325 KB394 KB111 KB2005Kokol-Bukovšek, DamjanaOmladič, Matjaž111 str., 30 cm111 straniCOBISSID:13777497URN:URN:NBN:SI:doc-H1NKEADXenD. Kokol Bukovšekvisokošolska delaDisertacijeHomomorfizmiMatrične polgrupemultiplikativne preslikavenerazcepnostHomomorphisms of matrix semigroups| doctoral dissertation|In this work we study non-degenerate homomorphisms from the multiplicative semigroup of all ?$n$?-by-?$n$? matrices over a field to the semigroup of ?$m$?-by-?$m$? matrices over the same field. A general introduction is given in the first chapter. In the second chapter we first state our main question and give some examples. Further we characterize all homomorphisms from the multiplicative semigroup of all ?$n$?-by-?$n$? matrices over an arbitrary field to the field and all non-degenerate homomorphisms from the multiplicative semigroup of all ?$n$?-by-?$n$? matrices over an arbitrary field to a semigroup of ?$m$?-by-?$m$? matrices over the same field, if ?$m \le n$?. In the third chapter we characterize all non-degenerate homomorphisms from the multiplicative semigroup of all 2-by-2 matrices over an arbitrary field to the semigroup of 3-by-3 matrices over the same field. If the characteristic of the field is not equal to 2 then we have two possibilities. Either it is a symmetric square, combined with a field homomorphism used entrywise and a matrix conjugation, or a direct sum of the identity and the determinant, combined with a field homomorphism, a homomorphism of the multiplicative semigroup of the field and a matrix conjugation. In the characteristic 2 a symmetric square gives rise to two different homomorphisms and we get three possibilities. In the case of the field of real numbers every irreducible non-degenerate homomorphism is a matrix conjugation of the symmetric square. In the fourth chapter we study non-degenerate irreducible homomorphisms from the multiplicative semigroup of all 2-by-2 matrices over an algebraically closed field of characteristic zero to the semigroup of ?$m$?-by-?$m$? matrices over the same field. If such a homomorphism maps a cyclic unipotent to a cyclic unipotent, it is the composition of a symmetric power, a field homomorphism used entrywise, and a matrix conjugation. In the case ?$m = 4$? we characterize all non-degenerate irreducible homomorphisms. In the fifth chapter we prove that every non-degenerate homomorphism from the multiplicative semigroup of all ?$n$?-by-?$n$? matrices over an algebraically closed field of characteristic zero to the semigroup of ?$(n+1)$?-by-?$(n+1)$? matrices over the same field when ?$n \le 3?$ is reducible and that every non-degenerate homomorphism from the multiplicative semigroup of all 3-by-3 matrices over an algebraically closed field of characteristic zero to the semigroup of 5-by-5 matrices over the same field isreducibleV delu študiramo nedegenerirane homomorfizme iz multiplikativne polgrupe vseh ?$n \times n$? matrik nad komutativnim obsegom v polgrupo ?$m \times m$? matrik nad istim obsegom. Najprej karakteriziramo vse homomorfizme iz matrične polgrupe ?$\mathcal{M}_n(\mathbb{F})$? v obseg ?$\mathbb{F}$? kot multiplikativno polgrupo. Glavni rezultat drugega poglavja je karakterizacija vseh nedegeneriranih homomorfizmov polgrup ?$\varphi : \matcal{M}_n(\mathbb{F}) \to \mathcal{M}_m(\mathbb{F})$?, kjer je ?$m \le n$?. Glavni rezultat tretjega poglavja je karakterizacija homomorfizmov iz polgrupe ?$2 \times 2$? matrik v polgrupo ?$3 \times 3$? matrik. V četrtem in petem poglavju se omejimo na primer, ko je komutativni obseg ?$\mathbb{F}$? algebraično zaprt in ima karakteristiko nič. Obravnavamo samo nerazcepne homomorfizme. Pokažemo, da nerazcepen nedegeneriran homomorfizem polgrup ?$\varphi : \matcal{M}_2(\mathbb{F}) \to \mathcal{M}_n(\mathbb{F})$?, preslika matrike ranga 1 v matrike ranga 1. V petem poglavju obravnavamo primer, ko je dimenzija matrik v polgrupi iz katere slikamo vsaj 3. Na koncu dodamo še nekaj primerovdoktorska disertacijaTEXTvisokošolska delatheses and dissertationsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza v Ljubljani, Fakulteta za matematiko in fiziko