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Endomorphisms of the poset of idempotent matricesŠemrl, PeterNaj bo ▫$\mathbb{D}$▫ ne nujno komutativen obseg, ▫$n \geq 3$▫ naravno število in ▫$P_n(\mathbb{D})$▫ delno urejena množica vseh ▫$n \times n$▫ idempotentnih matrik nad ▫$\mathbb{D}$▫, pri čemer je ... delna urejenost definirana s predpisom: ▫$P \leq Q$▫, če velja ▫$PQ = QP = P$▫. Naj bo ▫$T \in M_n(\mathbb{D})$▫ obrnljiva matrika in ▫$\sigma : \mathbb{D} \to \mathbb{D}$▫ endomorfizem (▫$\tau : \mathbb{D} \to \mathbb{D}$▫ anti-endomorfizem). Za vsak ▫$P \in P_n(\mathbb{D})$▫ označimo z ▫$P^\sigma$▫ (▫$P^\tau$▫) idempotentno matriko, ki jo dobimo tako, da vsak člen matrike ▫$P$▫ preslikamo s preslikavo ▫$\sigma(\tau)$▫. Preslikava ▫$\phi : P_n(\mathbb{D})\to P_n(\mathbb{D})$▫ definirana s predpisom ▫$\phi(P) = TP^\sigma T^{- 1},\ P \in P_n(\mathbb{D})\ (\phi(P) = T ^t(P^\tau) T^{- 1},\ P \in P_n(\mathbb{D}))$▫ je injektivna in ohranja urejenost v obe smeri. Vsaki taki preslikavi rečemo standardna. Že prej je bilo znano, da je v primeru, ko je ▫$\mathbb{D}$▫ EAS obseg, vsak injektiven ohranjevalec urejenosti ▫$\phi : P_n(\mathbb{D}) \to P_n(\mathbb{D})$▫ bodisi standarden bodisi ima zelo posebno degenerirano obliko. Z uporabo idej iz geometrije algebraičnih homogenih prostorov in teorije polj podamo primere, ki pokažejo, da ta rezultat brez EAS predpostavke ne drži. Nato definiramo posplošene standardne preslikave in z njihovo pomočjo opišemo splošno obliko injektivnih ohranjevalcev urejenosti na ▫$P_n(\mathbb{D})$▫ za poljuben obseg ▫$\mathbb{D}$▫. Ob EAS predpostavki dobimo krajši dokaz prej omenjenega izreka. Direktna posledica glavnega izreka sta dve karakterizaciji standardnih preslikav. S kontraprimeri pokažemo optimalnost vseh dobljenih rezultatov.Source: Journal of algebra. - ISSN 0021-8693 (Vol. 536, Oct. 2019, str. 1-38)Type of material - article, component part ; adult, seriousPublish date - 2019Language - englishCOBISS.SI-ID - 18728281
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Šemrl, Peter | 05953 |
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