delo diplomskega seminarja
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: |
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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 |