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УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА
Физико-математические пауки
UDK 512.56^12.57
ON LATTICES CONNECTED WITH VARIOUS TYPES OF CLASSES OF ALGEBRAIC STRUCTURES
A. Nurakunov, M. Semenova, A. Zamojska-Dzienio
Abstract
This survey paper reviews some recent, results related to various derived lattices connected with various types of classes of algebraic structures which were obtained by the authors.
Key words: axiomatizable class, variety, quasivariety, prevariety, finit.ary prevariety, identity, quasi-identity, lattice, subsemilattice lattice.
Introduction
This survey paper presents recent results obtained for lattices of subclasses of certain types. Mainly, we focus on representing lattices by lattices of relatively axiomatizable classes and those of (finitary) prevarieties. also mentioning some general algebraic and computational properties of those lattices.
Study of such lattices has a long history and goes back to G. Birkhoff and A.I. Malt-sev. In fl] and [2]. they have independently asked about which lattices can be represented as lattices of (quasi)varieties, that is. classes defined by (quasi-¡identities. It is one of the oldest and hardest problems in lattice theory. A number of remarkable results was obtained concerning this question of Birkhoff and Maltsev. An advance in the Birkhoff-Maltsev problem was made by K.V. Adaricheva. W. Dziobiak and V.A. Gor-bunov by describing algebraic atomistic lattices isomorphic to quasivariety lattices in [3]. see also [4. Theorem 5.3.17]. It is also known (V.A. Gorbunov [4]) that all atomistic algebraic quasivariety lattices are isomorphic to the so-called lattices of algebraic subsets of algebraic lattices. We also note that those lattices are dual to lattices of suitable first-order theories (cf. results of Iv. Adaricheva. J.B. Nation [5 7] and also the talk of G.F. McNnlty on lattices of eqnational theories [8]). For other results concerning this topic, we refer to the book [4. Chapter 5]. see also the survey paper [9]. as well as to the bibliography lists in those two. In addition to these, lattices of psendovarieties of finite algebras were investigated in a number of papers, see. for example. [10].
A.M. Nurakunov proved in [11] that there are qnasivarieties of algebras (structures with no relation in the signature) snch that the set of finite snblattices of their quasivariety lattices is not computable, see Section 6. This result shows in particular that finding a complete description of quasivariety lattices should be very hard. But there are some restricted versions of the Birkhoff Maltsev problem which are still of big interest.
While sub (quasi) variety lattices were studied in a considerable extent, lattices of other first-order axiomatizable classes remain almost untouched. In [12]. D.E. Pal'chunov has proved that any at most countable complete lattice is isomorphic to a lattice of relatively axiomatizable classes. In [12. Problem 1]. he asked whether the same result holds for an arbitrary complete lattice. We answer the latter question in the positive in Theorem 4. which is based on the result of V.A. Gorbunov [13].
All classes are abstract: that is. they are closed under isomorphic copies. For example, when writing {Ai | i G I} for a set I, we always mean the class of isomorphic copies of structures from the set {Ai | i G I}.
For all the concepts which are not defined here, we refer to [4].
1. Basic concepts
For an arbitrary signature a, let K(a) denote the class of all structures of signature a. Let also T(a) denote the variety of a-structures defined by the identity Vxy x = y.
Following [4], for a class K Ç K(a), let V(K) [Q(K), respectively] denote the least [quasi-]variety containing K. Let H(K) denote the class of structures from K(a) which are homomorphic images of structures from K; let P(K) [Pw (K), respectively] denote the class of structures from K(a) which are isomorphic to Cartesian products of [finitely many] structures from K; let Ps(K) [P" (K), respectively] denote the class K(a)
structures from K; let Ls(K) denote the class of structu res from K(a) which are isomorphic to superdirect limits of structures from K; and let S(K) denote the class of structures from K(a) which are isomorphic to substructures of structures from K. Finally, let K fin denote the class of finite members of K. According to Birkhoff's Theorem (see [4. Section 2.3]).
V(K) = HSP(K) = HPsS(K) = HPs(K),
while according to [14. Theorem 5.2] (see also [4. Theorem 2.3.6]).
Q(K) = LsPsS(K) = LSPS(K).
A class K Ç K(a) is a (finitary) prevariety if K = SP(K) = PsS(K) (K = SP"(K) = P"S(K), respectively). The notion of a finitary prevariety (in case of signature containing no relation symbols) was introduced by A. Vernitski in [15]. According to [16]. a class is a prevariety if and only if it can be defined by infinite implications.
Definition 1 [4, Section 2.5]. Let K' Ç K Ç K(a^en K' is K-(quasi-) equational if K' = K n Mod(S) for some set S of (quasi-)identities of signature a.
For the following concept, see [4] and also [17].
K' Ç K Ç K(a) K' K
K' = K n A for some (finitary) prevariety A Ç K(a); K' is a K-(quasi) variety if K' = K n A for some (quasi)variety A Ç K(a).
Equivalently, K' is a (finitary) K-prevariety if and only if K' = K n SP(K') (K' = K n SP"(K'), respectively). Similarly, K' is a K-(quasi)variety if and only if K' = K n V(K') (K' = K n Q(K'), respectively).
Definition 3. A class K Ç K(a)fin is a pseudo-quasivariety if it is a finitary prevariety.
Note that K Ç K(a)fin is a pseudo-quasivariety if and only if it is a (finitary) K f in -prevariety, and if and only if it is a K fin -quasivariety.
Let Lv(K) denote the set of all K-equational subclasses of K, while Lq(K) denotes the set of all K-quasi-equational subclasses of K. Let also Lp(K) (Lp"(K),
K
set inclusion, all the three form complete lattices. Note that in the case of (finitary) prevarieties, we also allow the case when the ground of a lattice is a proper class.
Definition 4. Let L be a complete lattice. A subset A Ç L is a complete meet subsemilattice of L, if /\ X G A for any X Ç A. A complete meet subsemilattice A Ç L is an algebraic subset of L if \J X G A for any non-empty up-directed subset X of A.
A binary relation R on a meet semilattice (S, A) is distributive if for any a, b, c G S relation (c, a A b) G R implies that c = a' A b' for some a', b' G S such that (a', a) G R and (b', b) G R. The equality relation = is obviously distributive.
For a meet semilattice (S, A, 1) with unit and for any binary relation R C S2, let Sub(S, R) denote the set of all R-closed subsemilattices of S; that is, X G Sub(S, R) if and only if the following conditions hold:
• AF G X for all finite F C X;
• b G X and (a, b) G R imply a G X.
For a complete lattice L, let Subc (L, R) denote the set of all co mplete R-closed meet subsemilattices of L, while Sp(L, R) denotes the set of all algebraic subsets of L which are R-closed. Let also F(L, R) denote the set of R-closed filters of L. We write Sub(L), Subc(L), Sp(L), and F(L) instead of Sub(L, =), Subc(L, =), Sp(L, =^d F(L, =), respectively. Ordered by inclusion, Sub(L, R), Subc(L, R^^d Sp(L, R) form complete
F(L, R)
2. Representing by congruence lattices
For a structure A G K(<r) and for a class K C K(a), let ConK A denote the set of congruences 0 on A such that A/0 G ^f K = K(<r), then we write Con A instead of ConK A. For 0,0' G Con A, we write 0' E 0 if A/0' embeds into A/0. Then E is called the embedding relation. Obviously, this relation is distributive.
The next theorem combines the characterization theorem proved for qnasivarieties by V.A. Gorbunov and V.I. Tumanov [14, 19], see also [4, Corollaries 5.2.2, 5.2.6] with its analogue for (finitary) prevarieties obtained in [17].
Theorem 1. Let A C K(a) be a prevariety, and let A G A. The following holds:
Lp(H(A) n A) = Subc(ConA A, E); Lp"(H(A) n A) = Sub(ConA A, E).
If A is [/] -projective in A, then
Lq(H(A) n A) = Sp(Cona A, E); Lv(H(A) n A) = F(Cona A, E).
In particular, one gets the following
Corollary 1 [4, Corollaries 5.2.2, 5.2.5]. Let A C K(a) be a prevariety, and let FK(w) G A be a K-free structure of countable rank. The following holds:
Lq(A) = Sp(Conx Fk(w), E); Lv(A) = F(Conx Fk(w), E).
For any class K C K(a) ^^d any cardinal k, let KK ^^^^^e the class of «-generated K
eties.
Corollary 2 [17]. For any prevariety K C K(a) and for any cardinal k, Lp(KK) = Subc (Conx Fk(k), E).
KA
Conk A K
In the next section, we will state a partial converse of Corollary 1. More precisely, any
complete lattice is isomorphic to the lattice of relative varieties of a prevariety, any lattice of algebraic subsets of an algebraic lattice is isomorphic to a quasivariety lattice, any lattice of complete subsemilattices of a complete lattice is isomorphic to a prevariety lattice, and any subsemilattice lattice is isomorphic to a finitary prevariety lattice, see Propositions 1. 2 and 3.
A well-known and long-standing problem in lattice theory asks whether any finite lattice is isomorphic to the congruence lattice of a finite algebra of finite signature. The next result proved by A.M. Nurakunov [20] shows that any finite lattice is isomorphic to a relative congruence lattice of a finite algebra of finite signature.
Theorem 2 [20]. For any finite lattice L, there is a quasivariety K of unars [pointed Abelian groups, respectively] and a finite alg ebra A G K such th at L = ConK(A).
The following result obtained by A.M. Nurakunov [21] gives a description of lattices of snbvarieties in terms of congruence lattices.
Theorem 3 [21]. A lattice is isomorphic to a variety lattice if and only if it is dually isomorphic to the congruence lattice of a monoid with two additional unary operations possessing certain properties.
Based on ideas from [21]. Iv. Adaricheva and J.B. Nation proved in [5] an analogue of Theorem 3 for quasivariety lattices: quasivariety lattices are exactly lattices dually isomorphic to congruence lattices of semilattices endowed with unary operations possessing certain properties. In addition to that. J.B. Nation proved in [7. Corollary 16] that the congruence lattice of any semilattice with operators is dually isomorphic to the lattice of snbprevarieties of a prevariety.
3. Representation by lattices of subclasses
3.1. Relation symbols. Let a = {pi | i G I} be a signature consisting of unary-relation symbols only. Furthermore, for any set X C I, let AX denote a structure from T(a) such th at AX = Vx pi(x) if and onl y if i G X. Obviously, T(a) consists of isomorphic copies of structures AX, X C I.
Let (X, C) be a closure space and L(X, C) be the closure lattice on X. We put
a(X) = {px | x G X}. Let £(X, C) consist of (in general infinite) implications of the form Vx /\ pa(x) ^ Pb(x), A C X, b G C(A).
aeA
Of course, if the set X is finite, then the signature a(X) is finite, while £(X, C) becomes a finite set of qnasi-identities.
The class Mod(E(X, C)) is obviously closed under substructures and Cartesian products, whence it is a prevariety. Therefore, the class K(X, C) = Mod(E(X, C)) n n T(a(X)) is also a prevariety.
(X, C) K ( X, C)
morphic copies of structures Ab, where B G L(X, C).
The following proposition shows, in particular, that any complete lattice is isomorphic to the lattice of relative eqnational classes of a prevariety. Originally, it was proved by V.A. Gorbunov [13, Example 4.9]. In [17], M. Semenova and A. Zamojska-Dzienio gave a short direct proof: a sketch of it is presented below.
Proposition 1. For any complete lattice L, there is a signature a consisting only of unary relation symbols, and a prevariety K Ç T(a) such that Ld = Lv(K) and, Subc(L) = Lp(K).
Sketch of proof. Since the lattice L is complete there is a closure space (X, C) such that L = L(X, C ) .Let a = a(X ) and K = K(X, C). Then K is a prevariety and a map < : L(X, C) ^ Lv(K) defined by the rule
<: B ^ {Af G T(a) | F G L(X, C) and B Ç F}, B G L(X, C),
establishes a dual lattice isomorphism. □
The following proposition is a finitary analogue of Proposition 1 for prevarieties.
Proposition 2 [17]. For any meet semilattice (S, A, 1) with unit, there is a signature a consisting only of unary relation symbols, and a finitary prevariety K Ç T(a) such that Sub(S) = Lp" (K).
Combining Propositions 1. 2. one gets the following proposition. A part of this result concerning relative (quasi)variety lattices was proved by V.A. Gorbunov and V.I. Tunianov [14. 19]. see also [4. Theorem 5.2.8]. In the present form, it was proved in [17].
La
consisting only of unary relation symbols, and a quasivariety K Ç T(a) such that Ld = Lv(K), Sp(L) = Lq(K) > Subc(L) = Lp(K), and Sub(L) = Lp" (K).
From Proposition 3. we get also the following statement which appeared in [17].
Corollary 3. The class of complete dually algebraic lattices coincides with the class of lattices of relative equational classes of quasivarieties.
La
consisting only of unary relation symbols, and a prevariety K Ç T(a) such that Sp(L) embeds into Lq(K).
In general, for a complete upper continuous lattice L, the lattice Sp(L) is not necessarily isomorphic Lq(K) , S66 L
as Proposition 3 above shows.
Remark 1. It is well-known that quasivariety lattices are completely join-semidist-ribntive and dually algebraic (cf. [4. Theorem 5.1.12 and Proposition 5.1.1]). In contrast.
Lq(K) Lp(K)
K
Corollary 4 [17]. There are prevarieties K such that neither Lq(K) nor Lp(K) embed into a quasivariety lattice.
Using similar methods one can also prove that any complete lattice is isomorphic to the lattice of relative equational classes of a class of signature with one unary relation symbol and constant symbols as well as of signature containing only constant symbols.
(X, C)
We consider the signature ap(X) = {p} U {cx | x G X}, where p is a unary relation symbol and cx is a constant symbol for any x G X.
Let K' Ç K(ap(X)) be the class of structures A = (A; ap(X)} such that for any a G A, there is x G X with a = cA, and satisfying the following first-order sentences:
Vxy cu = cv ^ x = y, u = v in X;
Vx cu = cv ^ p(x), u = v in X;
Vxy /\ p(cx) ^ x = y.
xex
Furthermore, for any set U Ç X, let VV denote a structure from K' such that Vu = p(cx) if and only if x G U. Obviously, K' consists of isomorphic copies of structures VV, U Ç X. Moreover, VX is the trivial structure.
X
(i) If A, B Ç X, then Va G H(Vb) if and only if B Ç A.
(ii) Let {Ai | i G I} Ç X mid A Ç X. Then the strueture A = Va G K' is
isomorphic to a substructure in B = n Va* if and onl y if A = p| ieI Ai.
iei
Let £p(X, C) consist of the following (in general infinite) implications of the form
/\p(cu) ^ p(cv), U Ç X, v G C(U).
ueu
Of course, if the set X is finite, then the signature ap(X) is finite, while Ep(X, C) becomes a finite set of quasi-identities. Let Kp(X, C) = K' n Mod(£p(X, C)).
Lemma 3. For any closure space (X, C}, the class Kp(X, C) consists of isomorphic copies of structures VB, where B G L(X, C).
Proposition 5. For any complete lattice L, there is a signature a consisting of one unary relation symbol and |L| many constant symbols, and there is a class K Ç K(a) such that L = Lv(K) and Subc(Ld) = Lp(K).
L (X, C}
such that Ld = L(X, C ) .Let a = ap(X ^d K = Kp(X, C ). It follows from Lemma 1 that the class K consists of isomorphic copies of structures V^(a), where a G L. Now, the map y : L(X, C) ^ Lv(K) defined by the rule
y: B ^ {VF G K' | B Ç F G L(X, C)}, B G L(X, C),
establishes a dual isomorphism. Moreover, the map y' : Subc(Ld) ^ Lp(K) defined by the rule
y' : B ^ {V^(6) G K' | b G B}, B G Subc(Ld), is a lattice isomorphism. □
Proposition 6. For any meet semilattice (S, A, 1} with unit, there is a signature a consisting of one unary relation symbol and |S| many constant symbols, and there is a, class K Ç K(a) such that Sub(S) = Lp"(K).
Sketch of proof. Let a = {p} U {cx | x G S} consist of a unary relation symbol p and constant symbols cx, x G S, and let the class K consist of isomorphic copies of structures V[a, wher e a G S. Define a map y : Sub(S) ^ Lp" (K) by the rule
y : B G K | b G B}, B G Sub(S).
It is a lattice isomorphism. □
3.3. Only constants. Let (X, C) be a fixed closure space. We consider the signature a(X) = {c} U {cx | x G X}, where cx is a constant symbol for any x G X as c
c
Let K' C K(a(X)) be the class of structures A = (A; a(X)) such that for any a G A, a = cA or there is x G X with a = cA and satisfying the following first-order sentences:
Vxy cu = cv ^ cu = c, u = v in X.
Furthermore, for any set U C X, let Fu denote a structure from K' such that Fu = cx = c if and only if x G U. Obviously, K' consists of isomorphic copies of structures Fu, U C X. Moreover, FX is the trivial structure.
X
(i) If A, B C X, then Fa G H(Fb) if and only if B C A.
(ii) Let {Ai | i G I} C X and A = p| i£l Ai. Then the struc ture A = FA G K' is isomorphic to a substructure in B = nieI FAi.
Let E(X, C) consist of the following (in general infinite) implications of the form
f\cu = c ^ cv = c, U C X, v G C(U).
ueu
Of course, if the set X is finite, then the signature a(X) is finite, while E(X, C) becomes a finite set of quasi-identities. Let K(X, C) = K' n Mod(E(X, C)).
Proofs of all the results presented in this section are similar to ones of corresponding results about the class Kp(X, C) presented in Subsection 3.2.
(X, C) K ( X, C)
copies of structures FB, where B G L(X, C).
Proposition 7. For any complete lattice L, there is a signature a consisting of |L| + 1 many constant symbols, and a class K C K(a) such that L = Lv(K) and Subc(Ld) = Lp(K).
Proposition 8. For any meet semilattice (S, A, 1) with unit, there is a signature a consisting of |S| + 1 many constant symbols, and a class K C K(a) such that Sub(S) = Lp" (K).
4. Relatively axiomatizable classes of structures
In [12, Theorem 8], D.E. Pal'chunov has proved that any at most countable complete lattice is isomorphic to a lattice of relatively axiomatizable classes. In [12, Problem 1], he asked whether the same result holds for an arbitrary complete lattice. M. Semenova and A. Zamojska-Dzienio answered the latter question in the positive in [17] for a signature consisting of unary relation symbols and a prevariety of trivial structures, see Theorem 4 below. We emphasize that this positive answer follows essentially by the results of V.A. Gorbnnov [13], see also [4] and Proposition 1.
Exposition here follows [17]. We also note that Theorem 4 can be inferred from the results of Subsections 3.2, 3.3 for a signature containing one unary relation symbol and constants as well as for a signature containing only constants.
K
a, and let A be a set of first-order sentences of the same signature. A class K' is axiomatizable in K relative to A if K' = K n Mod(E) for some set E C A.
It follows from Definition 5 that a class K C K(a) is axiomatizable if and only if it K(a)
any set A of sentences and any class K C K(a), the set of all classes, axiomatizable in K relative to A, forms a complete lattice. Following D.E. Pal'chunov [12], we denote this lattice by A(K, A). The following corollary shows that any complete lattice is a lattice of relatively axiomatizable classes.
Theorem 4. For any complete lattice L, there is a signature a, a prevariety K C C K(a), and a set A such that L = A(K, A), where A is a set of all identities of a
Now, we get from Corollary 3 and [4, Proposition 5.1.1]:
Corollary 5. The class of complete dually algebraic lattices coincides with the class A(K, A) K A
sentences.
L
signature a and a s et A of first-order senten ces of a such that L = A(K(a), A).
5. A reduction theorem
In [23] (see also his monograph [4]), V.A. Gorbunov has proved the so-called reduction theorems for lattices of qnasivarieties and lattices of varieties. For a class K C K(a), and for a positive n < w, let FK(n) denote the K-free structure of rank n.
Theorem 5 [4, Corollaries 5.5.2, 5.5.12]. Let K C K(a) be a prevariety. Then the following holds:
Lq(K) = lim Lq(H(FK(n)) n K) ^ lim Sp(ConK FK(n), E);
Lv(K) ^ lim Lv(H(FK(n)) n K) = liin F*(ConK FK(n)).
In particular, the following statements are true.
a
symbols, and let K C K(a) be a locally finite prevariety. Then
(i) Lq(K) = Jim Ln for a set {Ln | n < w} of finite lower bounded lattices;
(ii) Lv(K) = ljm Ln for a set {Ln | n < w} of finite lattices.
In particular, both Lq(K) and Lv(K) are residually finite lattices.
In [23], V.A. Gorbunov has also proved the following version of the reduction theorem for lattices of psendo-qnasivarieties.
a
bols, and let K C K(a) be a pseudo-quasivariety. Then there is a family {Ln | n < w} of finite lower bounded lattices such that Lp(K) = hm Ln.
In [17], M. Semenova and A. Zamojska-Dzienio proved a (finitary) prevariety analogue of Theorem 5. More precisely, the lattice of snbprevarieties of a prevariety is isomorphic to an inverse limit of complete snbsemilattice lattices of semilattices endowed with a distributive binary relation (see Theorem 7), while the lattice of finitary snbprevarieties of a finitary prevariety is isomorphic to an inverse limit of snbsemilattice lattices of semilattices endowed with a distributive binary relation (see Theorem 8). These results generalize Theorem 6.
To prove Theorems 7 and 8, one should assume the following class form of the Axiom of Choice, see (CAC 1) in [24, Section II.2]:
If S is a class of non-empty sets, there is a function F such that F(x) e x for each x e S.
Theorem 7 [17]. For any prevariety K C K(a), the lattice Lp(K) is isomorphic to an inverse limit of lattices of the form Subc(S, R), where S is a complete meet semilattice with unit, and, R is a distributive relation on S.
Sketch of proof. Let I be the class of all subsets of K ordered by inclusion, let Ai = n(A | A e i}, and let Ki = H(Ai) n K for all i e I. Moreover, as Ki C Kj, the map
j : Lp(Kj) ^ Lp(Kj), <fji: X ^ X n Kj
is a complete lattice homomorphism for all i C j in I. In addition, <pkj j = ¥ki and y>ii is just the identity map for all i C j C k in I. Therefore, the triple A = (I, Ki; is an inverse spectrnm.
Now, the map ^: Lp(K) ^ lim A defined as
p: X ^ (X n Ki | i e I), is a complete lattice isomorphism, and one obtains Lp(K) = lim A.
Finally, for any i e I, we have Lp(Ki) = L^H(A-j) n K = Sub^ConK Ai, E) according to Theorem 1, whence the statement of the theorem follows. □
The next statement is an analogue of Theorem 7 for finitary prevarieties.
Theorem 8 [17]. For any finitary prevariety K C K(ct), the lattice Lp"(K) is
Sub(S, R) S
RS
Now, Theorem 6 becomes an easy corollary of any of Theorems 7 and 8 according to the definition of a psendo-qnasivariety. We also note that to prove Theorem 8 for psendo-qnasivarieties, ordinary Axiom of Choice is sufficient.
K
a locally finite qnasivariety, then the map
^: Lq(K) ^ Lp(Kfin); ^: X ^ X/i„
defines an isomorphism. Therefore, Theorem 7 implies Corollary 7(i).
For a pseudo-quasivariety K C K(<r), let I ^e ^^e set of all finite subsets of K, let Ki = ^^{A | A e i}) n K for all i e I, and let
Lq = {Lq(Q(Ki)) | i e I}.
The following corollary generalizes V.A. Gorbnnov [4, Corollary 5.5.22].
Corollary 8. Let a contain finitely many relation symbols, and let K C K(a)fin be a pseudo-quasivariety. Then Lp(K) e SPuH(Lq) n SP^L^Q(K))). In particular, any universal sentence which holds in Lq(Q(K^) also holds in Lp(K).
The next theorem shows that a similar result for lattices of pseudo-varieties also holds. It was proved by P. Agliano and J.B. Nation [10] for psendo-varieties of algebras, but their proof remains valid for structures with finitely many relation symbols.
a
and let K C K(a)fin be a pseudo-variety. Then the lattice Lpv(K) of pseudo-varieties K
HSP„(Lv(V(A)) | Ae K).
In particular, any positive universal sentence which holds in Lv(V(K)) also holds in the lattice Lpv(K) of all pseudo-varieties contained in K.
6. Non-computability properties of relative subclass lattices
The following problem is duo to [8]. Is the set of all finite lattices of varieties computable? This problem is also mentioned in [25].
In [11, Theorem 1], A.M. Nurakunov has proved the following statement.
Theorem 10. Let a signature a contain at least one non-constant operation. Then there is a quasivariety K C K(a) such that the set of all finite sublattices of the quasi-variety lattice Lq(K) is not computable.
The latter result means that there is no algorithm to decide whether a given finite lattice embeds into such a quasivariety lattice. Therefore, it looks hopeless to find a complete structural description of lattices isomorphic to (quasi)variety lattices (cf. the Birkhoff-Maltsev problem).
We also note that from the proof of Theorem 10, it is possible to get an estimation of algorithmic complexity for certain quasivariety lattices as well as to compute the number of non-isomorphic quasivariety lattices having a non-compntable set of finite snblattices.
Corollary 9. There is a locally finite quasivariety such thai the set of all finite sub-lattices of its quasivariety lattice is not computable, while it is computably enumerable.
Corollary 10. There are continuum many locally finite quasivarieties such thai the set of finite sublattices of their quasivariety lattices is not computable.
While Theorem 10 and Corollaries 9, 10 deal with purely functional signature, there are their complete analogues for purely relational signature. In particular, it is proved in [17] (based on ideas from [11]) that there are qnasivarieties of one-element relation structures snch that their (quasi)variety lattices or (finitary) prevariety lattices have a non-compntable computably enumerable set of finite snblattices.
Theorem 11 [17]. The following statements hold.
(i) There is a countable relation signature t and a quasivariety K C T(t) such that the set of all finite sublattices of the relative variety lattice Lv(K) is computably enumerable but not computable.
(ii) There is a countable relation signature a and a quasivariety K C T(a) such that Lq(K) = Lp(K) = Lp" (K) and the set of all finite sublattices of this lattice is computably enumerable but not computable.
7. Open problems
As it has been already mentioned in Introduction, very little is known about lattices of first-order axiomatizable classes different from (qnasi)varieties. Tims the following general problem arises:
Problem 1. Study lattices of (relatively) axiomatizable classes and lattices of (finitary) prevariety lattices.
Remark 1 suggests the following problem.
Problem 2 [17]. Is there a nontrivial lattice property satisfied by all lattices of (finitary) prevarieties? Which lattices are isomorphic to lattices of (finitary) prevari-eties ?
Problem 2 is an analogue of the Birkhoff Maltsev problem. It is well-known (cf. [26, Theorem 2.84]) that finite bounded lattices generate the variety of all lattices. According
to [27], the lattice Subc(L) is finite lower bounded for any finite lattice L. Therefore, prevariety lattices of qnasivarieties generate the variety of all lattices. Tims according to Proposition 1, there is no nontrivial lattice identity which would hold on all prevariety lattices.
Dne to the results presented in Section 6, one can also pose the following problem.
Problem 3. For certain classes of structures, is the finite membership problem decidable ?
The second author was supported by the Presidential Grant Council of the Russian Federation, the Program for Support of Leading Scientific Schools (Grant No. NSh-3669.2010.1), by the Jözef Mianowski Fund, and by the Foundation for Polish Science. The third author was supported by the Warsaw University of Technology (Statutory Grant No. 504G/'1120/0054000).
Резюме
A.M. Нуракуиои, M.B. Семенова, А. Замойска-Дж.е.иио. О решетках, связанных с различными типами классов алгебраических структур.
В обзорной статье приводятся результаты, полученные авторами за последнее время, о различных производных решетках, связанных с различными типами классов алгебраических структур.
Ключевые слова: аксиоматизируемый класс, многообразие, квазимпогообразие, предмпогообразие. финитарное предмпогобразие, тождество, квазитожедество, решетка, подполурешетка решетки.
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Поступила в редакцию 27.10.10
Nurakunov, Anvar Mukhparovich PliD in Physics and Mathematics, Senior Research Fellow, Institute of Mathematics, National Academy of Sciences of Kyrgyzstan, Bishkek, Kyrgyzst.au.
Нуракунов Анвар Мухпарович кандидат физико-математических паук, старший научный сотрудник Института математики Национальной академии паук Кыргызской Республики, г. Бишкек. Кыргызстан.
E-mail: а.пигакипоьвдтай.сот
Semenova, Marina Vladimirovna Doctor of Physics and Mathematics, Leading Research Fellow, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia: Novosibirsk State University, Novosibirsk, Russia.
Семенова Марина Владимировна доктор физико-математических паук, ведущий паучпый сотрудник лаборатории алгебраических систем Института математики им. С.Л. Соболева Сибирского отделения РАН, г. Новосибирск, Россия: Новосибирский государственный университет, г. Новосибирск, Россия.
E-mail: udavl7Qgmail.com: semenovaemath.mc.ru
Zamojska-Dzienio, Anna PhD, Adjunct, Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland.
Замойска-Дженио, Анна доктор паук, адыопкт факультета математики и информатики Варшавского политехнического университета, г. Варшава, Польша.
E-mail: А.Zamojskaeelka.pw.etlu.pl
