Latinski kvadrati, porojeni z grupami

delo diplomskega seminarja

Ines Sovdat (Author), Primož Moravec (Mentor)

Abstract

V diplomski nalogi predstavimo latinske kvadrate, izotopijo, kvazigrupe in zanke. Dokažemo, da je vsaka kvazigrupa izotopna zanki, torej vsak izotopni razred vsebuje vsaj eno zanko. Posvetimo se odnosu med kvazigrupami in latinskimi kvadrati ter pokažemo, da je latinski kvadrat ekvivalenten Cayleyjevi tabeli kvazigrupe. Na protiprimeru pokažemo, zakaj trditve ne moremo razširiti na grupe. Predstavimo kriterije, ki zagotavljajo, da je latinski kvadrat izotopen grupi, torej porojen z grupo. Na primerih in protiprimerih podrobneje spoznamo njihovo delovanje. Seznanimo se s štirikotnim kriterijem in njegovimi različicami. Predstavimo Thomsenov pogoj, ki zagotavlja porojenost latinskega kvadrata z Abelovo grupo. Predstavimo tudi kriterij, ki je zasnovan na permutaciji stolpcev in vrstic Cayleyjevih tabel.

Keywords

matematika;latinski kvadrati;izotopije;kvazigrupe;zanke;porojenost z grupami;

Data

Language:	Slovenian
Year of publishing:	2019
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[I. Sovdat]
UDC:	512
COBISS:	18819929
Views:	1594
Downloads:	254
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Latin squares based on groups
Secondary abstract:	In this thesis we present Latin squares, isotopy, quasigroups and loops. We prove that each quasigroup is isotopic to a group, therefore each isotopy class contains at least one loop. We focus on a relationship between quasigroups and Latin squares and show equivalence between Latin squares and Cayley tables of a quasigroup. Reason why this can not be generalised to groups is shown on a counterexample. Criteria which ensure Latin square is isotopic to a group, therefore based on a group, are presented. Functioning of those criteria is closely explained using examples and counterexamples. Quadrangle criterion and his variations are presented. Thomsen condition, which ensures a Latin square is based on an Abelian group, is also presented. Criteria based on permutations of rows and columns of a Cayley tabele is also introduced.
Secondary keywords:	mathematics;Latin squares;isotopy;quasigroups;loops;based on groups;
Type (COBISS):	Final seminar paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	26 str.
ID:	11229069