Congruence properties of lattices of quasivarieties

The congruence properties close to being lower boundedness in the sense of McKenzie are treated. In particular, the affirmative answer is obtained to a known question as to whether finite lattices of quasivarieties are lower bounded in the case where quasivarieties are congruence-Noetherian and locally finite. Namely, we state that for every congruence-Noetherian or finitely generated locally finite quasivariety K, the lattice L_q(K) possesses the Day-Pudlak-Tuma property. But if a quasivariety is locally finite without the condition of being finitely generated), that lattice satisfies only the Pudlak-Tuma property. Translated fromAlgebra i Logika, Vol. 36, No. 6, pp. 605–620, Noember, 1997.     

Lattices with unique irreducible decompositions

We define the concept of a minimal decomposition in a lattice, and prove that all the currently known lattices with unique irreducible decompositions are in fact lattices with minimal ones. Also, the characterization of a class of lattices with minimal decompositions is given. A new proof of the Crawley-Dilworth characterization theorem for the class of coalgebraic strongly coatomic lattices with unique irreducible decompositions obtains as a consequence. In memory of Victor A. Gorbunov Supported by the RF State Committee of Higher Education (1998 project), by RFFR grant No. 99-01-00485, and jointly by RFFR and DFG grants Nos. 96-01-00097 and 436113/2670 respectively. Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 93–103, January–February, 2000.     

Maximal quasivarieties of groups

We consider a lattice L_q(qG) of quasivarieties contained in the quasivariety qG, generated by a polycyclic-by-finite group G. It is proved that the lattice contains a finite set of coatoms (i.e., proper maximal elements) and that each of its elements distinct from qG is contained in some coatom. We construct an example of a finitely generated solvable group B of derived length 3, whose quasivariety lattice L_q(qB) is freed of coatoms. Supported by RFFR grant No. 96-01-00088, and by the RF Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 279–290, May–June, 1998.     

Elementary equivalence for lattices of subalgebras of free algebras

A class of varieties V (including all finitely based lattice varieties) is determined for which the elementary equivalence of lattices of subalgebras of free V-algebras, F_v(X) and F_v(Y), is equivalent to sets X and Y being second-order equivalent. Supported by RFFR grant No. 99-01-00571. Supported by the National Research Foundation of the Republic of South Africa, and by the University of Cape Town Research Committee. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 595–601, September–October, 2000.     

Idempotent matrix lattices over distributive lattices

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an n-element set Qord(n) is not graduated for n ≥ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval ([I, J]_≤, ≤) of idempotent (n × n)-matrices over {ie879-01}-lattices is isomorphic to the lattice of quasi-orders Qord(n). Using this isomorphism, we calculate the lattice height of idempotent {ie879-02}-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 121–144, 2007.     

Intermediate subgroups of chevalley groups over a field of fractions of a ring of principal ideals

Let K be a field of fractions of a principal ideal ring R and G_K be a Chevalley group (of normal type) over K. For each subring P ⊂ K, denote by G_P a subgroup of all elements of G_K with coefficients in P. Let M be intermediate between G_R and G_K, i.e., G_R ⊆ M ⊆ G_K. We prove that M=G_P for some intermediate subring P (R ⊆ P ⊆ K). Supported by RFFR grant No. 96-01-00409. Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 347–358, May–June, 2000.     

A property of the lattice of equational theories

It had been conjectured that any algebraic lattice having a compact one could be represented as the lattice of equational theories extending some theory. However, we show that each lattice having such a representation satisfies a nontrivial quasidistributivity condition. In particular,M ₃ has no such representation.To the memory of András HuhnPresented by Walter Taylor.     

On finite lattice coverings

We consider finite lattice coverings of strictly convex bodies K. For planar centrally symmetric K we characterize the finite arrangements C _n such that conv , where C _n is a subset of a covering lattice for K (which satisfies some natural conditions). We prove that for a fixed lattice the optimal arrangement (measured with the parametric density) is either a sausage, a so-called double sausage or tends to a Wulff-shape, depending on the parameter. This shows that the Wulff-shape plays an important role for packings as well as for coverings. Further we give a version of this result for variable lattices. For the Euclidean d-ball we characterize the lattices, for which the optimal arrangement is a sausage, for large parameter. Received 19 May 1999.     

Sublattices of complete lattices with continuity conditions

Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice Id S of S is both algebraic and dually algebraic. Furthermore, if there are no infinite D-sequences in J(S), then Id S can be embedded into a direct product of finite lower bounded lattices. We also find a system of infinitary identities that characterize sublattices of complete, lower continuous, and join-semidistributive lattices. These conditions are satisfied by any (not necessarily finitely generated) lower bounded lattice and by any locally finite, join-semidistributive lattice. Furthermore, they imply M. Erné’s dual staircase distributivity.On the other hand, we prove that the subspace lattice of any infinite-dimensional vector space cannot be embedded into any ℵ₀-complete, ℵ₀-upper continuous, and ℵ₀-lower continuous lattice. A similar result holds for the lattice of all order-convex subsets of any infinite chain.Dedicated to the memory of Ivan RivalReceived April 4, 2003; accepted in final form June 16, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Lattice universality of free burnside groups

We generalize Whitman's theorem on the representation of lattices by partition lattices or, which is the same, by subgroup lattices of a suitable group. A sufficient condition is stated for a group variety to be lattice-universal (i.e., every lattice has a presentation by the subgroup lattice of a group in this variety). As a consequence, we infer that every couniable lattice is representable by the subgroup lattice of a finitely generated free Burnside group of a large enough odd exponent. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 587–611, September–October, 1996.     

Version of the Grothendieck theorem for subspaces of analytic functions in lattices

A version of Grothendieck’s inequality says that any bounded linear operator acting from a Banach lattice X to a Banach lattice Y acts from X(ℓ²) to Y (ℓ²) as well. A similar statement is proved for Hardy-type subspaces in lattices of measurable functions. Namely, let X be a Banach lattice of measurable functions on the circle, and let an operator T act from the corresponding subspace of analytic functions X_A to a Banach lattice Y or, if Y is also a lattice of measurable functions on the circle, to the quotient space Y/Y_A. Under certain mild conditions on the lattices involved, it is proved that T induces an operator acting from X_A(ℓ²) to Y (ℓ²) or to Y/Y_A(ℓ2), respectively. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 5–16.     

Layer-projective lattices. I

The class of layer-projective lattices is singled out. For example, it contains the lattices of subgroups of finite Abelianp-groups, finite modular lattices of centralizers that are indecomposable into a finite sum, and lattices of subspaces of a finite-dimensional linear space over a finite field that are invariant with respect to a linear operator with zero eigenvalues. In the class of layer-projective lattices, the notion of type (of a lattice) is naturally introduced and the isomorphism problem for lattices of the same type is posed. This problem is positively solved for some special types of layer-projective lattices. The main method is the layer-wise lifting of the coordinates. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 170–182, February, 1998.     

Quasivarieties of groups closed under wreath products and wreathZ-products

LetL _q(M) be a lattice of quasivarieties contained in a quasivarietyM. The quasivariety is closed under direct wreath Z-products if together with a group G, it contains its wreath product G ≀ Z with an infinite cyclic group Z. We prove the following: (a) ifM is closed under direct wreath Z-products then every quasivariety, which is a coatom inL _q(M), is likewise closed under these; (b) ifM is closed under direct wreath products thenL _q(M) has at most one coatom. An example of a quasivariety is furnished which is closed under direct wreath Z-products and whose subquasivariety lattice contains exactly one coatom. Also, it turns out that the set of quasivarieties closed under direct wreath Z-products form a complete sublatttice of the lattice of quasivarieties of groups. Supported by RFFR grant No. 96-01-00088, and by the RF Committee of Higher Education. Translated fromAlgebra is Logika, Vol. 38, No. 3, pp. 257–268, May–June, 1999.     

Completel-groupoids and their prime spectra

We argue for the validity of the infinite ∩-distributivity law in a lattice of radical elements of an arbitrary complete l-groupoid, in particular, in the lattice of radical ideals of an arbitrary ring. The notion of a prime spectrumSpec (L) is defined, and we point out a specific place that topological spaces of typeSpec (L) hold among all T_o-spaces. Supported by the AMS grant. Translated fromAlgebra i logika, Vol. 36, No. 3, pp. 341–355, May–June, 1997.     

Lattices of dominions of universal algebras

We fix a universal algebra A and its subalgebra H. The dominion of H in A (in a class M) is the set of all elements a ∈ A such that any pair of homomorphisms f, g: A → M ∈ M satisfies the following: if f and g coincide on H then f(a) = g(a). In association with every quasivariety, therefore, is a dominion of H in A. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that quasivariety. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 26–45, January–February, 2007.     

On the Subalgebra Lattice of Unary Algebras

We characterize pairs L, A, where Lis a lattice and Ais a unary partial algebra, such that the strong subalgebra lattice S_s(A) is isomorphic to L. Moreover, we find necessary and sufficient conditions for arbitrary unary partial algebras to have isomorphic strong subalgebra lattices. Observe, that for a total algebra A, the lattice S_s(A) is the usual well-known subalgebra lattice. Thus in particular we solve these two problems for total unary algebras and their lattices of (also total) subalgebras.For this purpose we apply some non-obvious connections between unary partial algebras and graphs from [9]. More precisely, we first characterize the pairs L, G, where Lis a lattice and Ga directed graph, such that the strong subdigraph lattice of Gis isomorphic to L. Next, we find a characterization of arbitrary digraphs with isomorphic strong subalgebra lattices. From these results we easily get solutions of our algebraic problems.     

Abstract isomorphisms of quasiminimal groups

A connected solvable algebraic group is called minimal if its center is trivial and it has no proper connected subgroups with trivial center. We show that abstract isomorphisms of minimal groups defined over fields of characteristic zero are standard. Supported by RFFR grants Nos. 96-01-01678 and 96-01-01675, and by the State Committee for Higher Education of Russia, grant No. 2. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 567–586, September–October, 1996.     

Distributivity conditions for lattices of dominions in quasivarieties of Abelian groups

Let ℳ be any quasivariety of Abelian groups, L_q(ℳ) be a subquasivariety lattice of ℳ, dom _G ^ℳ be the dominion of a subgroup H of a group G in ℳ, and G/dom _G ^ℳ (H) be a finitely generated group. It is known that the set L(G, H, ℳ) = {dom _G ^N (H)| N ∈ L_q(ℳ)} forms a lattice w.r.t. set-theoretic inclusion. We look at the structure of dom _G ^ℳ (H). It is proved that the lattice L(G,H,ℳ) is semidistributive and necessary and sufficient conditions are specified for its being distributive. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 484–499, July–August, 2006.     

Distributive Lattices with a Generalized Implication: Topological Duality

In this paper we introduce the notion of generalized implication for lattices, as a binary function ⇒ that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ⇒ defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized implication as a common abstraction of the notions of annihilator (Mandelker, Duke Math J 37:377–386, 1970), Quasi-modal algebras (Celani, Math Bohem 126:721–736, 2001), and weakly Heyting algebras (Celani and Jansana, Math Log Q 51:219–246, 2005). We introduce the suitable notions of morphisms in order to obtain a category, as well as the corresponding notion of congruence. We develop a Priestley style topological duality for the bounded distributive lattices with a generalized implication. This duality generalizes the duality given in Celani and Jansana (Math Log Q 51:219–246, 2005) for weakly Heyting algebras and the duality given in Celani (Math Bohem 126:721–736, 2001) for Quasi-modal algebras.     

A Distributive Lattice Structure Connecting Dyck Paths,Noncrossing Partitions and 312-avoiding Permutations

In [Ferrari, L. and Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference 135 (2005), 77–92] a natural order on Dyck paths of any fixed length inducing a distributive lattice structure is defined. We transfer this order to noncrossing partitions along a well-known bijection [Simion, R.: Noncrossing partitions. Discrete Math. 217 (2000), 367–409], thus showing that noncrossing partitions can be endowed with a distributive lattice structure having some combinatorial relevance. Finally we prove that our lattices are isomorphic to the posets of 312-avoiding permutations with the order induced by the strong Bruhat order of the symmetric group.