20.500.12556/RUL-150026 Kombinatorični modeli v teoriji inverznih monoidov magistrsko delo Combinatorial models in the theory of inverse monoids Inverzne polgrupe so polgrupe, v katerih ima vsak element $x$ enoličen inverz $x^{-1}$, tj. velja $x = xx^{-1}x$ in $x^{-1} =x^{-1}xx^{-1}$. Opremimo jih lahko z naravno delno urejenostjo $\leq$, kjer velja $a \leq b$ natanko tedaj, ko obstaja idempotent $e$, da je $a=be$. V takšnih polgrupah igrata pomembno vlogo dve relaciji. To sta kompatibilnostna relacija $\sim $, kjer je $s \sim t$ natanko tedaj, ko sta $ s^{-1}t$ in $ st^{-1}$ idempotenta, in najmanjša grupna kongruenca $\sigma$, kjer je $s \mathrel{\sigma} t$ natanko tedaj, ko obstaja $u$, da velja $u \leq s,t $. Razred inverznih polgrup, za katere relacija kompatibilnosti sovpada z najmanjšo grupno kongruenco, imenujemo $E$-unitarne inverzne polgrupe, razred, za katere ima vsak $\sigma$-razred maksimalen element, pa $F$-inverzni monoidi. Torej lahko $F$-inverzne monoide opremimo z dodatno unarno operacijo $a \mapsto m(a)$, kjer z $m(a)$ označimo maksimalni element razreda $[a]_\sigma$. Izkaže se, da ravno v tej razširjeni signaturi $F$-inverzni monoidi tvorijo variateto. V nadaljevanju se osredotočimo na kombinatorične modele grupnih razširitev, in sicer na Margolis-Meakinovo razširitev, Birget-Rhodesovo razširitev in modela $F$-inverznih ter popolnih $F$-inverznih monoidov. S pomočjo teh modelov lahko opišemo proste inverzne monoide, proste $F$-inverzne monoide in proste popolne $F$-inverzne monoide. Inverse semigroups are semigroups in which each element $x$ has a unique inverse $x^{-1}$, i.e. $x = xx^{-1}x$ and $x^{-1} = x^{-1}xx^{-1}$ hold. We can equip them with a natural partial order $\leq$, where $a \leq b$ holds if and only if there is an idempotent $e$ such that $a=be$. In such semigroups, two relations play an important role. These are the compatibility relation $\sim $, where $s \sim t$ if and only if $ s^{-1}t$ and $ st^{-1}$ are idempotent, and the smallest group congruence $\sigma$, where $s \mathrel{\sigma} t$ if and only if there is such a $u$ that $u \leq s,t $ holds. A class of inverse semigroups for which the compatibility relation coincides with the smallest group congruence is called a class of $E$-unitary inverse semigroups, and a class for which every $\sigma$-class has a maximal element is called a class of $F$-inverse monoids. Thus, $F$-inverse monoids can be equipped with an additional unary operation $a \mapsto m(a)$. It turns out that $F$-inverse monoids form a variety precisely in this extended signature. Next we focus on combinatorial models of group expansions, namely the Margolis-Meakin expansion, the Birget-Rhodes expansion, and models of $F$-inverse and perfect $F$-inverse monoids. Using these models, we can describe free inverse monoids, free $F$-inverse monoids, and free perfect $F$-inverse monoids. $E$-unitarne inverzne polgrupe $F$-inverzni monoidi popolni $F$-inverzni monoidi grupne razširitve Cayleyjev graf $E$-unitary inverse semigroups $F$-inverse monoids perfect $F$-inverse monoids group extensions Cayley graph true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2023-09-13 08:16:02 2023-09-13 08:16:06 2024-05-29 10:40:15 0000-00-00 00:00:00 2023 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 135855 164180739 9169.pdf 9169.pdf 1 00164EDAEBC3E33F65A13CDC12F0A496 6126e51c8df65eae5e9b6d0e629edc30f37d205cadfa24a57f0820823437f219 060d8a3f-51fd-11ee-916e-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=174558 Fakulteta za matematiko in fiziko 0 0 0