we have the homomorphism µ : W (X) → H and, hence, the affine space is viewed as the set of homomorphisms Hom(W (X), H).
Every point µ has also the logical kernel LKer(µ). Along with the algebra W (X) we will consider the algebra of formulas Φ(X). Logical kernel LKer(µ) consists of all formulas u ∈ Φ(X) valid on the point µ.
The algebra Φ(X) will be defined later on, but let us note now that it is an extended Boolean algebra (Boolean algebra in which quantifiers ∃x, x ∈ X act as operations, and equalities (Θ-equalities) w ≡ w ′ , w, w ′ ∈ W (X) are defined). It is also defined what does it mean that the point µ satisfies a formula u ∈ Φ(X). These u are treated as equations. For T ⊂ Φ(X) in Hom(W (X), H) we have an elementary set (definable set) consisting of points µ which satisfy every u ∈ T .
Each kernel LKer(µ) is a Boolean ultrafilter in Φ(X). Note that where M X is the set of all w ≡ w ′ , w, w ′ ∈ W (X).
Here the numerals 0 and 1 are zero and unit of the Boolean algebra B and a, b are arbitrary elements of B.
As usual, the quantifiers ∃ and ∀ are coordinated by: ¬(∃a) = ∀(¬a), and (∀a) = ¬(∃(¬a)). Now suppose that a variety of algebras Θ is fixed and W (X) is the free in Θ algebra over the set of variables X. These data allow to define the extended Boolean algebra. This is a Boolean algebra where the quantifiers ∃x are defined for every x ∈ X and ∃x∃y = ∃y∃x, for every x and y from X. Besides that, for every pair of elements w, w ′ ∈ W (X) in an extended Boolean algebra the equality w ≡ w ′ is defined. These equalities are considered as nullary operations, that is as constants. Each equality satisfies conditions of an equivalence relation, and for every operation ω from the signature of algebras from Θ we have

Note that quantifiers and Boolean connectives satisfy
Algebra of formulas Φ(X) is an example of extended Boolean algebra in Θ. Now consider another example.
2.3. Important example. Let us start from an affine space Hom(W (X), H). Let Bool(W (X), H) be a Boolean algebra of all subsets in Hom(W (X), H). Extend this algebra by adding quantifiers ∃x and equalities. For A ∈ Bool(W (X), H) we set: B = ∃xA is a set of points µ : W (X) → H such that there is ν : It is indeed an existential quantifier for every x ∈ X.
Define an equality in Remark 2.1. The set [w ≡ w ′ ] H can be empty. Thus we give the following definition. The equality [w ≡ w ′ ] H is called admissible for the given Θ if for every H ∈ Θ the set [w ≡ w ′ ] H is not empty. If Θ is the variety of all groups, then each equality is admissible. The same is true for the variety of associative algebras with unity over complex numbers. However, for the field of real numbers this is not the case. Here x 2 + 1 = 0 is not an admissible equality.
We assume that in each algebra of formulas Φ(X) lie all Θ-equalities. To arbitrary equality w ≡ w ′ corresponds either a non-empty equality [w ≡ w ′ ] H in H ∈ Θ, or the empty set in H ∈ Θ which is the zero element of this Boolean algebra.
We arrived to an extended Boolean algebra, denoted now by Hal X Θ (H). This algebra and the algebra of formulas Φ(X) have the same signature.
2.4. Homomorphism V al X H . We will proceed from the homomor- non-empty, or 0 otherwise. This homomorphism will be defined in subsection 2.9. The existence of such homomorphism is not a trivial fact, since the equalities M X does not generate (and of course does not generate freely) the algebra Φ(X). If, further, u ∈ Φ(X), then V al X H (u) is a set of points in the affine space Hom(W (X), H). We say that a point µ satisfies the formula u if µ belongs to V al X H (u). Thus, V al X H (u) is precisely the set of points satisfying the formula u. Define the logical kernel LKer(µ) of a point µ as a set of all formulas u such that µ ∈ V al X H (u). We have also Ker(µ) = LKer(µ) ∩ M X .
Here Ker(µ) is the set of all formulas of the form w = w ′ , w, w ′ ∈ W (X), such that the point µ satisfies these formulas. In parallel, LKer(µ) is the set of all formulas u, such that the point µ satisfies these formulas. Then, Here , H) and thus T h X (H) is an X-component of the elementary theory of the algebra H.
In general we have a multi-sorted representation of the elementary theory It follows from the previous considerations that the algebra of formulas Φ(X) can be embedded in Hal X Θ (H) modulo elementary theory of the algebra H. This fact will be used in the sequel.

2.5.
Multi-sorted logic: first approximation. Let, further, X 0 be an infinite set of variables and Γ a system of all finite subsets X in X 0 .
So, in the logic under consideration we have an infinite system Γ of finite sets instead of one infinite X 0 . This leads to multi-sorted logic. This approach is caused by relations with UAG. We distinguish UAG, equational UAG and LG in UAG. Correspondingly, we have algebraic sets of points and definable sets of points in the affine space. In the third part of the paper along with the system of sorts Γ we use also a system of sorts Γ where one initial infinite set X 0 is added to the system Γ.
2.6. Algebra Hal Θ (H). All these algebras and corresponding categories present universal semantics for the logic concerning with a variety Θ. Syntax of this logic is given by the algebraΦ. The homomorphism V al H :Φ → Hal Θ (H) gives the correspondence between syntax and semantics. This homomorphism and the homomorphism V al X H :Φ(X) → Hal X Θ (H) will be defined at the end of the section.
We start with the category Θ * (H) of affine spaces. Its objects are spaces Hom(W (X), H), where X ∈ Γ. Morphismss and s act in the opposite direction. Note that if s is surjective thens is injective, and if s is injective thens is surjective. Proof. The condition of duality implies that if s 1 = s 2 for the given morphisms s 1 , s 2 : W (Y ) → W (X) then s 1 = s 2 .
Let us assume that V ar(H) = Θ and the categories are not dual, so there are morphisms s 1 and s 2 such that s 1 = s 2 but s 1 = s 2 . Take some y ∈ Y such that s 1 (y) = w 1 , s 2 (y) = w 2 and w 1 = w 2 . We will show that in the algebra H there is the non-trivial identity w 1 ≡ w 2 . Take an arbitrary homomorphism ν : W (X) → H. The equalitys 1 =s 2 impliess 1 (ν) =s 2 (ν) or νs 1 = νs 2 . We apply this morphism to the variable y: νs 1 (y) = νs 2 (y) or νw 1 = νw 2 .
Since ν : W (X) → H is an arbitrary homomorphism, then w 1 ≡ w 2 is an identity of the algebra H. But V ar(H) = Θ, which means that there are no non-trivial identities in H. We have a contradiction and the condition V ar(H) = Θ implies duality of the given categories. Now we show that if V ar(H) ⊂ Θ, then there is no duality. Let w 1 ≡ w 2 be some non-trivial identity of the algebra H. Take Y = {y 0 } and let s 1 (y 0 ) = w 1 , s 2 (y 0 ) = w 2 . For any ν : W (X) → H we have Since the set Y contains only one element y 0 , then s 1 (ν) = s 2 (ν). As ν is arbitrary, then s 1 = s 2 and there is no duality of the categories.
Define further a category of all Hal Θ (H). Its objects are algebras Hal X Θ (H). Proceed from s : W (X) → W (Y ) and pass tos : for every object A ⊂ Hom(W (X), H). We have µ ∈ B if and only if µs =s(µ) ∈ A. This determines a morphism Here s * is well coordinated with the Boolean structure, and relations with quantifiers and equalities are coordinated by identities from Definition 2.3. The category Hal Θ (H) can be also treated as a multi-sorted algebra Hal Θ (H) = (Hal X Θ (H), X ∈ Γ).

2.7.
Variety of Halmos algebras Hal Θ . Algebras in Hal Θ have the form L = (L X , X ∈ Γ). Here all domains L X are X-extended Boolean algebras. The unary operation s * : L X → L Y corresponds to each homomorphism s : W (X) → W (Y ). Besides, we will define a category L of all L X , X ∈ Γ with morphisms s * : L X → L Y . The transition s → s * determines a covariant functor Θ 0 → L. Informally, operations of s * type make logics dynamical.
Every L X is an X-extended Boolean algebra. Denote its signature by Here M X stands for the set of all symbols of relations of equality of the form w ≡ w ′ .
Denote by S X,Y the set of symbols of operations s * of the type τ = (X; Y ), where X, Y ∈ Γ. Define the signature The signature L Θ is multi-sorted. We take L Θ as the signature of an arbitrary algebra from the variety of multi-sorted algebras Hal Θ . The constructed multi-sorted algebras Hal Θ (H) possess this signature with the natural realization of all operations from L Θ .
There is a bunch of axioms which determine algebras from the variety Hal Θ . For example, every s * respects Boolean operations in L X and L Y . Correlations of s * with equalities and quantifiers are described by more complex identities. Below we give the complete list of axioms for Hal Θ (see also [31], [29]). Definition 2.3. We call an algebra L = (L X , X ∈ Γ) in the signature L Θ a Halmos algebra, if (1) Every domain L X is an extended Boolean algebra in the signature L X .
(b) s * ∃xa = ∃(s(x))(s * a), a ∈ L(X), if s(x) = y and y is a variable which does not belong to the support of s(x ′ ), for every x ′ ∈ X and x ′ = x. This condition means that y does not participate in the shortest expression of the element s(x ′ ) ∈ W (Y ). (4) Conditions controlling the interaction of s * with equalities are as follows: (a) s * (w ≡ w ′ ) = (s(w) ≡ s(w ′ )).
Remark 2.4. We should note that all conditions from the definition of a Halmos algebra can be represented as identities, and this is why the class of Halmos algebras is indeed a variety.
Define Hal Θ to be the variety of all Halmos algebras, that is every algebra from Hal Θ satisfies Definition 2.3. In view of Theorem 2.6 one could define from the very beginning the variety Hal Θ as the variety, generated by all algebras Hal Θ (H).
Recall, that every ideal of an extended Boolean algebra is a Boolean ideal invariant with respect to the universal quantifiers action. Extended Boolean algebra is called simple if it does not have non-trivial ideals. In the multi-sorted case an ideal is a system of one-sorted ideals which respects all operations of the form s * . A multi-sorted Halmos algebra is simple if it does not have non-trivial ideals. Algebras Hal Θ (H) and their subalgebras are simple Halmos algebras, see [32]. Moreover, these algebras are the only simple algebras in the variety Hal Θ . Finally, every Halmos algebra is residually simple, see [32]. This fact is essential in the next subsection. Note, that all these facts are true because of the clever choice of the identities in the variety Hal Θ .
2.8. Multi-sorted algebra of formulas. We shall define the algebra of formulas Φ = (Φ(X), X ∈ Γ). We define this algebra as the free over the multi-sorted set of equalities M = (M X , X ∈ Γ) algebra in Hal Θ . Assuming this property denote it as Hal 0 Θ = (Hal X Θ , X ∈ Γ). So, Hal X Θ = Φ(X) and Φ = Hal 0 Θ . In order to define Hal 0 Θ we start from the absolutely free over the same M algebra L 0 = (L 0 (X), X ∈ Γ).
This free algebra is considered in the signature of the variety Hal Θ . Algebra L 0 can be viewed as the algebra of pure formulas of the corresponding logical calculus. Then, Φ is defined as the quotient algebra of L 0 modulo the verbal congruence of identities of the variety Hal Θ . The same algebra Φ can be obtained from L 0 using the Lindenbaum-Tarski approach. Namely, basing on identities of Hal Θ we distinguish in L 0 a system of axioms and rules of inference. For every X ∈ Γ consider the formulas where u, v ∈ L 0 (X). Here u → v means ¬u ∨ v. We assume that every is deducible from the axioms if and if the pair (u, v) belongs to the X-component of the given verbal congruence.
So, Φ can be viewed as an algebra of the compressed formulas modulo this congruence. Note that this homomorphism is a unique homomorphism from Φ → Hal Θ (H), since equalities are considered as constants.
We have V al X H : Φ(X) → Hal X Θ (H), i.e., V al H acts componentwise for each X ∈ Γ.
Recall that for every u ∈ Φ(X) the corresponding set V al X H (u) is a set of points µ : W (X) → H satisfying the formula u (see Subsection 2.4). The logical kernel LKer(µ) was defined in Subsection 2.1 in these terms. Now we can say, that if a formula u belongs to Φ(X) and a point µ : W (X) → H is given, then u ∈ LKer(µ) if and only if µ ∈ V al X H (u). We shall note that a formula u can be, in general, of the form u = s * (v), where v ∈ Φ(Y ), Y is different from X. This means that the logical kernel of the point is very big and it gives a rich characterization of the whole theory.
Recall further that LKer(µ) is a Boolean ultrafilter containing the elementary theory T h X (H). Any ultrafilter with this property will be considered as an X-type of the algebra H.
It is clear that This remark is used, for example, in Recall that the algebra Φ is residually simple. This fact implies two important observations: 1. Let u, v be two formulas in Φ(X). These formulas coincide if and only if for every algebra H ∈ Θ the equality 2. Let a morphism s : W (X) → W (Y ) be given. It corresponds the morphism s * : Φ(X) → Φ(Y ). Let us take formulas u ∈ Φ(X) and v ∈ Φ(Y ). The equality s * (u) = v holds true if and only if for every algebra H in Θ we have We finished the survey of the notions of multi-sorted logic needed for UAG and in the next section we will relate these notions with the ideas of one-sorted logic used in Model Theory. Note also that we cannot define algebras of formulas Φ(X) individually. They are defined only in the multi-sorted case of algebrasΦ = (Φ(X), X ∈ Γ).
In fact, the definition of the algebra of formulas Φ and the system of algebras Φ(X) is the main result of the first part of the paper. They are essentially used throughout the paper.

2.10.
Identities of the variety Hal Θ for algebras Hal Θ (H). We have given already the definition of the algebras Hal Θ (H). Now we show that these algebras satisfy the axioms of Definition 2.3 and thus belong to the variety Hal Θ . In fact we should check the correspondences between s * and quantifiers and between s * and equalities.
First we consider interaction of s * with quantifiers. This interaction is determined by two following propositions.
Then the equality We also have the following equalities: Taking A to be a point a we obtain the axiom (3.a) of Definition 2.3. Proposition 2.9. Let s : W (X) → W (Y ) be morphism. Take x ∈ X and let s(x) = y for some y ∈ Y . We assume also that y does not contain in the support of each s(x ′ ), x ′ = x. Then the equality for every x ′ = x. So we have µs ∈ ∃x(A) and µ ∈ s * ∃x(A) .
Before proving of the inverse inclusion we give some remarks. In first we generalize this situation. Instead of the one variable x we will consider a set of variables I. Define the quantifier ∃(I) by: µ ∈ ∃(I)A if there is a point ν in A such that µ(y) = ν(y) for y ∈ I. Then we are interested in the following equality Let assume that s(I) = J and I ⊂ s −1 (J), and consider the equality s * ∃(s −1 (J))A = ∃(J)s * A. We will prove that it is true under the condition: s(x) = s(y) ∈ J if and only if x = y. Note that the latter condition follows from the assumption of our proposition.
As before we check that if µ ∈ ∃(J)s * A then µ ∈ s * ∃(s −1 (J))A. Let now µ ∈ s * ∃(s −1 (J))A. We will show that µ ∈ ∃(J)s * A. We have µs ∈ ∃(s −1 (J))A and ν ∈ A with µs(y) = ν(y) for all y ∈ s −1 (J) = I. Now we choose the certain element γ ∈ s * A. We assume that where x is an arbitrary element from the set I.
Let now x ′ ∈ I and s(x ′ ) = x does not belong to J. Then As a result we have that We have started the proof of this equality with the set I and then turned to the set s(I) = J. Using the condition s(x) = s(y) implies x = y we have s −1 (J) = I. Now we can rewrite the equality above as follows: s * ∃(I)A = ∃(s(I))s * A. If the set I consist of only one element x then we get that the statement of Proposition 2.9 holds. Now we consider the correspondence between morphisms and equalities. Here we have two conditions to check in Hal Θ (H): We show that the first condition holds. Let µ : Now we show that the second condition is true. Let Then µs x w ∈ A and w µ = (w ′ ) µ . From the last condition follows that µs x w (x) = µs x w ′ (x) and µs x w (y) = µs x w ′ (y) for y = x. This gives that µs x w = µs x w ′ . Since µs x w ∈ A then µs x w ′ ∈ A and µ ∈ s x w ′ * (A). Thus the correspondence between morphisms and equalities is verified.
So each algebra Hal Θ (H) satisfies the identities of the variety Hal Θ .

Logical geometry
3.1. Introduction. The setting of logical geometry looks as follows.
As before, we fix a variety of algebras Θ. Let X = {x 1 , . . . , x n } be a finite set of variables, W (X) the free in Θ algebra over X, H an algebra in Θ. The set Hom(W (X), H) of all homomorphisms µ : W (X) → H is viewed as the affine space of the sort X over H. Take the algebra of formulas Φ(X) which was defined in Subsection 2.8. Consider various subsets T of Φ(X). We establish a Galois correspondence between such T and sets of points A in the space Hom(W (X), H). This Galois correspondence gives rise to logical geometry in the given Θ.
The notion of the logical kernel plays a major role in this correspondence. Recall (see Subsection 2.4), that for every point µ : W (X) → H in the algebra Φ(X) there exists it logical kernel LKer(µ), which is a Boolean ultrafilter in Φ(X), containing the elementary theory T h X (H).
Having in mind the context of the theory of models (see the next section), we view LKer(µ) as a LG-type (that is logically-geometric) type of the point µ. Denote LKer(µ) = LG X H (µ). Note that the variety Θ is arbitrary and, correspondingly, the system of notions and statements of problems are of the universal character. However, even in the classical situation Θ = Com − P of the commutative and associative algebras with unit over the field P , a bunch of new problems and new results appear.

3.2.
Galois correspondence in the Logical Geometry. Let us start with particular case when the set of formulas T in Φ(X) is a set of equations of the form w = w ′ , w, w ′ ∈ W (X), X ∈ Γ.
We set Here A is an algebraic set in Hom(W (X), H), determined by the set T .
Let, further, A be a subset in Hom(W (X), H). We set Congruences T of such kind are called H-closed in W (X). We have also Galois-closures T ′′ H and A ′′ H . Let us pass to general case of logical geometry. Let now T be a set of arbitrary formulas in Φ(X). We set Here A is called a definable set in Hom(W (X), H), determined by the set T (cf., Section 4.5). We use the term "definable" for A of such kind, meaning that A is defined by some set of formulas T . For the set of points A in Hom(W (X), H) we set We have also Here T is a Boolean filter in Φ(X) determined by the set of points A. Filters of such kind are Galois-closed and we can define the Galoisclosures of arbitrary sets T in Φ(X) and A in Hom(W (X), H) as T LL and A LL .
Intersection of H-closed filters is also H-closed filter.

AG-equivalent and LG-equivalent algebras.
LG-isotypic algebras. Let us formulate two key definitions and two results (see, for example, [27], [30] Let now be a quasiidentity. We will also write In particular, if H 1 and H 2 are AG-equivalent then they generate the same quasi-variety. The inverse statement is not true (see [17]). Recall that quasi-varieties are generated by systems of finitary quasiidentities.
Consider the following formula: The set T can be infinite and then we speak about infinitary formula.
From this proposition follows: Denote by ImT h(H) the implicative theory of the algebra H. Remind that the implicative theory is the set of all formulas of the form T → u, for different X ∈ Γ, which hold true in the algebra H. So algebras H 1 and H 2 are LG-equivalent if their implicative theories are coincided i.e., ImT h(H 1 ) = ImT h(H 2 ). Now we give one more approach to the LG-equivalence. Let T be a set of formulas from Φ(X) and let T ∨ be the set of all disjunctions of the formulas u ∈ T and T ∨ be the set of all disjunctions of the formulas ¬u for u ∈ T . Here we have the following properties We want to consider the disjunctive theory of the algebra H. The disjunctive theory of the algebra H is the set of all possible formulas T ∨ , for all T ⊂ Φ(X) and different X ∈ Γ, which hold true in the algebra H.
Note that the formula T → v holds true in the algebra H if and only if the formula T ∨ ∨ v is true in H. Thus if the disjunctive theories of two algebras H 1 and H 2 coincide then these algebras are LG-equivalent. Moreover there is the following Proof. Let algebras H 1 and H 2 be LG-equivalent. We take a set of formulas T ⊂ Φ(X) and consider the formula  Proof. Let H 1 and H 2 be LG-equivalent algebras. By definition for any finite set X, and any H 1 -closed filter T from Φ(X) we have: Let T = LKer(µ) be the logical kernel of a point µ : Hence, T is an H 2 -closed filter. Since T = LKer(µ), the filter T is maximal. Since H 1 and H 2 are LGequivalent, there exists a set B in Hom(W (X), H 2 ) such that B L H 2 = T . Then T = ν∈B LKer(ν). Since the filter T is maximal, LKer(ν) = T = LKer(µ) for all points ν ∈ B.
Note that we used the fact that T L H 2 is not empty. Indeed, if we assume that T L In the similar way one can prove that if T = LKer(µ) is the logical kernel of a point ν : W (X) → H 2 , then there exists a point µ : W (X) → H 1 such that LKer(ν) = LKer(µ). Hence, H 1 and H 2 are isotypic.
Let, further, H 1 and H 2 be isotypic algebras. This means that if T = LKer(µ) is the logical kernel of a point µ : W (X) → H 1 , then T = LKer(ν) is the logical kernel for some ν : W (X) → H 2 as well, and vice versa. Recall, that every logical kernel is a closed filter, so T is H 1 -and H 2 -closed filter.
Let, now, T be an arbitrary H 1 -closed filter in Φ(X). We will show that T is H 2 -closed. Let Since H 1 and H 2 are isotypic, there exist points ν : W (X) → H 2 such that ν∈Hom(W (X),H) LKer(ν) = µ∈A LKer(µ). According to Proposition 3.1, the intersection of H-closed filters is also H-closed filter, hence T is an H 2 -closed filter.
Similarly, we can prove that each H 2 -closed filter is H 1 -closed. Hence, H 1 and H 2 are LG-equivalent. 3.4. Categories of algebraic and definable sets for a given algebra H. Recall that we introduced (Section 2.6) the category of affine spaces Θ * (H). It is natural to assume that V ar(H) = Θ. If this condition does not hold, the situation when for two different morphisms s 1 : W (Y ) → W (X) and s 2 : W (Y ) → W (X) the corresponding morphisms s 1 and s 2 in Θ * (H) coincide , is possible. This breaks duality between Θ 0 and Θ * (Proposition 2.2) and, as we will see, leads to a lot of other disadvantages. The condition V ar(H) = Θ plays also a crucial role in the problem of sameness of geometries over different algebras.
Define now a category of algebraic sets AG Θ (H) and a category of definable sets LG Θ (H).
Define first a category Set Θ (H). Its objects are pairs (X, A) with A a subset in Hom(W (X), H) and X ∈ Γ.
A morphism s * takes (X, A) to (Y, B), where s : W (Y ) → W (X) and B contains the points ν : W (Y ) → H such that ν = µs, for µ ∈ A. Now, AG Θ (H) is a full subcategory in Set Θ (H), whose objects are pairs (X, A), where A is an algebraic set.
If for A we take definable sets, then we have the category LG Θ (H) which is a full subcategory in Set Θ (H).
Two key results are as follows (see, for example, [27], [30]).  Morphisms between AG X Θ (H) and AG Y Θ (H), as well as between LG X Θ (H) and LG Y Θ (H) are defined towards the maps s : W (Y ) → W (X). We will describe these morphisms in more detail.
First of all, recall that objects in the categories Θ 0 and Φ Θ are free algebras W (X) and algebras of formulas Φ(X), respectively. Every homomorphism s : W (Y ) → W (X) gives rise to a morphism s * : Φ(Y ) → Φ(X). In particular, s * acts on equalities as follows: s * (w 1 ≡ w 2 ) = (s(w 1 ) ≡ s(w 2 )) (action of s * is regulated by Definition 2.3). Note that equalities of the form w ≡ w ′ , w, w ′ in W (X), can be treated as formulas in Φ(X). This correspondence s → s * allows us to define morphisms s and s * in AG Θ (H) and LG Θ (H).
Consider the commutative diagram (♦) (follows from Section 2.9). Here T 2 and T 1 are H-closed congruences in W (Y ) and W (X), respectively. In particular (♦) implies that s * : This diagram gives rise to the category C Θ (H) of all H-closed congruences.
It is important to get another look at the morphisms s * in C Θ (H). Let the H-closed congruences T 2 in C Y Θ (H) and T 1 in C X Θ (H) be given. A morphism s * takes T 2 to T 1 , if and only if s * satisfies the diagram (♦). So, s * assigns T 1 to T 2 if and only if we have (♦). Moreover, if one knows s * and T 1 , then (♦) recovers T 2 . Thus, one can define the category C Θ (H) −1 , with the same objects as C Θ (H) and morphisms acting opposite-wise. Proof. The correspondence C Θ (H) → AG Θ (H) is one-to-one. The condition V ar(H) = Θ provides that the correspondence s * → s is also one-to-one (see Proposition 2.2).
We shall repeat the similar construction using L-Galois correspondence. We have the diagram (♦♦) (whose particular case is the diagram (♦) : Here T 2 and T 1 are H-closed filters in Φ(X) and Φ(Y ), respectively. It gives rise to the categories of H-closed filters F Θ (H) and In other words, let the H-closed filters T 2 and T 1 in F Y Θ (H) and F X Θ (H), respectively, be given. Take T L 1H = A and T L 2H = B. The diagram (♦♦) determines when s * takes T 2 to T 1 . In particular, T 1 defines uniquely T 1 by T 2 = s −1 * (T 1 ), that is T 2 is the inverse image of T 1 .  The following problems is our main target: Problem 1. Find necessary and sufficient conditions on algebras H 1 and H 2 in Θ that provide algebraic similarity of these algebras. and H 2 in Θ that provide logical similarity of these algebras.
We start with examples of specific varieties, where necessary and sufficient conditions for isomorphism of the categories of algebraic sets can be formulated solely in terms of properties of algebras H 1 and H 2 . Afterwards we will dwell on a some general approach. In what follows, all fields and rings are assumed to be infinite.  [6]) [16], [42], [40]).
(2) Let Θ = Com − P or Lie − P and σ ∈ Aut(P ). Define a new algebra H σ . In H σ the multiplication on a scalar • is defined through the multiplication in H by the rule: λ • a = λ σ · a, λ ∈ P, a ∈ H.
We will make some preparations, basing on the idea of isomorphism of functors. Definition 3.20. Let ϕ 1 , ϕ 2 be two functors from a category C 1 to C 2 . We say that an isomorphism of functors S : ϕ 1 → ϕ 2 is defined if for any morphism ν : A → B in C 1 the following commutative diagram takes place Here S A is the A-component of S, that is, a function which makes a bijective correspondence between ϕ 1 (A) and ϕ 2 (A). The same is valid for S B .
Note that S A and S B are not necessarily morphisms in C 2 . Thus, this definition is different from the standard one, where all S A have to be morphisms in C 2 . The commutative diagram above can be reformulated as A . An invertible functor from a category to itself is an automorphism of a category. The notion of isomorphism of functors gives rise to the notion of the inner automorphism of a category. An automorphism ϕ of the category C is called inner (see [27]) if ϕ is isomorphic to the identity functor 1 C . This provides the commutative diagram A . The following Proposition plays an important role in the proof of Theorem 3.18: So, studying automorphisms of Θ 0 play a crucial role in Problem 1 related to geometric similarity of algebras. The latter problem is reduced by Reduction Theorem to studying the group Aut(End(W (X)) of automorphisms of the endomorphism semigroup of a free algebra (see [27], [9], [14]). Now we will treat the general problem using the Galois-closure functors.

For every algebra H ∈ Θ consider two functors
Cl A H : Θ 0 → P oSet, Cl L H : Φ Θ → Lat, where A and L stand for the functors of algebraic and logical closures, respectively. We will suppress these indexes in the sequel, assuming that the type of Cl-functor is clear in each particular case.
In fact, P oSet is the category C Θ (H) of partially ordered sets of Hclosed congruences C X Θ (H), while Lat is the category F Θ (H) of lattices of H-closed filters F X Θ (H). So, Cl H assigns to every object W (X) in Θ 0 the poset C X Θ (H) of all H-closed congruences on W (X). If s : The correspondence s → s * gives rise to the contravariant Cl H functor Φ → F Θ (H). Apply these notions to Problem 1 and Problem 2. Consider two commutative diagrams: Lat where ϕ is an automorphism of Θ 0 or Φ. Commutativity of these diagrams means that there exists an isomorphism of functors In its turn, this isomorphism of functors means that the diagram Cl H 2 (ϕ(W (X))), and the diagram Cl H 2 (ϕ(Φ(X))), are commutative.
where the automorphism of categories ϕ : Θ 0 → Θ 0 is coordinated with the lattice structures of close congruences by additional conditions (see [25], [26], [27] where the automorphism of categories ϕ : Φ Θ → Φ Θ is coordinated with the lattice structures of closed filters by additional conditions (see [33]).
Then the algebras H 1 and H 2 are logically similar.
In the particular case ϕ = id we have the diagrams: Cl H 2 (W (X)), and Cl Cl H 2 (Φ(X)), which imply that  Now we shall formulate several problems related to logical geometry. Let us start with the case when Θ = Grp. Problem 3. It is known [36], [45], that any group H which is LGequivalent to a free group W (X), is isomorphic to it. What is the situation, if H is semi-LG equivalent to W (X).

Problem 4.
What can be said about a group H which is logically similar to a free group W (X).
Problem 5. If two groups are LG-equivalent, then they are isotypic and, hence, elementary equivalent. What is the relation between the elementary equivalence of groups and their logical similarity? Problem 6. Are their logically similar groups H 1 i H 2 , such that the functors Cl H 1 and Cl H 2 ϕ are not isomorphic for any automorphism ϕ.

The next problem deals with logical invariants associated with semi-
LG-equivalence. Problem 7. Propositions 3.5 and 3.6 provide implicative and disjunctive criteria for algebras to be logically equivalent. Find criteria which provide semi-LG-equivalence of algebras.
As it was said above, the group of automorphisms of the category Θ 0 plays an exceptional role in problems related to geometrical similarity. The following problems are directed to find out what is the situation in the case of logical geometry. Problem 8. Study the group of automorphisms of the category Φ Θ . Problem 9. Study the group of automorphisms Aut(End(Φ(X))).

3.6.
Logically perfect and logically regular varieties. Up to now we assumed that the variety Θ is arbitrary. Now we distinguish the classes of varieties which are characterized by specific logical properties.
Let H be an algebra in Θ. Definition 3.27. A variety of algebras Θ is called logically perfect if every finitely generated free in Θ algebra W (X), X ∈ Γ is logically homogeneous.
Definition 3.28. An algebra H in Θ is called logically separable, if every H ′ ∈ Θ which is LG-equivalent to H is isomorphic to H.

Definition 3.29.
A variety Θ is called logically regular if every free in Θ algebra W (X), X ∈ Γ is logically separable.
The following theorem is valid: If the variety Θ is logically perfect, then it is logically regular.
Proof. Let the variety Θ be logically perfect and W = W (X) a free in Θ algebra of the rang n, X = {x 1 , . . . , x n }. Rewrite W = H =< a 1 , . . . , a n >, where a 1 , . . . , a n are free generators in H. Let H and G ∈ Θ be isotypic.
The algebras H and B are isomorphic by the isomorphism a i → b i , i = 1, . . . , n.
Let us prove that B = G. Let B = G and there is a b ∈ G which doesn't lie in B.
Problem 10. Is the converse statement true? That is, whether every logically regular algebra is logically perfect?
It seems to us that the answer can be negative and the logical regularity of a variety Θ doesn't imply its logical perfectness. This leads to the problem Problem 11. Find a logically regular but not logically perfect variety Θ. In particular, consider this problem for different varieties of groups and varieties of semigroups.
• The variety of all groups is logically perfect, and, hence, is logically regular. • The variety of abelian groups is logically perfect, and, hence, is logically regular. • The variety of all nilpotent groups of class n is logically perfect, and, hence, is logically regular. • The variety of all semigroups is logically regular • The variety of all inverse semigroups is logically regular. Now we can specify Problem 11 to the case of semigroups.
Problem 12. Check whether the variety of all semigroups and of all inverse semigroups are logically perfect?
We shall emphasize two following problems regarding solvable groups. Problem 15. Let Θ be a classical variety Com − P , a variety of commutative and associative algebras with unit over a field P . The problem is to verify its logical regularity and logical perfectness.
The same question stands with respect to some other well-known varieties. So, are the following varieties logically perfect or logically regular.
Problem 16. The variety Ass − P of associative algebras over a field P . It is also important to find out how the passage from a semigroup/group to a semigroup/group algebra behaves with respect to logical regularity and logical perfectness. This leads to the problem: Problem 21. Let S be a semigroup/group and P a field, both logically homogeneous. Whether it is true that the semigroup/group algebra P S is logically homogeneous as well? 3.7. Logically noetherian and saturated algebras. Definition 3.31. An algebra H is called logically noetherian if for any set of formulas T ⊂ Φ(X), X ∈ Γ there is a finite subset T 0 in T determining the same set of points A that is determined by the set T .
Definition 3.32. An algebra H ∈ Θ is called LG-saturated if for every X ∈ Γ, each ultrafilter T in Φ(X) containing T h X (h) has the form T = LKer(µ) for some u : W (X) → H. Each finite algebra H is logically noetherian. Therefore, every finite H is saturated. This holds for every Θ.
3.8. Automorphically finitary algebras. We have already mentioned that the group Aut(H) acts in each space Hom(W (X), H), X ∈ Γ. Let us make some comments regarding Problem 24. According to Theorem 3.9, LG-equivalent abelian groups are isotypic. As we know (Corollary 3.10), isotypeness of algebras implies their elementary equivalence. Classification of abelian groups with respect to elementary equivalence had been obtained by W.Szmielew in her classical paper [Sz]. So, Problem 24 asks how one should modify the list from [Sz] in order to obtain the isotypic abelian groups.
We had considered two important characteristics of varieties of algebras, namely, their logical perfectness and logical regularity. Let us introduce one more characteristic.
We call a variety Θ exceptional if • any two free in Θ algebras W (X) and W (Y ) of a finite rank, generating the whole Θ, are elementarily equivalent, and • if they are isotypic then they are isomorphic. In what follows we will distinguish between model theoretical types (MT-types) and logically geometric types (LGtypes). Both kinds of types are oriented towards some algebra H ∈ Θ, where Θ is a fixed variety of algebras. Generally speaking, a type of a point µ : W (X) → H is a logical characteristic of the point µ. Model-theoretical idea of a type and its definition is described in many sources, see, in particular, [8], [12]. We consider this idea from the perspective of algebraic logic (cf., [31]) and give all the definitions in the corresponding terms.
Proceed from the algebra of formulas Φ(X 0 ), where X 0 is an infinite set of variables. It arrives from the algebra of pure first-order formulas with equalities w ≡ w ′ , w, w ′ ∈ W (X 0 ) by Lindenbaum-Tarski algebraization approach (cf. Section 2.8). Φ(X 0 ) is an X 0 -extended Boolean algebra, which means that Φ(X 0 ) is a Boolean algebra with quantifiers ∃x, x ∈ X 0 and equalities w ≡ w ′ , where w, w ′ ∈ W (X 0 ), where W (X 0 ) is the free over X 0 algebra in Θ. All these equalities generate the algebra Φ(X 0 ). Besides, the semigroup End(W (X 0 )) acts in the Boolean algebra Φ(X 0 ) and we can speak of a polyadic algebra Φ(X 0 ). However, the elements s ∈ End(W (X 0 )) and the corresponding s * are not included in the signature of the algebra Φ(X 0 ).
Since Φ(X 0 ) is a one-sorted algebra, one can speak, as usual, about free and bound occurrences of the variables in the formulas u ∈ Φ(X 0 ).
Define further X-special formulas in Φ(X 0 ), X = {x 1 , . . . , x n }. Take X 0 \X = Y 0 . A formula u ∈ Φ(X 0 ) is X-special if each its free variable is occurred in X and each bound variable belongs to Y 0 . A formula u ∈ Φ(X 0 ) is closed if it does not have free variables. Only finite number of variables occur in each formula.
Denoting an X-special formula u as u = u(x 1 , . . . , x n ; y 1 , . . . , y m ) we solely mean that the set X consists of variables x i , i = 1, . . . n, and those of them who occur in u, occur freely. We call such type an X-MT-type (Model Theoretic type) over H. An X-MT-type is called complete if it is maximal with respect to inclusion. Any complete X-MT-type is a Boolean ultrafilter in the algebra Φ(X 0 ). Hence, for every X-special formula u ∈ Φ(X 0 ), either u or its negation belongs to a complete type. So, any X-MT-type lies in the one-sorted algebra Φ(X 0 ). Any X-LG-type lies in the domain Φ(X) of the multi-sorted algebra Φ.
We denote the MT-type of a point µ : W (X) → H by T p H (µ), while the LG-type of the same point is, by definition, its logical kernel LKer(µ).
The type T p H (µ) consists of all X-special formulas satisfied on µ. It is a complete X-MT-type over H.
By definition, the formula v = u(a 1 , . . . , a n ; y 1 , . . . , y m ) is closed. Thus, if it is satisfied one a point, then its value set V al H X (v) is the whole Hom(W (X), H).
Note that in our definition of an X-MT-type the set of free variables in the formula u is not necessarily the whole X = {x 1 , . . . , x n } and can be a part of it. In particular, the set of free variables can be empty. In this case the formula u belongs to the type if it is satisfied in H.
In the previous sections the algebra Φ was built basing on the set Γ of all finite sunsets of the set Γ. In fact, one can take instead of Γ the system Γ * = Γ X 0 and construct the corresponding multisorted algebra. Then, to each homomorphism s : W (X 0 ) → W (X) it corresponds a morphism s * : Φ(X 0 ) → Φ(X) and, vice versa, s : W (X) → W (X 0 ) induces s * : Φ(X) → Φ(X 0 ). In this setting the extended Boolean algebra Hal X 0 Θ (H) and the homomorphism V al X 0 H : Φ(X 0 ) → Hal X 0 Θ (H) are defined in a usual way. A point µ : W (X 0 ) → H satisfies u ∈ Φ(X 0 ) if µ ∈ V al X 0 H (u). One more remark. Since Φ(X 0 ) is generated by equalities, when we say that a variable occur in a formula u ∈ Φ(X 0 ), this means that it occur in one of the equalities w = w ′ , participating in u. The set of variables occurring in u determines a subalgebra Φ(X ∪ Y ) in Φ(X 0 ), such that u ∈ Φ(X ∪ Y ).
If we stay in one-sorted logic, this is a subalgebra in the signature of the one-sorted algebra Φ(X 0 ).
On the other hand, we can view algebra Φ(X ∪ Y ) as an object in the multi-sorted logic. Here, to every homomorphism s : For u ∈ Φ(X ∪Y ) we have s * u ∈ Φ(X ′ ∪Y ′ ). Let u be an X-special formula. It is important to know for which s the formula s * u is X ′ -special.

4.2.
Another characteristic of the type T p H (µ). We would like to relate an MT-type of a point to its LG-type.
Consider a special homomorphism s : W (X 0 ) → W (X) for an infinite set X 0 and its finite subset X = {x 1 , . . . , x n }, such that s(x) = x for each x ∈ X, i.e., s is identical on the set X. According to the transition from s to s * , we have Proof. We need one more look at a formula u ∈ T p H (µ). Given a point µ, consider a set A µ : W (X) → H of the points η : W (X 0 ) → H defined by the rule η(x i ) = µ(x i ) = a i for x i ∈ X and, η(y) is an arbitrary element in H for y ∈ Y 0 . Denote Here, as usual, LKer(η) is the ultrafilter in Φ(X 0 ), consisting of formulas u valid on a point η. It is proved [31], that a special formula u belongs to the type T p H (µ) if and only if u ∈ T µ , which is equivalent to V al X 0 H (u) ⊃ A µ . Note that the formula u of the kind belongs to each LKer(η) if the closed formula v(y 1 , . . . , y m ) is satisfied in the algebra H. This means also that T µ is not empty for every µ.
Return to the special homomorphism s : W (X 0 ) → W (X) and consider the point µs : W (X 0 ) → H. For x i ∈ X we have µs(x i ) = µ(x i ) = a i . Hence, the point µs belongs to A µ .
Observe that for the formula u = u(x 1 , . . . , x n ; y 1 , . . . , y m ), the formula u(a 1 , . . . , a n ; y 1 , . . . , y m ) is satisfied in the algebra H if the set A µ lies in V al X 0 H (u). Thus, µs belongs to V al X 0 H (u). By definition of s * we have that µ lies in s * V al X 0 H (u) = V al X H (s * u), which means that s * u ∈ LKer(µ).
We proved the statement in one direction.
Conversely, let s * u ∈ LKer(µ). Then µ ∈ V al X H (s * u) = s * V al X 0 H (u) and µs ⊂ V al X 0 H (u). Since the formula u(a 1 , . . . , a n ; y 1 , . . . , y m ) is satisfied in H, then every point from the set A µ belongs to V al X 0 H (u) (see also [5]). This means that the formula u belongs to T p H (µ).
Recall that we have mentioned the notion of a saturated algebra. It was LG-saturation. In the Model Theory MT-saturation is defined. MT-saturation of the algebra H means that for any X-type T there is a point µ : W (X) → H such that T ⊂ T p H (µ).

Theorem 4.5. If algebra H is LG-saturated then it is MT-saturated.
Proof. Let algebra H be LG-saturated and T be X-MT-type correlated with T h X 0 (H). We can assume that the theory T h X 0 (H) is contained in the set of formulas T .
Take a special homomorphism s : W (X 0 ) → W (X) and pass to s * : Φ(X 0 ) → Φ(X). Take a formula s * u ∈ Φ(X) for each formula u ∈ T and denote the set of all such s * u by s * T . This set is a filter in Φ(X) containing the elementary theory T h X (H), since, if u ∈ T h X 0 (H) then s * u ∈ T h X (H).
Further we embed the filter s * T into the ultrafilter T 0 in Φ(X) which contains the theory T h X (H). By the LG-saturation of the algebra H condition, T 0 = LKer(µ) for some point µ : W (X) → H. Thus, s * u ∈ LKer(µ) for each formula u ∈ T . Hence (Theorem 4.4), u ∈ T p H (µ) for each u ∈ T , and T ⊂ T p H (µ). This gives MT-saturation of the algebra H.
We do not know whether MT-saturation implies LG-saturation. It seems that not. If it is the case, then LG-saturation of an algebra H is stronger than its MT-saturation.

4.3.
Correspondence between u ∈ Φ(X) and u ∈ Φ(X 0 ). Definition 4.6. A formula u ∈ Φ(X) is called correct, if there exists an X-special formula u in Φ(X 0 ) such that for every point µ : W (X) → H we have u ∈ LKerµ if and only if u ∈ T H p (µ). Now, for the sake of completeness and for the aims of clarity we give a proof of the principal Theorem 4.7 of G.Zhitomiskii (see [45] for the original exposition). This fact will be essentially used in Theorem 4.8 and Theorem 4.12. We hope this will help to reveal ties between two approaches to the idea of a type of a point: the one-sorted model theoretic approach and the multi-sorted logically-geometric approach. Note that the proofs are sometimes different from that of [45].
such that u ∈ T H p (µ) if and only if u ∈ LKer(µ), where µ : W (X) → H. Define The formula ∃x u is not X-special since x is bound (we assume that x coincides with one of x i , say x n ). Take a variable y ∈ X 0 , such that y is different from each x i ∈ X and y j ∈ Y 0 .
Define ∃y u y to be a formula which coincides with ∃x u modulo replacement of x by y. So, ∃y u y has one less free variable and one more bound variable than ∃x u.
Consider endomorphism s of W (X 0 ) taking s(x) to y and leaving all other variables from X 0 unchanged. Let s * be the corresponding automorphism of the one-sorted Halmos algebra Φ(X 0 ). Then s * (∃x u) = ∃s * (x)s * ( u) = ∃y u y .
Thus, in order to check that ∃xu is correct, we need to verify that for every µ : W (X) → H the formula ∃xu lies in LKer(µ) if and only if ∃y u y ∈ T H p (µ). Let ∃xu lies in LKer(µ). Thus, there exits a point ν : W (X) → H such that u ∈ LKer(ν) and µ coincides with ν for every variable x ′ = x, x ′ ∈ X. Consider X y = {x 1 , . . . , x n−1 , y}.
If algebras H 1 and H 2 are isotypic then they are locally isomorphic. This means that if A is a finitely generated subalgebra in H, then there exists a subalgebra B in H 2 which is isomorphic to A and, similarly, in the direction from H 2 to H 1 .
On the other hand, local isomorphism of H 1 and H 2 does not imply their isotypeness: the groups F n and F m , m, n > 1 are locally isomorphic, but they are isotypic only for n = m.
Isotypeness imply elementary equivalence of algebras, but the same example with F n and F m shows that the opposite is wrong.
In Section 2 we pointed out several problems related to isotypic algebras. Let us give some other problems: Problem 26. Let H 1 and H 2 be two finitely generated isotypic algebras. Are they always isomorphic?
In particular: Problem 27. Let G 1 and G 2 be two finitely generated isotypic groups. Are they always isomorphic? Problem 28. Let H 1 be a finitely generated algebra and H 2 is an isotypic to it algebra. Is H 2 also finitely generated?
The next problem is connected with the previously named problems on isotypeness and isomorphism of free algebras.

4.5.
LG and MT-geometries. Compare, first, different approaches to the notion of a definable set in the affine space Hom(W (X), H).
Suppose that the variety of algebra Θ, an algebra H ∈ Θ and the finite set X = {x 1 , . . . , x n } are fixed.
In the affine space Hom(W (X), H) consider subsets A, whose points have the form µ : W (X) → H. Each point µ : W (X) → H has a classical kernel Ker(µ), a logical kernel LKer(µ) and a type (T p H (µ)). Correspondingly, we have three different geometries: algebraic geometry (AG), logical geometry (LG), and the model-theoretic geometry (MT G).
For AG consider a system T of equations w ≡ w ′ , w, w ′ ∈ W (X). For LG we take a set of formulas T in the algebra of formulas Φ(X). For MT G we proceed from an X-type T . In all these cases the set can be infinite. • A set A in Hom(W (X), H) is definable in MT G (i.e., A is MTdefinable) if there exists an X-type T such that Besides that, we have three closures: T ′′ H for AG, T LL H for LG, and T L 0 L 0 H for MT G. In the reverse direction the Galois correspondence for each of three cases above is as follows T ′ consisting of all u which correspond u ∈ T . The points µ ∈ A satisfy every formula from T ′ . This means that T ′ is a consistent set of X-special formulas. Thus T ′ is an X-type, such that A ⊂ T ′L 0 H . Let now the point ν lies in T ′L 0 H . Then ν satisfies every formula u. Hence it satisfies every formula u ∈ T . Thus ν lies in T L H = A. This means that T ′L 0 H = A and the theorem is proved.
Consider now the case when algebra H is logically homogeneous and A is an Aut(H)-orbit over the point µ : W (X) → H. We have A = (LKer(µ)) L H . The equality LKer(µ) = LKer(ν) holds if and only if a point ν belongs to A. The same condition is needed for the equality T p H (µ) = T p H (ν). Now, ν ∈ (T p H (µ)) L 0 H by the definition of L 0 . Thus, A = (T p H (µ)) L 0 H . We proved that the orbit A is MTdefinable and LG-definable.
Recall that we defined two full sub-categories K Θ (H) and LK Θ (H) in the category Set Θ (H). Let us take one more sub-category denoted by L 0 K Θ (H). In each object (X, A) of this category the set A is an X-MT-type definable set. The category L 0 K Θ (H) is a full subcategory in LK Θ (H). In view of Theorem 4.12 categories LK Θ (H) and L 0 K Θ (H) coincide.