http://hdl.handle.net/11089/21993 2026-04-06T05:13:43Z http://hdl.handle.net/11089/22189 Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras Dzik, Wojciech; Radeleczki, Sándor We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, and G operations as well as expansions of some commutative integral residuated lattices with successor operations. 2016-01-01T00:00:00Z http://hdl.handle.net/11089/22188 Some Algebraic and Algorithmic Problems in Acoustocerebrography Kolany, Adam; Wrobel, Miroslaw Progress in the medical diagnostic is relentlessly pushing the measurement technology as well with its intertwined mathematical models and solutions. Mathematics has applications to many problems that are vital to human health but not for all. In this article we describe how the mathematics of acoustocerebrography has become one of the most important applications of mathematics to the problems of brain monitoring as well we will show some algebraic problems which still have to be solved. Acoustocerebrography ([4, 1]) is a set of techniques of visualizing the state of (human) brain tissue and its changes with use of ultrasounds, which mainly rely on a relation between the tissue density and speed of propagation for ultrasound waves in this medium. Propagation speed or, equivalently, times of arriving for an ultrasound pulse, can be inferred from phase relations for various frequencies. Since, due to Kramers-Kronig relations,the propagation speeds depend significantly on the frequency of investigated waves, we consider multispectral wave packages of the form W (n) = ∑Hh=1 Ah · sin(2π ·fh · n/F + ψh), n = 0, . . . , N – 1 with appropriately chosen frequencies fh, h = 1, . . . ,H, amplifications Ah, h = 1, . . . ,H, start phases ψh, h = 1, . . . , H and sampling frequency F. In this paper we show some problems of algebraic and, to some extend, algorithmic nature which raise up in this topic. Like, for instance, the influence of relations between the signal length and frequency values on the error on estimated phases or on neutralizing alien frequencies. Another problem is finding appropriate initial phases for avoiding improper distributions of peaks in the resulting signal or finding a stable algorithm of phase unwinding which is resistant to sudden random disruptions. 2016-01-01T00:00:00Z http://hdl.handle.net/11089/22187 Irredundant Decomposition of Algebras into One-Dimensional Factors Staruch, Bogdan We introduce a notion of dimension of an algebraic lattice and, treating such a lattice as the congruence lattice of an algebra, we introduce the dimension of an algebra, too. We define a star-product as a special kind of subdirect product. We obtain the star-decomposition of algebras into one-dimensional factors, which generalizes the known decomposition theorems e.g. for Abelian groups, linear spaces, Boolean algebras. 2016-01-01T00:00:00Z http://hdl.handle.net/11089/22186 Decomposition of Congruence Modular Algebras into Atomic, Atomless Locally Uniform and Anti-Uniform Parts Staruch, Bogdan; Staruch, Bożena We describe here a special subdirect decomposition of algebras with modular congruence lattice. Such a decomposition (called a star-decomposition) is based on the properties of the congruence lattices of algebras. We consider four properties of lattices: atomic, atomless, locally uniform and anti-uniform. In effect, we describe a star-decomposition of a given algebra with modular congruence lattice into two or three parts associated to these properties. 2016-01-01T00:00:00Z