Intrinsic metrics for lattice ordered groups

Swamy and Jakubik studied the metric ¦x y¦ on lattice ordered groups, and isometries which presere it. We show the only intrinsic metrics on lattice ordered groups are the multiplesn ¦x–y ¦ of theirs, and that the triangle inequality is satisfied by such a metric iff the group is abelian. We show that there are isometries for each of these metrics, but they are rare. We give a simpler proof via permutation groups of the following augmented version of a theorem of Jakubik. IfT is an isometry of the lattice ordered groupG with respect to the metric ¦x¥¦ andT(0)=0, thenG=AB, B is abelian, andT(a+b)=a–b; conversely, any suchT is an isometry.To Paul Conrad on his 60th birthdayPresented by L. Fuchs.     

Topological groups with the coalgebraic lattice of closed subgroups

Locally compact groups with a coalgebraic lattice of closed subgroups are investigated and Abelian and zero-dimensional nilpotent groups with a coalgebraic lattice of closed subgroups are described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 7 and 8, pp. 1102–1107, July–August, 1991.     

Embedding Finite Lattices into Finite Biatomic Lattices

For a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively,– the class of all finite lattices;– the class of all finite lower bounded lattices (solved by the first author's earlier work);– the class of all finite join-semidistributive lattices (this problem was, until now, open).We solve the latter problem by finding a quasi-identity valid in all finite, atomistic, biatomic, join-semidistributive lattices but not in all finite join-semidistributive lattices.     

Finite sublattices in the lattice of clones

The lattice of clones of functions over a k-element set is studied. It is shown that every lattice which is a countable direct product of finite lattices is embedded (up to isomorphism) in and, hence, in for k 4. This directly implies that every finite and any countable residually finite lattice is embedded in , k 4, and that no nontrivial quasi-identity holds in , k 4. A number of particular lattices (which are free in some lattice varieties) embeddable in , k 4,are presented. Translated fromAlgebra i Logika, Vol. 33, No. 5, pp. 514–549, September–October, 1994.     

Two classes of finite groups whose Chermak-Delgado lattice is a chain of length zero

It is an open question in the study of Chermak-Delgado lattices precisely which finite groups G have the property that 𝒞𝒟(G) is a chain of length 0. In this note, we determine two classes of groups with this property. We prove that if G = AB is a finite group, where A and B are abelian subgroups of relatively prime orders with A normal in G, then the Chermak-Delgado lattice of G equals {AC_B(A)}, a strengthening of earlier known results.     

A lattice in more than two Kac-Moody groups is arithmetic

Let Γ < G ₁ × … × G _n be an irreducible lattice in a product of infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n ≥ 3, then each G _i is a simple algebraic group over a local field and Γ is an S-arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n ≥ 2: either Γ is an S-arithmetic (hence linear) group, or Γ is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.     

Enumeration of lattice paths and generating functions for skew plane partitions

n-dimensional lattice paths not touching the hyperplanesX _i–X _i+1=–1,i=1,2,...,n, are counted by four different statistics, one of which is MacMahon's major index. By a reflection-like proof, heavily relying on Zeilberger's (Discrete Math. 44(1983), 325–326) solution of then-candidate ballot problem, determinantal expressions are obtained. As corollaries the generating functions for skew plane partitions, column-strict skew plane partitions, reverse skew plane plane partitions and column-strict reverse skew plane partitions, respectively, are evaluated, thus establishing partly new results, partly new proofs for known theorems in the theory of plane partitions.     

Finite groups in which all p-subnormal subgroups form a lattice

O. Kegel, in 1962, introduced the concept of p-subnormal subgroups of a finite group as the subgroups whose intersections with all Sylow p-subgroups of the group are Sylow p-subgroups of the subgroup. The set of p-subnormal subgroup of a finite group is not a lattice in general. In this paper, the class of all finite groups in which all p-subnormal subgroups from a lattice is determined. This is the class of all finite p-soluble groups whose p-length and p′-length, both, are less or equal to 1. The join-semilattice case and the meet-semilattice case are analyzed separately. The authors are supported by Proyecto PB 94-1048 of DGICYT, Ministerio de Educación y Ciencia of Spain.     

Auslander-Reiten quivers of local,orders of finite lattice type

The classification of the Auslander-Reiten quivers of the local orders of finite lattice type is completed. For this purpose, it is shown—using the results of [7]—that to the list of the known Auslander-Reiten quivers of the local Bäckström orders of finite lattice type [11], [14] and of the local Gorenstein orders of finite type and their minimal overorders [18] one has to add two remaining types of valued translation quivers to obtain a complete list of all Auslander-Reiten quivers of the local orders of finite lattice type.     

Modules lattice isomorphic to linearly compact modules

We study modules that are lattice isomorphic to linearly compact modules (in the discrete topology). In particular, we establish the equivalence of the following properties of a moduleM: 1)M satisfies the Grothendieck property AB5^* and all its submodules are Goldie finite-dimensional; 2)M is lattice isomorphic to a linearly compact module; 3)M is lattice antiisomorphic to a linearly compact module. We show that a linearly compact module cannot be determined in terms of the lattice of its submodules.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 174–181, February, 1996.The author wishes to thank A. V. Mikhalev and A. A. Tuganbaev for valuable discussions and remarks.     

Elasticity of the crystal lattice of polyethylene terephthalate

The elasticity of the crystal lattice of polyethylene terephthalate was studied along and across the axes of the polymer molecules. The elastic modulus across the chains depended on the degree of crystallinity and the interplane distances in the directions of thea and b parameters of the unit cell. The nature of the elastic deformation in the crystal lattice was analyzed, and its elastic modulus along and across the axes of the chains was calculated. On loading biaxially oriented amorphous-crystalline polymers with a tensile stress applied in the direction of one of the orientation axes of the polymer, the stresses in the crystallites oriented along and across the external applied force and the amorphous regions in series with them were equal to the stress in the sample averaged over the cross section.Deceased.Translated from Mekhanika Polimerov, No. 6, pp. 982–986, November–December, 1972.     

Characterization of infinite Chernikov groups

It is shown that an infinite locally finite group is a Chernikov group if and only if its Cartesian square G × G contains a subgroup T of finite index such that Aut T possesses the four group V as subgroup with Chernikov centralizer C_T(v) and the centralizers of involutions in V are weakly isolated in T.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 674–677, May, 1990.     

On the structure of subsets of a vector lattice that are closed with respect to addition

The structure of additive semigroups generated by finite sets of n -dimensional vectors with positive integral coordinates is investigated. It is proved that each such semigroup contains a sub-semigroup isomorphic to a convex hull (in the lattice) of this semigroup.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 99–103, 1992.     

Algebraic form of a theorem of Hahn—Banach type for lattice manifolds

Let E be a vector lattice. A linear functionalf on E is called a lattice homomorphism iff(sup (x, y)) = max (f(x),f(y)) for all x, y E. For lattice homomorphisms a theorem of Hahn—Banach type is valid. In this note we prove an algebraic analog of this theorem.Translated from Matematicheskie Zametki, Vol. 16, No. 4, pp. 595–600, October, 1974.In conclusion the author expresses his thanks to D. A. Raikov for his statement of the problem and his interest in my work.     

Classifying spaces of Kač-Moody groups

We study the structure of classifying spaces of Kač-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes'T functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kač-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kač-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kač-Moody groups, and centralizers of finite p-subgroups. Received: 15 June 2000 / in final form: 20 September 2001 / Published online: 29 April 2002     

On the algebra associated with a geometric lattice

Let L be a geometric lattice. Following P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. math. 56 (1980), 167–189, we associate with L a graded commutative algebra A(L). In this paper we introduce a new invariant ψ of the algebra A(L) which suffices to distinguish algebras for which all other known invariants coincide. This result is applied to the study of arrangements of complex hyperplanes, with L being the intersection lattice. In this case A(L) is isomorphic to the cohomology algebra of the associated hyperplane complement. The goal is to find examples of arrangements with non-isomorphic lattices but homotopy equivalent complements. The invariant introduced here effectively narrows the list of candidates. Nevertheless, we exhibit combinatorially inequivalent arrangements for which all known invariants, including ψ, coincide.     

The property of being algebraic for the lattice of all τ-closed totally saturated formations

It is proved that the lattice of all τ-closed totally saturated formations of finite groups is algebraic. __________ Translated From Algebra I Logika, Vol. 45, No. 5, pp. 620–626, September–October, 2006.     

Fixed point theorems for nonexpansive mappings and Suzuki-generalized nonexpansive mappings on a Banach lattice

We prove the existence of fixed points for multivalued nonexpansive nonself-mappings on a weakly orthogonal reflexive Banach lattice with uniformly monotone norm. Moreover, for single-valued mappings, we extend Betiuk-Pilarska and Prus’s result [A. Betiuk-Pilarska, S. Prus, Banach lattices which are order uniformly noncreasy, J. Math. Anal. Appl. 342 (2008) 1271–1279] on the weak fixed point property to continuous mappings satisfying condition (C) on a w-weakly orthogonal OUNC Banach lattice.     

Minimal number of generators of the lattice of subspaces of a finite-dimensional linear space

It is proved that a minimal generating system of the lattice of all subspaces of a finite-dimensional vector space over a finite field of q elements contains at most max(q+3) elements. This bound does not depend on the dimension of the space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 148–149, 1982.     

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.