A representation of m-algebraic lattices

In this paper, we give a short proof of the following result of G. Grätzer and E. T. Schmidt: every m-algebraic lattice can be represented as the lattice of m-complete congruence relations of some m-complete modular lattice.Dedicated to Bjarni Jonsson on his 70th birthdayThe research of the first author was supported by the NSERC of Canada.The research of the third author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

Congruence lattices ofp-algebras

T. Kartriák proved the following theorem:Every finite distributive lattice is the congruence lattice of a finite p-algebra. We provide a short proof, and a generalization, of this result.Presented by J. Berman.The research of the first author was supported by the NSERC of Canada.The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

Multipasting of lattices

In this paper we introduce a lattice construction, calledmultipasting, which is a common generalization of gluing, pasting, andS-glued sums. We give a Characterization Theorem which generalizes results for earlier constructions. Multipasting is too general to prove the analogues of many known results. Therefore, we investigate in some detail three special cases: strong multipasting, multipasting of convex sublattices, and multipasting with the Interpolation Property.Presented by R. Freese.The research of the first and the third authors was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1813. The research of the second author was supported by the NSERC of Canada.     

On isotone functions with the Substitution Property in distributive lattices

A universal algebra isaffine complete if all functions satisfying the Substitution Property are polynomials (composed of the basic operations and the elements of the algebra). In 1962, the first author proved that a bounded distributive lattice is affine complete if and only if it does not contain a proper Boolean interval. Recently, M. Ploica generalized this result to arbitrary distributive lattices.In this paper, we introduce a class of functions on a latticeL, we call themID-polynomials, that derive from polynomials on the ideal lattice (resp., dual ideal lattice) ofL; they are isotone functions and satisfy the Substitution Property. We prove that for a distributive latticeL, all unary functions with the Substitution Property are ID-polynomials if and only ifL contains no proper Boolean interval.The research of the first author was supported by the NSERC of Canada. The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

A counterexample to the finite height conjecture

There is an infinite subdirectly irreducible lattice which generates a variety that contains only finitely many subvarietes.The author was supported in part by NSF Grant DMS 94-00511     

The Hardy-Littlewood property of Banach lattices

We study a property of a Banach lattice which is characterized by the boundedness in several classical spaces of a version of the Hardy-Little-wood maximal function obtained by taking the supremum of averages in the order of the lattice. This property is related to the well known U.M.D. condition. The first and third authors were partially supported by DGICYT, Spain, under Grant PB 86-108. The second author was supported by Ministerio de Educación, Spain, under a Sabbatical Grant (1989).     

The complex of maximal lattice free simplices

The simplicial complexK(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form (x: Axb), withA a fixed generic (n + 1) ×n matrix. The topological space associated withK(A) is shown to be homeomorphic to ⁿ , and the space obtained by identifying lattice translates of these simplices is homeorphic to then-torus.Corresponding author.The first author was partially supported by Hungarian NSF grants 1907 and 1909, and also by U.S. NSF grant CCR-9111491. The research of the second author was supported by DMS9103608 and the third author by NSF grant SES9121936.     

Congruence lattices of function lattices

Thefunction lattice L ^P is the lattice of all isotone maps from a posetP into a latticeL.D. Duffus, B. Jónsson, and I. Rival proved in 1978 that for afinite poset P, the congruence lattice ofL ^P is a direct power of the congruence lattice ofL; the exponent is |P|.This result fails for infiniteP. However, utilizing a generalization of theL ^P construction, theL[D] construction (the extension ofL byD, whereD is a bounded distributive lattice), the second author proved in 1979 that ConL[D] is isomorphic to (ConL) [ConD] for afinite lattice L.In this paper we prove that the isomorphism ConL[D](ConL)[ConD] holds for a latticeL and a bounded distributive latticeD iff either ConL orD is finite.The research of the first author was supported by the NSERC of Canada.The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

Monotone clones and the varieties they determine

A finite, nontrivial algebra is order-primal if its term functions are precisely the monotone functions for some order on the underlying set. We show that the prevariety generated by an order-primal algebra P is relatively congruence-distributive and that the variety generated by P is congruence-distributive if and only if it contains at most two non-ismorphic subdirectly irreducible algebras. We also prove that if the prevarieties generated by order-primal algebras P and Q are equivalent as categories, then the corresponding orders or their duals generate the same order variety. A large class of order-primal algebras is described each member of which generates a variety equivalent as a category to the variety determined by the six-element, bounded ordered set which is not a lattice. These results are proved by considering topological dualities with particular emphasis on the case where there is a monotone near-unanimity function.This research was carried out while the third author held a research fellowship at La Trobe University supported by ARGS grant B85154851. The second author was supported by a grant from the NSERC.     

The canonical join complex for biclosed sets

The canonical join complex of a semidistributive lattice is a simplicial complex whose faces are canonical join representations of elements of the semidistributive lattice. We give a combinatorial classification of the faces of the canonical join complex of the lattice of biclosed sets of segments supported by a tree, as introduced by the third author and McConville. We also use our classification to describe the elements of the shard intersection order of the lattice of biclosed sets. As a consequence, we prove that this shard intersection order is a lattice.     

Amalgamation in the pentagon variety

The amalgamation class Amal (N) of a lattice variety generated by a pentagon is considered. It is shown that Amal (N) is closed under reduced products and therefore is an elementary class determined by Horn sentences. The above result is based on a new characterization of Amal (N). The lattice varieties whose amalgamation classes contain Amal (N) as a subclass are considered.Presented by G. Grätzer.The work of the first and third authors was supported by grants from the University of Cape Town Research Committee. That of the third author was supported by a grant from the South African Council for Scientific and Industrial Research.     

Morse theory on spaces of braids and Lagrangian dynamics

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index.?In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits. Oblatum 11-V-2001 & 13-XI-2002?Published online: 24 February 2003 RID="*" ID="*"The first author was supported by NSF DMS-9971629 and NSF DMS-0134408. The second author was supported by an EPSRC Fellowship. The third author was supported by NWO Vidi-grant 639.032.202.     

Chains of Modular Elements and Lattice Connectivity

We show that the order complex of any finite lattice with a chain of modular elements is at least (r−2)-connected. The first author was supported during part of this work by a postdoctoral fellowship from the Mathematical Sciences Research Institute. The second author was supported by NSF grant DMS-0300483.     

Solution of the linear inverse vector optimization problem by a single linear program

It is shown that finding a solution to a linear vector optimization problem which is efficient with respect to the constraints as well as to the objectives is equivalent to solving a single linear program.The research of this author was supported by NSF Grant DCR74-20584.The research of this author was partially supported by Canada Council Grant W760467.     

Quasi-Modal Lattices

We introduce the class of bounded distributive lattices with two operators, Δ and ∇, the first between the lattice and the set of its ideals, and the second between the lattice and the set of its filters. The results presented can be understood as a generalization of the results obtained by S. Celani. Jorge Castro: The work of the first author was partially supported by Grant BFM2001-3329 of D.G.I.C.Y.T. of Spain.     

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).     

On Stability and Stabilizability of Singular Stochastic Systems with Delays

This paper deals with the class of continuous-time singular linear systems with Markovian jump parameters and time delays. Sufficient conditions on the stochastic stability and stochastic stabilizability are developed. A design algorithm for a state feedback controller which guarantees that the closed-loop dynamics will be regular, impulse free, and stochastically stable is proposed in terms of the solutions to linear matrix inequalities. The research of this author was supported by NSERC Grant RGPIN36444-02. The research of this author was supported by the Program for a New Century of Excellent Talents in the Universities and by the Foundation for the Authors of National Excellent Doctoral Dissertations of P. R. China, Grant 200240. The research of this author was supported by HKU Grant RGC 7029/05P.     

Characterizations of Complete Sublattices of a Given Complete Lattice

To characterize all complete sublattices of a given complete lattice is an important problem. In this paper we will give three different characterizations of complete sublattices of a complete lattice by using closure operators, kernel operators, and by using Galoisclosed subrelations.The research was supported by a grant of the faculty of Science ChiangMai University Thailand.     

Lattices of Optimal Finite Lattice Packings

We consider finite lattice ball packings with respect to parametric density and show that densest packings are attained in critical lattices if the number of translates and the density parameter are sufficiently large. A corresponding result is not valid for general centrally symmetric convex bodies.The second author was partially supported by a DAAD Postdoc fellowship and the hospitality of Peking University during his work.     

John domains for the control distance of diagonal vector fields

We study John domains in the metric space associated with a system of diagonal vector fields. First author was supported by the Swiss National Science Foundation. Second author was supported by University of Bologna, funds for selected research topics.