Index transforms forn-dimensional DFT's

Summary In this paper fundamental theorems are proved for computing the so called index transforms which mapm-dimensional discrete Fourier transform operator (DFT) ton-dimensional one. These transforms are based on mapping indexes of input and output sequences of DFT operators. The general form of these transforms is found and the necessary and sufficient conditions of their existence are described.     

Definable relations in finite-dimensional subspace lattices with involution

For a large class of finite dimensional inner product spaces V, over division \(*\)-rings F, we consider definable relations on the subspace lattice \(\mathsf{L}(V)\) of V, endowed with the operation of taking orthogonals. In particular, we establish translations between the relevant first order languages, in order to associate these relations with definable and invariant relations on F—focussing on the quantification type of defining formulas. As an intermediate structure we consider the \(*\)-ring \(\mathsf{R}(V)\) of endomorphisms of V, thereby identifying \(\mathsf{L}(V)\) with the lattice of right ideals of \(\mathsf{R}(V)\), with the induced involution. As an application, model completeness of F is shown to imply that of \(\mathsf{R}(V)\) and \(\mathsf{L}(V)\).     

Generalization of relations between Brieskorn lattices for nonisolated singularities

Moscow State Academy of Printing. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 4, pp. 78–80, October–December, 1995.     

Finite distributive lattices are congruence lattices of almost-geometric lattices

A semimodular lattice L of finite length will be called an almost-geometric lattice if the order J(L) of its nonzero join-irreducible elements is a cardinal sum of at most two-element chains. We prove that each finite distributive lattice is isomorphic to the lattice of congruences of a finite almost-geometric lattice.     

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.     

Free lattices over join-trivial partial lattices

Presented by Alan Day.     

Idempotent matrix lattices over distributive lattices

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an n-element set Qord(n) is not graduated for n ≥ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval ([I, J]_≤, ≤) of idempotent (n × n)-matrices over {ie879-01}-lattices is isomorphic to the lattice of quasi-orders Qord(n). Using this isomorphism, we calculate the lattice height of idempotent {ie879-02}-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 121–144, 2007.     

Complete atomistic lattices are classification lattices

We prove that each complete atomistic lattice G is isomorphic to the lattice of classification systems of an appropriate complete atomistic lattice L. This implies an affirmative solution to a problem raised by S. Radeleczki in 2002.