Algebra in mreža zveznih funkcij C(X)

delo diplomskega seminarja

Jan Malej (Author), Marko Kandić (Mentor)

Abstract

V delu diplomskega seminarja spoznamo strukturi mreže in Rieszovega prostora ter dokažemo nekaj njunih osnovnih lastnosti. Definiramo Rieszove homomorfizme in mrežne ideale. Posebej izpostavimo še mrežne praideale in maksimalne ideale. Nato se osredotočimo na množici zveznih funkcij $C(X)$ in omejenih zveznih funkcij $C_b(X)$ na Hausdorffovem prostoru $X.$ Gre za Rieszova prostora, ki sta hkrati realni algebri, zato obravnavamo relacije med njunimi mrežnimi in algebraičnimi objekti. Dokažemo, da je vsak homomorfizem algeber $\varphi\colon C(X)\to C(Y)$ Rieszov homomorfizem. Za kompakten Hausdorffov prostor $K$ dokažemo, da je vsak mrežni ideal v $C(K)$ algebraičen ideal. Karakteriziramo zaprte mrežne ideale, ki se v tem primeru ujema\-jo z algebraičnimi. Dokažemo, da je vsak algebraičen praideal v $C(K)$ mrežni pra\-ideal. Karakteriziramo maksimalne ideale v $C(K).$ Nazadnje konstruiramo Stone-Čechovo kompaktifikacijo Hausdorffovega prostora $X$ in z njeno pomočjo opišemo maksimalne ideale še v prostoru omejenih zveznih funkcij $C_b(X).$

Keywords

Rieszovi prostori;Rieszovi homomorfizmi;mrežni ideali;praideali;maksimalni ideali;zvezne funkcije;popolnoma regularni prostori;Stone-Čechova kompaktifikacija;

Data

Language:	Slovenian
Year of publishing:	2025
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[J. Malej]
UDC:	517.9
COBISS:	241616643
Views:	170
Downloads:	64
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Algebra and lattice of continuous functions $C(X)$
Secondary abstract:	In this thesis, we familiarize ourselves with lattices and Riesz spaces and prove some of their basic properties. We define Riesz homomorphisms and order ideals. Extra attention is given to order prime and maximal ideals. Our focus then shifts towards the sets of continuous functions $C(X)$ and bounded continuous functions $C_b(X)$ on a Hausdorff space $X.$ These are Riesz spaces and also real algebras, which is why their order and algebraic objects are compared. We prove that every algebra homomorphism $\varphi\colon C(X)\to C(Y)$ is a Riesz homomorphism. Moreover, for a compact Hausdorff space $K,$ we prove that every order ideal in $C(K)$ is an algebraic ideal. Additionally, we characterize closed order ideals, which in this case coincide with algebraic ones. It is shown that every algebraic prime ideal in $C(K)$ is an order prime ideal. A characterization of maximal ideals in $C(K)$ is also dealt with. Finally, we construct the Stone-Čech compactification of a Hausdorff space $X$ and use it to describe maximal ideals in the space of bounded continuous functions $C_b(X).$
Secondary keywords:	Riesz spaces;Riesz homomorphisms;order ideals;prime ideals;maximal ideals;continuous functions;completely regular spaces;Stone-Čech compactification;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	44 str.
ID:	26719712