Filters and congruences in sectionally pseudocomplemented lattices and posets

Together with J. Paseka we introduced so-called sectionally pseudocomplemented lattices and posets and illuminated their role in algebraic constructions. We believe that—similar to relatively pseudocomplemented lattices—these structures can serve as an algebraic semantics of certain intuitionistic logics. The aim of the present paper is to define congruences and filters in these structures, derive mutual relationships between them and describe basic properties of congruences in strongly sectionally pseudocomplemented posets. For the description of filters in both sectionally pseudocomplemented lattices and posets, we use the tools introduced by A. Ursini, i.e., ideal terms and the closedness with respect to them. It seems to be of some interest that a similar machinery can be applied also for strongly sectionally pseudocomplemented posets in spite of the fact that the corresponding ideal terms are not everywhere defined.

Homotopical and operator algebraic twisted K-theory

Using the framework for multiplicative parametrized homotopy theory introduced in joint work with C. Schlichtkrull, we produce a multiplicative comparison between the homotopical and operator algebraic constructions of twisted K-theory, both in the real and complex case. We also improve several comparison results about twisted K-theory of $$C^*$$ C ∗ -algebras to include multiplicative structures. Our results can also be interpreted in the $$\infty $$ ∞ -categorical setup for parametrized spectra.

Algebraic constructions of rotated unimodular lattices and direct sum of Barnes–Wall lattices

In this paper, we construct some families of rotated unimodular lattices and rotated direct sum of Barnes–Wall lattices [Formula: see text] for [Formula: see text] and [Formula: see text] via ideals of the ring of the integers [Formula: see text] for [Formula: see text] and [Formula: see text]. We also construct rotated [Formula: see text] and [Formula: see text]-lattices via [Formula: see text]-submodules of [Formula: see text]. Our focus is on totally real number fields since the associated lattices have full diversity and then may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. The minimum product distances of such constructions are also presented here.

THE CLASS OF PERFECT TERNARY ARRAYS

In recent decades, perfect algebraic constructions are successfully being use to signal systems synthesis, to construct block and stream cryptographic algorithms, to create pseudo-random sequence generators as well as in many other fields of science and technology. Among perfect algebraic constructions a significant place is occupied by bent-sequences and the class of perfect binary arrays associated with them. Bent-sequences are used for development of modern cryptographic primitives, as well as for constructing constant amplitude codes (C-codes) used in code division multiple access technology. In turn, perfect binary arrays are used for constructing correction codes, systems of biphase phase- shifted signals and multi-level cryptographic systems. The development of methods of many-valued logic in modern information and communication systems has attracted the attention of researchers to the improvement of methods for synthesizing many-valued bent-sequences for cryptography and information transmission tasks. The new results obtained in the field of the synthesis of ternary bent-sequences, make actual the problem of researching the class of perfect ternary arrays. In this paper we consider the problem of extending the definition of perfect binary arrays to three-valued logic case, as a result of which the definition of a perfect ternary array was introduced on the basis of the determination of the unbalance of the ternary function. A complete class of perfect ternary arrays of the third order is obtained by a regular method, bypassing the search. Thus, it is established that the class of perfect ternary arrays is a union of four subclasses, in each of which the corresponding methods of reproduction are determined. The paper establishes the relationship between the class of ternary bent-sequences and the class of perfect ternary arrays. The obtained results are the basis for the introduction of perfect ternary arrays into modern cryptographic and telecommunication algorithms.

On finite Taylor algebras

We establish a new hereditary characterization of finite idempotent Taylor algebras. This generalizes the algebraic constructions which figured in the recent successful characterization of the correctness of the bounded width algorithm for constraint satisfaction problems by finite idempotent algebras generating congruence meet-semidistributive varieties. We introduce the cyclic reduct of a finite Taylor algebra and prove a collapse of Mal’cev conditions.