Sendvič polugrupe u lokalno malim kategorijama

Abstract: Let S be a locally small category, and fix two (not necessarily distinct) objects i, j in S. Let Sij and Sji denote the set of all morphisms i → j and j → i, respectively. Fix a ∈ Sji and define (Sij , ? a), where x?a y = xay for x, y ∈ Sij . Then, (Sij , ? a ) is a semigroup, known as a sandwich semigroup, and denoted by S a ij . In this thesis, we conduct a thorough investigation of sandwich semigroups (in locally small categories) in general, and then apply these results to infer detailed descriptions of sandwich semigroups in a number of categories. Firstly, we introduce the notion of a partial semigroup, and establish a framework for describing a category in "semigroup language". Then, we prove various results describing Green’s relations and preorders, stability and regularity of S a ij . In particular, we emphasize the relationships between the properties of the sandwich semigroup and the properties of the category containing it. Also, we highlight the significance of the properties of the sandwich  element a. In this process, we determine a natural condition on a called sandwich regularity which guarantees that the regular elements of S a ij form a subsemigroup tightly connected to certain non-sandwich semigroups. We explore these connections in detail and infer major structural results on Reg(S a ij) and the generation mechanisms in it. Finally, we investigate ranks and idempotent ranks of the regular subsemigroup Reg(S a ij ) and idempotent-generated subsemigroup E(S a ij ) of S a ij . In general, we are able to infer expressions for lower bounds for these values. However, we show that in the case when Reg(S a ij ) is MI-dominated (a property which has to do with the "covering power" of certain local monoids), the mentioned lower bounds are sharp. We apply the general theory to sandwich semigroups in various transformation categories (partial maps P T  , injective maps I , totally defined maps T , and matrices M(F) − corresponding to linear transformations of vector spaces over a field F) and diagram categories (partition P ,  lanar partition PP , Brauer B, partial Brauer PB, Motzkin M , and Temperley-Lieb T L categories), one at a time. In each case, we investigate the partial semigroup itself in terms of Green’s relations and regularity and then focus on a sandwich semigroup in it. We apply the general results to thoroughly describe its structural and combinatorial properties. Furthermore, since in each category that we consider all elements are sandwich-regular, we may apply the theory concerning the regular subsemigroup in all of these cases. In particular, Reg(S a ij ) turns out to be tightly connected to a certain nonsandwich monoid for each category S we consider, and we are able to describe  eg(S a ij ) and E(S a ij ). However, we conduct the combinatorial part of the investigation only for the sandwich semigroups in transformation categories (P T , I , T , and M(F)) and sandwich semigroups in the Brauer category B since only these have MI-dominated regular subsemigroups (and some other properties that make them more amenable to  investigation). For these sandwich semigroups, we enumerate regular Green’s classes and idempotents, and we calculate the ranks (and idempotent ranks, where appropriate) of Reg(S a ij ), E(S a ij ) and S a ij .Neka je S lokalno mala kategorija. Fiksirajmo proizvoljne (ne nužno različite) objekte i i j iz S. Neka Sij i Sji označavaju skupove svih morfizama i → j i j → i, redom. Fiksirajmo morfizam a ∈ Sji i definišimo strukturu (Sij , ? a ), gde je x ? a y = xay za sve x, y ∈ Sij . Tada je (Sij , ? a ) sendvič polugrupa, koju označavamo sa S a ij . U tezi ćemo sprovesti detaljno ispitivanje sendvič polugrupa (u lokalno maloj kategoriji) u opštem slučaju, a zatim ćemo primeniti dobijene rezultate u cilju opisivanja sendvič polugrupa u konkretnim kategorijama. Najpre uvodimo pojam parcijalne polugrupe i postavljamo osnovu koja nam omogu-ćava da opišemo kategoriju na "jeziku polugrupa". Zatim slede brojni rezultati koji opisuju Grinove relacije i poretke, kao i stabilnost i regularnost polugrupe (Sij , ? a ). Tu posebno ističemo veze između osobina sendvič polugrupe i parcijalne polugrupe koja je sadrži. Takođe, posebnu pažnju posvećujemo uticaju sendvič elementa a na osobine sendvič polugrupe (Sij , ? a ). Kao najbitniji primer se izdvaja osobina sendvič-regularnosti ; naime, dokazujemo da, ako je a sendvič- regularan, onda regularni elementi iz S a ij formiraju podgrupu koja je usko povezana sa određenim "ne-sendvič" polugrupama. U tezi detaljno ispitujemo te veze i dobijamo važne rezultate o strukturi polugrupe Reg(Sij , ? a ) i mehanizmima generisanja u njoj. Za kraj, ispitujemo rangove i idempotentne rangove regularne potpolugrupe Reg(Sij , ? a ) i idempotentno-generisane potpolugrupe E(Sij , ? a ). U opštem slučaju možemo dati donja ograničenja za ove vrednosti. Međutim, u slučaju kada je regularna polugrupa Reg(Sij , ? a ) MI-dominirana (što znači da je određeni lokalni monoidi pokrivaju), ta donja ograničenja su dostignuta. U ostatku teze, primenjujemo opštu teoriju na sendvič polugrupe u brojnim kategorijama transformacija (parcijalne funkcije P T , injektivne parcijalne funkcije I , potpuno definisane funkcije T i matrice M(F), koje predstavljaju linearne transformacije vektorskih prostora nad poljem F) i kategorijama dijagrama (particije P , planarne particije PP , Brauerove B,parcijalne Brauerove PB, Mockinove M , i Temperli-Lib T L particije). U svakom od ovih slučajeva, prvo istražujemo parcijalnu polugrupu iz aspekta Grinovih relacija i regularnosti, a zatim se fokusiramo na (proizvoljnu) sendvič polugrupu u njoj. Pri tome, primenjujemo opšte rezultate da bismo detaljno opisali njenu strukturu i kombinatorne osobine. Osim toga, u svim slučajevima primenjujemo i teoriju vezanu za regularnu potpolugrupu, pošto su svi elementi u našim kategorijama sendvič-regularni. To znači da je u svakoj kategoriji S koju razmatramo, Reg(Sij , ? a ) usko povezana sa određenim monoidom, i preko te veze možemo opisati  polugrupe Reg(Sij , ? a ) i E(Sij , ? a ). Ipak, kombinatorni deo ispitivanja sprovodimo samo za sendvič polugrupe u kategorijama transformacija (P T , I , T i M(F)) i sendvič polugrupe u Brauerovoj kategoriji B, pošto samo one imaju MI-dominirane regularne potpolugrupe (i još neke osobine koje ih čine pogodnijim za ispitivanje). U ovim sendvič polugrupama računamo broj regularnih Grinovih klasa i idempotenata, i izračunavamo rangove (i idempotentne rangove, ako postoje) polugrupa Reg(Sij , ? a ), E(Sij , ? a ) i S a ij

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