20.500.12556/RUL-104460 O rangih matrik nad polkolobarjem magistrsko delo Ranks of matrices over semirings Polkolobar $S$ je algebrska struktura z dvema notranjima operacijama $\oplus$ in $\odot$. Pri tem je $(S,\oplus)$ komutativni monoid z enoto $0_{S}$, $(S,\odot)$ monoid z enoto $1_{S}$, množenje je distributivno nad seštevanjem ter velja, da se vsak element iz $S$ pri množenju z $0_{S}$ zmnoži v $0_{S}$. Vendar pa ne obstajajo nujno inverzni elementi niti za $\oplus$ niti za $\odot$. V magistrskem delu bomo obravnavali polkolobarje, njihove lastnosti, različne primere in nekatere primere uporabe, predvsem optimizacijske. Osredotočili se bomo na polkolobarje matrik. Pri teh bomo definirali različne range, ki v splošnem med seboj niso enaki. Vemo, da je nad poljem veliko definicij rangov matrik, ki so med seboj ekvivalentne. Te ekvivalence pa ne držijo nujno za range matrik nad polkolobarjem. V delu bomo dokazali nekatere neenakosti ter pokazali na primerih, da enakosti v splošnem ne veljajo. Dokazali bomo tudi nekatere neenakosti, ki veljajo za rang vsote dveh matrik in za rang produkta dveh matrik. The semiring $S$ is an algebraic structure with two binary operations $\oplus$ and $\odot$, such that $(S, \oplus)$ is an Abelian monoid with identity $0_{S}$, $(S, \odot)$ is a monoid with identity $1_{S}$, multiplication is distributive over addition and $0_{S}$ is an absorbing element for multiplication. However, there might not exist inverse elements for $\oplus$ or for $\odot$. In this work we present semirings, their properties, various examples and some examples of applications, in particular in optimization. We concentrate on the semirings of matrices. We define different ranks of matrices over semirings, which in general do not coincide. We know that there are many equivalent definitions of ranks of matrices over the field. These equivalences need not hold for matrix ranks over the semiring. In this work we prove some inequalities among ranks and show by examples that the strict inequalities might apply. We also prove some inequalities that hold for the rank of the sums of two matrices and for the rank of the product of two matrices. polkolobar rang matrika polje tropski polkolobar semiring rank matrix field tropical semiring true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2018-10-07 07:45:16 2018-10-07 07:45:18 2022-08-17 03:43:58 0000-00-00 00:00:00 2018 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512 93188 18458201 292.pdf 292.pdf 1 33CD30560720A367B931D9156673129A eb7bfcee2978e156352b8e1be44aa98c686e6e6ff186bdad3eedef59206b3a2a fc8cdcf7-a1b5-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=114983 Fakulteta za matematiko in fiziko 0 0 0