Involutivne algebre

Abstract

<p>Tema ove disertacije je&nbsp; involucijau algebarskim strukturama. Involucije su bijektivna preslikavanja koja se poklapaju sa svojim inverznim funkcijama. One se pojavljuju u skoro svim oblastima matematike: podsetimo se samo projektivne geometrije, teorije algebarskih krivih, inverzije u euklidskoj geometriji i njenog značaja za modele hiperboličke geometrije, teorije matrica i drugih disciplina. Cilj disertacije je da prikaže teoriju involutivnih algebri, tj. neke rezultate u okviru te teorije. Najvi&scaron;e su istraženi odnosi između algebarskih zakona i involucije, i ti odnosi daju jednu sasvim novu algebarsku teoriju. Materijal je podeljen u četiri dela. U prvom delu se posmatraju tzv. Plonkine sume. Ispostavilo se da su mnoge klasične konstrukcije u algebri samo specijalni slučajevi Plonikih suma. Kako bismo ih prilagodili izučavanju involutivnih algebri, ove sume su modifikovane, tako da dobijamo involutivne Plonkine sume. U radu su ispitane neke osobine takvih suma. U drugom delu istražujemo involutivne polugrupe. Između ostalog, dokazano je da je klasa regularnih *-traka globalno određena. Treći deo prikazuje neke od najnovijih rezultata u oblasti involutivnih poluprstena. Najzad, poslednji, četvrti deo govori o involutivnim prstenima. Posmatrani su neki poddirektno nesvodljivi prsteni sa involucijom, i dokazan je involutivni analogon čuvene teoreme N. Jacobsona.</p>

<p>The topic o f this dissertation is&nbsp; involutionin algebraic structures. Involutions are bijective mappings which coincide with their inverse functions. They appear in almost all mathematical disciplines: recall projective geometries, theory of algebraic curves, inversion in euclidean geometry and its importance in the models of hyperbolic geometry, theory of matrices and other parts of mathematics. The aim of this dissertation is to present the theory o f involution algebras, i.e. some results in the frame o f that theory. We are investigating the relationship of algebraic laws and involution, which together give a new algebraic theory. The material is divided into four parts. In the first part, we are considering the so called Plonka sums. It turned out that many classical constructions in algebra are special cases o f Plonka sums. We modify these sums in order to make them applicable to involution algebras, and so we obtain the involutorial Plonka sums, whose properties are explored. In the second part, we investigate involution semigroups. Among other things, it is shown that the class o f regular &lsquo; -bands is globally determined. The third part is about semirings with involution. We review some of the latest results in the area o f involution semirings. The final, fourth part is about rings with involution. We are considering some subdirectly irreducible involution rings and prove an involutorial analogue of the wellknown theorem o f N. Jacobson.</p>