Nilpotentne matrike nad antikolobarji

Trojer, Klarisa (author), Dolžan, David (mentor)

Glavna tema diplomske naloge je opis lastnosti nilpotentnih matrik nad antikolobarji. V prvem poglavju se posvetimo abstraktnim strukturam, natančneje, grupam in polkolobarjem. Pogledamo si bomo lastnosti, ki veljajo za polgrupe, monoide in grupe, nato pa preidemo še na lastnosti polkolobarjev. Nadaljujemo s kratkim opisom zgodovine študije matrik nad polkolobarji ter podamo nekaj definicij in lem, ki jih uporabimo v nadaljevanju. Definiramo, kaj je antikolobar in obravnavamo nilpotentnost elementov in matrik, prav tako definiramo permanento ter z njo povezane permanentne minorje. Sledi poglavje, na katerega se navezuje naslov diplomske naloge, lastnosti nilpotentnih matrik. Nekaj strani namenimo še simultani nilpotenci in nilpotentnemu indeksu. V zadnjem delu pa podamo metodo, s katero ta indeks tudi poiščemo.

• Language: Slovenian
• Keywords: antikolobar, nilpotentna matrika, permanenta, permanentni minor, digraf, simultana nilpotenca, nilpotentni indeks.
• Work type: Bachelor thesis/paper
• Organization: FRI - Faculty of Computer and Information Science
• Year: 2018
• PID: 20.500.12556/RUL-103928
• Publication date in RUL: 28.09.2018

TROJER, Klarisa, 2018, Nilpotentne matrike nad antikolobarji [online]. Bachelor’s thesis. [Accessed 8 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=103928

Secondary language

• Language: English
• Title: Nilpotent matrices over antirings

Abstract

The main goal of this thesis is to decribe the properties of nilpotent matrices over antirings. We start with the introduction of the abstract structures, more precisely, groups and semirings. We observe the properties that apply to semigroups, monoids and groups, and also the properties that apply to semirings. After a brief description of the history of previous studies done on matrices over semirings, we give some denitions and lemmas that are used later on. We dene what an antiring is and talk about nilpotence of elements and matrices, we also dene the permanent and the associated permanent minors. Next, we dene and describe the properties of the nilpotent matrices. We also write about simultaneous nilpotency and the nilpotent index. At the end we provide a method for calculationg the nilpotent index.

• Keywords: antiring, nilpotent matrix, permanent, permanent minor, digraph, simultaneously nilpotency, nilpotent index.