20.500.12556/RUL-147347 Linearna algebra nad polkolobarji delo diplomskega seminarja Linear algebra over semirings Delo obravnava algebraično strukturo polkolobarja z dodatno pozornostjo dano na posebni primer dioida. Množica $R$ je polkolobar za binarni notranji operaciji $\oplus$ in $\otimes$, če je $(R,\oplus)$ komutativen monoid z enoto $0$, $(R,\otimes)$ monoid z enoto $1$, med $\otimes$ in $\oplus$ velja leva ali desna distributivnost ter velja, da $0$ izniči $\otimes$, torej ⠀$\forall a \in R; a \otimes 0 = 0\otimes a = 0$. Če je $(R,\oplus)$ poleg tega še delno urejen s kanonično relacijo $\le$, polkolobarju $(R,\oplus,\otimes)$ pravimo dioid. Tako pojem dioida kot pojem polkolobarja obstajata že nekaj časa in mnogo klasičnih vprašanj v povezavi s temi strukturami, s stališča linearne algebre, že ima odgovore. V nalogi bodo obravnavana zgolj osnovna izmed teh. Obravnavana vprašanja so predvsem centrirana na lastnostih polkolobarjev in dioidov ter posplošitvah konceptov iz klasične linearne algebre nad polji, kot so obstoj in lastnosti baz $R$-polmodula nad polkolobarjem $R$, lastnosti in obrnljivost matrik nad polkolobarjem $R$ ter Cayley-Hamiltonov izrek. This paper discusses the algebraic structure of a semiring, with additional attention given to the special case of a dioid. The set $R$ is a semiring for the binary internal laws $\oplus$ and $\otimes$ if $(R,\oplus)$ is a commutative monoid with the neutral element $0$, $(R,\otimes)$ is a monoid with the neutral element $1$, $\otimes$ is left- or right-distributive with respect to $\oplus$ and if $0$ is absorbing for $\otimes$, i. e. $\forall a \in R; a \otimes 0 = 0\otimes a = 0$. If additionally $(R,\oplus)$ is also ordered with the canonical order relation $\le$, we instead call $(R,\oplus,\otimes)$ a dioid. Both terms have existed for some time now and most of the classical questions relating to the structures, from the perspective of linear algebra, have already been answered. From among those, this paper will present some of the more elementary results. In particular, the focus will be on properties of semirings and dioids and on generalizations of concepts from classical linear algebra over fields such as the existence and properties of bases of an $R$-semimodule, the properties and invertibility of a matrix over a semiring $R$ and the Cayley-Hamilton theorem. Linearna algebra algebra polkolobar polmodul dioid pideterminanta karakteristični pipolinom posplošeni Cayley-Hamiltonov izrek Linear algebra algebra semiring semimodule dioid pideterminant characteristic pipolynomial generalized Cayley-Hamilton theorem true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2023-07-02 08:15:01 2023-07-02 08:15:07 2024-05-29 11:10:15 0000-00-00 00:00:00 2023 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512 132449 157581827 7751.pdf 7751.pdf 1 C1F273FE47BA5EECFC2444EC730593B0 d5605b6ce2fd4d260bde583517df1b0314d88f7d64509dc80b55b36ec9fe167c 61e2356c-189f-11ee-a314-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=170883 Fakulteta za matematiko in fiziko 0 0 0