We discuss orthogonal Chebyshev-Frolov lattices, their generating matrices and use in Frolov cubature formula. give a detailed account on coordinate-permuted systems that lead to fast computation enumeration of such lattices. In particular, we explain the recurrences identified (K. Suzuki T. Yoshiki, Hiroshima Math. J., 49(1):139–159, 2019) via plain constructive approach exhibiting new hierarc...

In nonlinear disordered Hamiltonian lattices, where there are no propagating phonons, the spreading of energy is of subdiffusive nature. Recently, the universality class of the subdiffusive spreading according to the nonlinear diffusion equation (NDE) has been suggested and checked for one-dimensional lattices. Here, we apply this approach to two-dimensional strongly nonlinear lattices and find...

We use graph-algebraic results proved in [8] and some results of the graph theory to characterize all pairs 〈L1,L2〉 of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to L1 and L2, respectively. Next, we describe other pairs of subalgebra lattices (weak and relative, etc.) of a finite unary algebra. Finally, necessary a...

Abstract. There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the number of full-rank integer lattices of index at most V in Z. This set of lattices L can naturally be partitioned with respect to the factor group Z/L. Accordingly, we count the number of full-rank integer lattices L ⊆ Z such that Z/L is cyclic and of order at most V , and deduce that these co-cyclic lattice...

Rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in 2009. By a patch lattice we mean a rectangular lattice whose weak corners are coatoms. As a sort of gluings, we introduce the concept of a patchwork system. We prove that every glued sum indecomposable planar semimodular lattice is a patchwork of its maximal patch lattice intervals “sewn togeth...

We develop a theory of stratified $LM$-filters which generalizes the theory of stratified $L$-filters. Our stratification condition explicitly depends on a suitable mapping between the lattices $L$ and $M$. If $L$ and $M$ are identical and the mapping is the identity mapping, then we obtain the theory of stratified $L$-filters. Based on the stratified $LM$-filters, a general theory of lattice-v...

Several papers relate different alternative approaches to classical concept lattices: such as property-oriented and object-oriented concept lattices and the dual concept lattices. Whereas the usual approach to the latter is via a negation operator, this paper presents a fuzzy generalization of the dual concept lattice, the dual multi-adjoint concept lattice, in which the philosophy of the multi...

In Archimedean vector lattices we show that each element of the band generated by a finite element is also finite. In vector lattices with the (PPP) and in Banach lattices we obtain some characterizations of finite elements by using the generalized order units for principal bands. In the case of Banach lattices with order continuous norm the ideal of all finite elements coincides with the linea...

By using Construction A modulo 4 the following remarkable unimodular lattices have been constructed: the Gosset lattice E8, the Leech lattice, the 23 Niemeier lattices in dimension 24, the two extremal even unimodular lattices in dimension 32 with an automorphism of order 31, all the extremal unimodular lattices and the odd Leech lattice. In this survey, we review basic facts of life in the Z4 ...