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On real-valued homomorphisms in countably generated differential structures

Cukrowski M. J., Pasternak-Winiarski Z., Sasin W.

Demonstratio Mathematica

2012

Vol. 45, nr 3

665-676

Real valued homomorphisms on the algebra of smooth functions on a differential space are described. The concept of generators of this algebra is emphasized in this description.

Differential completions and compactifications of a differential space

Dziewa-Dawidczyk D., Pasternak-Winiarski Z.

Demonstratio Mathematica

2012

Vol. 45, nr 4

975-992

Hopf-Sikorski algebras

Heller M., Odrzygóźdź Z., Pysiak L., Sasin W.

Demonstratio Mathematica

2011

Vol. 44, nr 2

213-221

The dual category with respect to the category of differential groups is defined and investigated. The objects of this category are algebras, called Hopf-Sikorski (H-S) algebras, the axioms of which combine the axioms of Sikorski's algebras with modified axiomas of Hopf algebras. Morphisms of this category are structural mappings corresponding to Hopf algebras that are smooth in the sense of Sikorski. As an example, we discuss the H-S algebra of the Lorentz group.

Quotient structured spaces

Sasin W., Lipiński P.

Demonstratio Mathematica

2008

Vol. 41, nr 2

455-471

In this paper we investigate some properties of quotient structured spaces. The notion of a structured space was originally considered by Mostow [2]. Some foundations of structured spaces with applications to relativistic physics are presented in [1]. In the beginning we present some basic notions and definitions from structured space theory. Then we discuss some properties of quotient structured spaces. In the third part we present a space-time as a quotient space. At the end of this paper we consider F-quasiregular equivalence relation and the structured space with malicious singularity.

Pseudogroups in premanifolds

Lipińska J.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2002

Vol. 1, nr 1

93--95

In [1] it was proved that the set of all diffeomorphisms of a quasi-algebraic space which was introduced by W. Waliszewski is the Ehresmann’s pseudogroup. Proving this theorem we did not use properties of a quasi-algebraic space, so it was possible to generalize this theorem and formulate it for any set of functions. It was noticed in [2]. The concept of an analytical premanifold was introduced by Waliszewski in [3]. In [4] he also called it a general differential space. In this paper we show how to use the above theorem in analytical premanifolds.