On 3-generated lattices with a completely modular element among generators

We consider a lattice generated by three elements, one of which is completely modular. The free lattice with this property is proved to be finite. It is not modular and contains exactly 39 elements. We have also found a finite set of defining relations for the generating elements of this lattice.     

3-Generated Lattice with Left Modular and Separating Generators

Mathematical Notes - We consider a lattice generated by three elements, two of which are a left modular element and a separating element. It is proved that such a lattice is finite and contains at...     

Finitely generated lattices with <Emphasis Type="Italic">M</Emphasis>-standard elements among generators

We prove that a lattice is modular if it is generated by three elements, two of which are M-standard. We also show that a lattice generated by n, n > 3, M-standard elements should not necessarily be modular.     

Varieties of Lattices with Geometric Descriptions

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann et al. proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem for free lattices. This solves two problems stated by Huhn in 1985. We prove that every modular (resp., n-distributive) lattice embeds within its variety into some strongly spatial lattice. Every lattice which is either algebraic modular spatial or bi-algebraic is strongly spatial. We also construct a lattice that cannot be embedded, within its variety, into any algebraic and spatial lattice. This lattice has a least and a largest element, and it generates a locally finite variety of join-semidistributive lattices.     

The Schur property for subgroup lattices of groups

A classical theorem of Schur states that if the centre of a group G has finite index, then the commutator subgroup G′ of G is finite. A lattice analogue of this result is proved in this paper: if a group G contains a modularly embedded subgroup of finite index, then there exists a finite normal subgroup N of G such that G/N has modular subgroup lattice. Here a subgroup M of a group G is said to be modularly embedded in G if the lattice is modular for each element x of G. Some consequences of this theorem are also obtained; in particular, the behaviour of groups covered by finitely many subgroups with modular subgroup lattice is described. Received: 16 October 2007, Final version received: 22 February 2008     

Polycyclic groups with modular finite homomorphic images

A group G is said to be a modular group if it has modular subgroup lattice. We will prove in this paper that a polycyclic group G is modular if and only if all its finite homomorphic images are modular groups. Similar results will also be obtained for other conditions of modular type.     

Large characteristic subgroups with modular subgroup lattice

It is known that if a group contains an abelian subgroup of finite index, then it also has an abelian characteristic subgroup of finite index. The aim of this paper is to prove that corresponding results hold when abelian subgroups are replaced either by subgroups having a modular subgroup lattice or by quasihamiltonian subgroups.     

分配格与模格的另一二条件等价定义

针对分配格与模格的格等式定义问题,得知了二条件是定义分配格与模格的最少条件,并进一步证明了Sholander's basis是定义分配格的最短最少变量格等式,最后又从分配格和模格的基本定义出发给出了新的分配格的二条件和三条件等价定义等式及模格的二条件与三条件等价定义等式.     

Impossible specifications for the modular group

With any subgroup of the modular group we associate a set of non-negative integers satisfying two conditions. This set is the specification of the subgroup. Several sets were known which satisfied both conditions, but which did not correspond to subgroups. Here we find several infinite families of such sets. We also investigate situations where a given set corresponds to just one conjugacy class of subgroups. These are related to the well-known lattice subgroups. The method involves the use of coset diagrams, described by Atkin and Swinnerton-Dyer in [1].     

On modular lattices generated by chains

We describe the free modular lattice generated by two chains and a single point, under the assumption that there are few meets. Received February 11, 2005; accepted in final form August 11, 2005.     

A note on the finiteness of certain cuspidal divisor class groups

We prove the finiteness of the subgroup of the Jacobian generated by the cusps of a Drinfeld modular curve. This is an analogue in positive characteristic of a classical result of Drinfeld and Manin about elliptic modular curves, and generalizes earlier work on special cases.     

Groups with Few Nonmodular Subgroups

Let G be a Tarski-free group such that the join of all nonmodular subgroups of G is a proper subgroup in G. It is proved that G contains a finite normal subgroup N such that the quotient group G/N has a modular subgroup lattice.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1419 – 1423, October, 2004.     

Lower-Modular Elements of the Lattice of Semigroup Varieties

We call a semigroup variety modular [upper-modular, lower-modular, neutral] if it is a modular [respectively upper-modular, lower-modular, neutral] element of the lattice of all semigroup varieties. It is proved that if V is a lower-modular variety then either V coincides with the variety of all semigroups or V is periodic and the greatest nil-subvariety of V may be given by 0-reduced identities only. We completely determine all commutative lower-modular varieties. In particular, it turns out that a commutative variety is lower-modular if and only if it is neutral. A number of corollaries of these results are obtained.     

关于有1模格的等价定义

进一步讨论有1模格的等价定义问题,得到并证明了一个(2,2,0)型代数成为有1模格的一个充分必要条件.这样大大简化了有1模格的等价定义.     

Cohen-Macaulay types of subgroup lattices of finite abelianp-groups

For a minimal free resolution of a Stanley-Reisner ring constructed from the order complex of a modular lattice. T. Hibi showed that its last Betti number (called the Cohen-Macaulay type) is computed by means of the Möbius function of the given modular lattice. Using this result, we consider the Stanley-Reisner ring of the subgroup lattice of a finite abelianp-group associated with a given partition, and show that its Cohen-Macaulay type is a polynomial inp with integer coefficients.     

格中两类分配等式的关系研究

针对离散数学经典教材中提出的"交运算对并运算的分配等式和并运算对交运算的分配等式是等价的"这一结论,分析了一种常见的错误证明,通过一个反例说明该结论在一般的格中不一定成立,进一步证明这两个分配等式在且仅在模格中是等价的,并提出利用定义判断一个模格是否是分配格的简便算法.作为一个应用,重新证明了该教材中的一条定理.     

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.     

Non-Axiomatizability of the amalgamation class of modular lattice varieties

We prove that if v is the variety generated by a finite modular lattice, then v is not an elementary class. We also consider the same question for the variety generated by N ₅.Research partially supported by National Science Foundation grant DMS-8701643.     

Minimal Paths between Maximal Chains in Finite Rank Semimodular Lattices

We study paths between maximal chains, or flags, in finite rank semimodular lattices. Two flags are adjacent if they differ on at most one rank. A path is a sequence of flags in which consecutive flags are adjacent. We study the union of all flags on at least one minimum length path connecting two flags in the lattice. This is a subposet of the original lattice. If the lattice is modular, the subposet is equal to the sublattice generated by the flags. It is a distributive lattice which is determined by the Jordan-Hölder permutation between the flags. The minimal paths correspond to all reduced decompositions of this permutation. In a semimodular lattice, the subposet is not uniquely determined by the Jordan-Hölder permutation for the flags. However, it is a join sublattice of the distributive lattice corresponding to this permutation. It is semimodular, unlike the lattice generated by the two flags, which may not be ranked. The minimal paths correspond to some reduced decompositions of the permutation, though not necessarily all. We classify the possible lattices which can arise in this way, and characterize all possibilities for the set of shortest paths between two flags in a semimodular lattice.     

Finite modular congruence lattices

We consider the variety of modular lattices generated by all finite lattices obtained by gluing together some M₃’s. We prove that every finite lattice in this variety is the congruence lattice of a suitable finite algebra (in fact, of an operator group). Received February 26, 2004; accepted in final form December 16, 2004.