Z-Join Spectra of Z-Supercompactly Generated Lattices

The main result of this paper is a generalization of the classical equivalence between the category of continuous posets and the category of completely distributive lattices, based on the fact that the continuous posets are precisely the spectra of completely distributive lattices. Here we show that for so-called hereditary and union complete subset selections Z, the category of Z-continuous posets is equivalent (via a suitable spectrum functor) to the category of Z-supercompactly generated lattices; these are completely distributive lattices with a join-dense subset of certain Z-hypercompact elements. By appropriate change of the morphisms, these equivalences turn into dualities. We present two different approaches: the first one directly uses the Z-join ideal completion and the Z-below relation; the other combines two known equivalence theorems, namely a topological representation of Z-continuous posets and a general lattice theoretical representation of closure spaces.     

On Jump-Critical Posets with Jump-Number Equal to Width

We investigate the structure of m-jump-critical posets P with w(P) = m. We prove that the size of such posets satisfies |P| 3m ². For the special when the maximum antichain occurs as the maximal (or minimal) elements, we have the sharp upper bound |P| 3m - k; where k = min {|{Max}(P)|, |{Min}(P)|}. We give examples of posets which illustrate the explored structure of these m-jump-critical posets P with width m.     

The structure of a class of Z-local rings

A local ring R is called Z-local if J(R) = Z(R) and J(R)² = 0. In this paper the structure of a class of Z-local rings is determined.     

An Assmus–Mattson-Type Approach for Identifying 3-Designs from Linear Codes over Z 4

The complete weight enumerator of the Delsarte–Goethals code over Z ₄ is derived and an Assmus–Mattson-type approach at identifying t-designs in linear codes over Z ₄ is presented. The Assmus–Mattson-type approach, in conjunction with the complete weight enumerator are together used to show that the codewords of constant Hamming weight in both the Goethals code over Z ₄ as well as the Delsarte–Goethals code over Z ₄ yield 3-designs, possibly with repeated blocks.     

ComplexK-theory ofBSL3(Z)

ComplexK-theory ofBSL₃(Z) andBSt₃(Z) are computed. We first study a Brown-Peterson (BP) version of Soulé's arguments. Then we consider complexK-theory by using a Conner-Floyd type theorem.     

3-Designs from the Z 4-Goethals Codes via a New Kloosterman Sum Identity

Recently, active research has been performed on constructing t-designs from linear codes over Z ₄. In this paper, we will construct a new simple 3 – (2 ^m , 7, 14/3 (2 ^m – 8)) design from codewords of Hamming weight 7 in the Z ₄-Goethals code for odd m 5. For 3 arbitrary positions, we will count the number of codewords of Hamming weight 7 whose support includes those 3 positions. This counting can be simplified by using the double-transitivity of the Goethals code and divided into small cases. It turns out interestingly that, in almost all cases, this count is related to the value of a Kloosterman sum. As a result, we can also prove a new Kloosterman sum identity while deriving the 3-design.     

Codes with the Same Weight Distributions as the Goethals Codes and the Delsarte-Goethals Codes

The Goethals code is a binary nonlinear code of length 2^m+1 which has codewords and minimum Hamming distance 8 for any odd . Recently, Hammons et. al. showed that codes with the same weight distribution can be obtained via the Gray map from a linear code over Z ₄of length 2^m and Lee distance 8. The Gray map of the dual of the corresponding Z ₄ code is a Delsarte-Goethals code. We construct codes over Z ₄ such that their Gray maps lead to codes with the same weight distribution as the Goethals codes and the Delsarte-Goethals codes.     

A Criterion for Designs in ℤ4-codes on the Symmetrized Weight Enumerator

The Assmus–Mattson theorem is known as a method to find designs in linear codes over a finite field. It is an interesting problem to find an analog of the theorem for Z ₄-codes. In a previous paper, the author gave a candidate of the theorem. The purpose of this paper is to give an improvement of the theorem. It is known that the lifted Golay code over Z ₄ contains 5-designs on Lee compositions. The improved method can find some of those without computational difficulty and without the help of a computer.     

Möbius Polynomials of Face Posets of Convex Polytopes

The Möbius polynomial is an invariant of ranked posets, closely related to the Möbius function. In this paper, we study the Möbius polynomial of face posets of convex polytopes. We present formulas for computing the Möbius polynomial of the face poset of a pyramid or a prism over an existing polytope, or of the gluing of two or more polytopes in terms of the Möbius polynomials of the original polytopes. We also present general formulas for calculating Möbius polynomials of face posets of simplicial polytopes and of Eulerian posets in terms of their f-vectors and some additional constraints.     

Lim-inf convergence in partially ordered sets

The lim-inf convergence in a complete lattice was introduced by Scott to characterize continuous lattices. Here we introduce and study the lim-inf convergence in a partially ordered set. The main result is that for a poset P the lim-inf convergence is topological if and only if P is a continuous poset. A weaker form of lim-inf convergence in posets is also discussed.     

The Cohen--Lusk Conjecture

The question of Cohen and Lusk about the partial gluing of an orbit under a map of a free Z _p-space to ⁿ is answered in part.     

Dense Families of Selections and Finite-Dimensional Spaces

A characterization of n-dimensional spaces via continuous selections avoiding Z _n -sets is given, and a selection theorem for strongly countable-dimensional spaces is established. We apply these results to prove a generalized Ostrand's theorem, and to obtain a new alternative proof of the Hurewicz formula. It is also shown that our selection theorem yields an easy proof of a Michael's result.     

The Involutory Dimension of Involution Posets

The involutory dimension, if it exists, of an involution poset P:=(P,,) is the minimum cardinality of a family of linear extensions of , involutory with respect to , whose intersection is the ordering . We show that the involutory dimension of an involution poset exists iff any pair of isotropic elements are orthogonal. Some characterizations of the involutory dimension of such posets are given. We study prime order ideals in involution posets and use them to generate involutory linear extensions of the partial ordering on orthoposets. We prove several of the standard results in the theory of the order dimension of posets for the involutory dimension of involution posets. For example, we show that the involutory dimension of a finite orthoposet does not exceed the cardinality of an antichain of maximal cardinality. We illustrate the fact that the order dimension of an orthoposet may be different from the involutory dimension.     

q-Weighted log-concavity and q-product theorem on the normality of posets

We show that q-weighted log-concavity and the strict normalized matching property are preserved under the q-direct product over weighted posets. As consequences, two classes of weighted posets including the finite linear lattices are strictly q-weighted log-concave and strictly normal.     

Z n -Orthograded Monocomposition Algebras

We study NQM algebras A having an orthogonal automorphism of finite order n 3 (called Z _n -orthograded NQM algebras). The Z ₃-orthograded NQM algebras of dimension 7 are treated in more detail. In particular, we find all algebras A which are not bi-isotropic in this class, and for every algebra A, determine an automorphism group Aut,A and an orthogonal automorphism group Ortaut,A. In constructing and classifying (up to isomorphism) NQM algebras, use is made of orthogonal decompositions of the algebras.     

Existence of Z-cyclic 3PTWh (p) for any Prime p≡ 1 (mod 4)

We show that a Z-cyclic triplewhist tournament on p players with three-person property, briefly 3PTWh(p), exists for any prime p≡ 1 (mod 4) with the only exceptions of p=5,13,17.     

Commuting involutions with fixed point set of constant codimension

Special generators of the unoriented cobordism ringMO _* are constructed to determine the groupsJ _n,κ ^r ofn-dimensional cobordism classes inMO _n containing a representativeM ⁿ admitting a (Z ₂) ^k -action with fixed point set of constant codimension. This work is supported by HNSF     

Lattices That are Embeddable in Suborder Lattices

Various types of lattices are embedded in suborder lattices of posets possessing certain properties. In particular, it is shown that the class of lattices isomorphic to sublattices of suborder lattices of posets of length at most n is a variety, for any n < ω.Supported by INTAS-YSF grant 2001/1-65, by RFBR-DFG grant No. 01-01-04003, by RF Ministry of Education grant No. E02-1.0-32, by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2112.2003.1, and by a grant from the Russian Science Support Foundation.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 483–511, July–August, 2005.     

Socles of Buchsbaum modules, complexes and posets

The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel's conjecture for the maximum value of the Euler characteristic of a 2k-dimensional simplicial manifold on n vertices as well as Kalai's conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension, number of vertices, and the first Betti number.