A Compact Representation for Modular Semilattices and Its Applications

Order - A modular semilattice is a semilattice generalization of a modular lattice. We establish a Birkhoff-type representation theorem for modular semilattices, which says that every 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.     

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.     

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...     

Cutset Condition for Geometric Lattices

In this paper we prove that an atomistic lattice L of finite length is geometric if it has the nontrivial modular cutset condition, that is, every maximal chain of L contains a modular element which is different from the minimum element and the maximum element of L. The first author is partially supported by the National Natural Science Foundation of China (Grant no. 10471016).     

An extension theorem for modular measures on effect algebras

We prove an extension theorem for modular measures on lattice ordered effect algebras. This is used to obtain a representation of these measures by the classical ones. With the aid of this theorem we transfer control theorems, Vitali-Hahn-Saks, Nikodym theorems and range theorems to this setting.     

Defining relations of a free modular lattice of rank 3

For a 3-generated free modular lattice we obtain a set of 11 defining relations and prove that this set is minimal.     

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.     

Cancellable Elements of the Lattice of Epigroup Varieties

We completely describe all commutative epigroup varieties that are cancellable elements of the lattice EPI of all epigroup varieties. In particular, we prove that a commutative epigroup variety is a cancellable element of the lattice EPI if and only if it is a modular element of this lattice.     

A condition for modular lattices

This paper proves that a geometric lattice of rank n is a modular lattice if its every maximal chain contains a modular element of rank greater than 1 and less than n. This result is generalized to a more general lattices of finite rank. The first author is partially supported by the National Natural Science Foundation of China (Grant No. 10471016).     

Ojective ideals in modular lattices

The concept of an extending ideal in a modular lattice is introduced. A translation of module-theoretical concept of ojectivity (i.e. generalized relative injectivity) in the context of the lattice of ideals of a modular lattice is introduced. In a modular lattice satisfying a certain condition, a characterization is given for direct summands of an extending ideal to be mutually ojective. We define exchangeable decomposition and internal exchange property of an ideal in a modular lattice. It is shown that a finite decomposition of an extending ideal is exchangeable if and only if its summands are mutually ojective.     

Modular and lower-modular elements of lattices of semigroup varieties

The paper contains three main results. First, we show that if a commutative semigroup variety is a modular element of the lattice Com of all commutative semigroup varieties then it is either the variety $\mathcal{COM}$ of all commutative semigroups or a nilvariety or the join of a nilvariety with the variety of semilattices. Second, we prove that if a commutative nilvariety is a modular element of Com then it may be given within $\mathcal{COM}$ by 0-reduced and substitutive identities only. Third, we completely classify all lower-modular elements of Com. As a corollary, we prove that an element of Com is modular whenever it is lower-modular. All these results are precise analogues of results concerning modular and lower-modular elements of the lattice of all semigroup varieties obtained earlier by Je?ek, McKenzie, Vernikov, and the author. As an application of a technique developed in this paper, we provide new proofs of the ??prototypes?? of the first and the third our results.     

Matchings and radon transforms in lattices. I. Consistent lattices

An element in a lattice is join-irreducible if x=ab implies x=a or x=b. A meet-irreducible is a join-irreducible in the order dual. A lattice is consistent if for every element x and every join-irreducible j, the element xj is a join-irreducible in the upper interval [x, î]. We prove that in a finite consistent lattice, the incidence matrix of meet-irreducibles versus join-irreducibles has rank the number of join-irreducibles. Since modular lattices and their order duals are consistent, this settles a conjecture of Rival on matchings in modular lattices.     

The lattice of closure relations on a poset

In this paper we show that the set of closure relations on a finite posetP forms a supersolvable lattice, as suggested by Rota. Furthermore this lattice is dually isomorphic to the lattice of closed sets in a convex geometry (in the sense of Edelman and Jamison [EJ]). We also characterize the modular elements of this lattice (whenP has a greatest element) and compute its characteristic polynomial.Presented by R. W. Quackenbush.     

Extremal Bases for the Adjoint Representations of the Simple Lie Algebras

We construct n distinct weight bases, which we call extremal bases, for the adjoint representation of each simple Lie algebra 𝔤 of rank n: One construction for each simple root. We explicitly describe actions of the Chevalley generators on the basis elements. We show that these extremal bases are distinguished by their “supporting graphs” in three ways. (In general, the supporting graph of a weight basis for a representation of a semisimple Lie algebra is a directed graph with colored edges that describe the supports of the actions of the Chevalley generators on the elements of the basis.) We show that each extremal basis constructed is essentially the only basis with its supporting graph (i.e., each extremal basis is solitary), and that each supporting graph is a modular lattice. Each extremal basis is shown to be edge-minimizing: Its supporting graph has the minimum number of edges. The extremal bases are shown to be the only edge-minimizing as well as the only modular lattice weight bases (up to scalar multiples) for the adjoint representation of 𝔤. The supporting graph for an extremal basis is shown to be a distributive lattice if and only if the associated simple root corresponds to an end node for a “branchless” simple Lie algebra, i.e., type A, B, C, F, or G. For each extremal basis, basis elements for the Cartan subalgebra are explicitly expressed in terms of the h _i Chevalley generators.     

Finite Intervals in the Lattice of Topologies

We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T ₀-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive.     

Reducible classes of finite lattices

In this paper we study a notion of reducibility in finite lattices. An element x of a (finite) lattice L satisfying certain properties is deletable if L-x is a lattice satisfying the same properties. A class of lattices is reducible if each lattice of this class admits (at least) one deletable element (equivalently if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step). First we characterize the deletable elements in a pseudocomplemented lattice what allows to prove that the class of pseudocomplemented lattices is reducible. Then we characterize the deletable elements in semimodular, modular and distributive lattices what allows to prove that the classes of semimodular and locally distributive lattices are reducible. In conclusion the notion of reducibility for a class of lattices is compared with some other notions like the notion of order variety.     

Skein construction of idempotents in Birman-Murakami-Wenzl algebras

We give skein theoretic formulas for minimal idempotents in the Birman-Murakami-Wenzl algebras. These formulas are then applied to derive various known results needed in the construction of quantum invariants and modular categories. In particular, an elementary proof of the Wenzl formula for quantum dimensions is given. This proof does not use the representation theory of quantum groups and the character formulas. Received: 26 September 2000 / Published online: 17 August 2001     

Free 3-Generated Lattices with Two Semi-Normal Generators

We consider a lattice generated by three elements, two of which are semi-normal. We have proved it is infinite and have also found an additional condition for the lattice to be modular. As a result, it is proved that the sublattice generated in the subgroup lattice by two normal subgroups and any subgroup is modular.     

Computational efficiency analysis of Wu et al.’s fast modular multi-exponentiation algorithm

Very recently, for speeding up the computation of modular multi-exponentiation, Wu et al. presented a fast algorithm combining the complement recoding method and the minimal weight binary signed-digit representation technique. They claimed that the proposed algorithm reduced the number of modular multiplications from 1.503k to 1.306k on average, where the value k is the maximum bit-length of two exponents. However, in this paper, we show that their claim is unwarranted. We analyze the computational efficiency of Wu et al.’s algorithm by modeling it as a Markov chain. Our main result is that Wu et al.’s algorithm requires 1.471k modular multiplications on average.