In this research announcement we outline the methods used in our recent proof that the Leech lattice is the unique densest lattice in R24. Complete details will appear elsewhere, but here we illustrate our techniques by applying them to the case of lattice packings in R2, and we discuss the obstacles that arise in higher dimensions.

In this paper we introduce fuzzy congruence and fuzzy strong congruence of semi-hypergroups as well as polygroups. We obtain some results in this respect. We also prove that FC(H), the set of all fuzzy strong congruences on a polygroup H , consists a complete lattice and this lattice is isomorphic to FN (H), the lattice of fuzzy normal subpolygroups.

We investigate the complexity of the lattice of local clones over a countably infinite base set. In particular, we prove that this lattice contains all algebraic lattices with at most countably many compact elements as complete sublattices, but that the class of lattices embeddable into the local clone lattice is strictly larger than that.

Let E be a Dedekind σ -complete Banach lattice, let Ea denote the order continuous part of E, and suppose that Ea is order dense in E. If E contains a lattice copy of ∞ (equivalently, E = Ea) then E/Ea contains a lattice copy of ∞/c0, and hence a lattice-isometric copy of ∞. This is one of the consequences of more general results presented in this paper.

An orthomodular lattice L is said to be interval homogeneous (resp. centrally interval homogeneous) if it is σ-complete and satisfies the following property: Whenever L is isomorphic to an interval, [a, b], in L then L is isomorphic to each interval [c, d] with c ≤ a and d ≥ b (resp. the same condition as above only under the assumption that all elements a, b, c, d are central in L). Let us den...

A finite lattice is representable if it is isomorphic to the congruence lattice of a finite algebra. In this paper, we develop methods by which we can construct new representable lattices from known ones. The techniques we employ are sufficient to show that every finite lattice which contains no three element antichains is representable. We then show that if an order polynomially complete latti...

We establish an asymptotic stability result for Toda lattice soliton solutions, by making use of a linearized Bäcklund transformation whose domain has codimension one. Combining a linear stability result with a general theory of nonlinear stability by Friesecke and Pego for solitary waves in lattice equations, we conclude that all solitons in the Toda lattice are asymptotically stable in an exp...

In this paper for any dimension n we give a complete description of lattice convex polytopes in R that are regular with respect to the group of affine transformations preserving the lattice.

In this paper for any dimension n we give a complete list of lattice convex polytopes in R that are regular with respect to the group of affine transformations preserving the lattice.

In this paper for any dimension n we give a complete description of lattice convex polytopes in R that are regular with respect to the group of affine transformations preserving the lattice.