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We begin the investigation of the variety of semilattices of Mal'cev blocks, which we call SMB algebras.

Problem upravljanja bespilotnom letjelicom kroz niz točaka u prostoru može se postaviti kao problem generiranja trajektorija baziranih na polinomijalnim funkcijama višeg reda ovisnih o vremenu. Takve trajektorije mogu osigurati kontinuiranost brzine i akceleracije letjelice kroz među-točke putanje, čime se eliminira potreba za zaustavljanjem. Dodatno, minimizacija polinoma po četvrtoj derivaciji položaja i po vremenu osigurava optimalno glatke trajektorije. Za generiranje polinoma optimiziranih po četvrtoj derivaciji položaja koristi se kvadratno programiranje postavljeno s uvjetima jednakosti. Zatim se optimiraju omjeri vremena leta između susjednih među-točaka korištenjem metode najstrmijeg spusta. U konačnici se trajanje trajektorije normira ovisno o zahtjevu na maksimalnu brzinu ili akceleraciju, čime se regulira trajanje i agresivnost trajektorije. The problem of navigating an Unmanned Aerial Vehicle (UAV) through a collection of checkpoints can be converted to the problem of generating time-based, high-order polynomial functions as trajectories. This method ensures continuity of derivatives of path, such as velocity and acceleration, between the aforementioned checkpoints. Minimising the fourth derivative of path, i. e. snap, and the duration of the trajectory leads to an optimal path. Minimum - snap polynomials are generated using Equality Constrained Quadratic Programming. Segment time ratios are then adjusted using Gradient Descent and finally, the duration of the trajectory is scaled dependent on requirements of maximum velocity or acceleration, which regulates the aggresivness and duration of the flight.

Ultrastrukturne promene rožnjače kod osoba obolelih od dijabetes melitusa (DM) opisane su u ranijim radovima. Cilj istraživanja je bio poređenje centralne debljine rožnjače (CDR) osoba obolelih od DM bez retinopatije u stadijumu neproliferativne i proliferativne dijabetičke retinopa-tije sa CDR zdravih osoba kontrolne grupe. Metode rada: Ukupno je ispitana 121 osoba sa DM i 125 zdravih osoba. Kontrolnu grupu su činile osobe individualno uparene prema polu i starosti sa bolesnicima iz studijske grupe. Svi ispitanici su podvrgnuti oftalmološkom pregledu, koji je obuhvatio pregled očnog dna i merenje CDR ultrazvučnim pahimetrom. Oči ispitanika sa DM su, prema kriterijumima Early Treatment Diabetic Retinopathy Study, podeljene u tri grupe: grupu bez dijabetičke retinopatije (NDR), sa neprolife-rativnom dijabetičkom retinopatijom (NPDR) i proliferativnom dijabetičkom retinopatijom (PDR). U istraživanje je uključeno samo jedno oko svakog ispitanika. Rezultati: Prosečna vrednost CDR bolesnika sa DM bila je 570,52±31,81 μm, a zdravih ispitanika 541,42±27,82 μm. Razlika u CDR između dve posmatrane grupe bila je statistički visoko značajna (p0,05). Zaključak: Veća CDR je utvrđena kod osoba sa DM u poređenju sa zdravim ispitanicima. Najveća CDR je ustanovljena u grupi očiju sa PDR; slede grupe NPDR i NDR. Introduction Ultrastructural changes in corneas of patients with diabetes mellitus have been previously described. Objective The aim of this study was to compare central corneal thickness (CDR) values in diabetic patients without retinopathy at the stage of diabetic nonproliferative and proliferative retinopathy and CDR in a control group of healthy subjects. Methods The study included 121 diabetic patients and 125 healthy subjects matched according to gender and age. Each patient underwent ophthalmological examination involving a dilated fundus examination and CDR measurement using the ultrasound pachymeter. The eyes of diabetic patients were classified according to Early Treatment Diabetic Retinopathy Study into three groups: without diabetic retinopathy (NDR), with nonproliferative diabetic retinopathy (NPDR) and a group with proliferative diabetic retinopathy (PDR). Only one eye of each subject was chosen for the study. Results The mean CDR value was significantly higher in the diabetic group (570.52 +/- 31.81 mu m) compared with the control group (541.42 +/- 27.82 mu m). The difference between the two groups was statistically significant (p lt 0.0001). The highest mean CDR value was recorded in the PDR group (585.97 +/- 28.58 mu m), followed by the NPDR group (570.84 +/- 30.27 mu m), whereas the lowest mean CDR value was recorded in the NDR group (559.80 +/- 31.55 mu m). There was a statistically significant difference in CDR between the NDR and PDR groups, as well as between the NPDR and PDR groups (p lt 0.001, p lt 0.05 respectively). No significant difference was recorded between the NDR and NPDR groups (p>0.05). Conclusion CDR of diabetic patients was higher compared to healthy subjects. The highest mean value of CDR was registered in the PDR group, followed by the NPDR and the NDR groups.

U ovoj tezi opisujemo linearne, idempotentne, jake Maljcevljeve uslove za kongruencijsku ^-poludistributivnost u lokalno konačnim varijetetima. U [40] je predstavljen jedan takav jak Maljcevljev uslov i tehnika koju su autori koristili je uopštenje jednog od glavnih rezultata iz [46]. Glavna razlika je u tome što jednostavna primjena Dirihleovog principa u [46] postaje dosta komplikovaniji argument Remzijevskog tipa u [40]. Mi ovdje dodatno uopštavamo taj argument i prezentujemo dokaz pomenute karakterizacije. Svi ovi radovi koriste snažan rezultat [4] L. Barta o rješivosti Problema zadovoljenja uslova metodama provjere lokalne konzistencije uslova, pa je treća glava teze posvećena detaljnoj prezentaciji tog rada. Takođe, dokazujemo da neki jak Maljcevljev uslov karakteriše kongruencijsku ^-poludistributivnost u lokalno konačnim varijetetima ako i samo ako je realizovan u određenoj četvoroelementnoj algebri. Na kraju, bavimo se i problemom pronalaženja optimalnog jakog Maljcevljevog uslova koji karakteriše egzistenciju Tejlorovog terma u opštem slučaju. U [53] M. Olšak predstavio je iznenađujući rezultat da je egzistencija Tejlorovog terma jako Maljcevljevo svojstvo. Term iz prvobitne verzije [53] ima arnost 12, dok mi ovdje prezentujemo dokaz da se arnost može redukovati na 9., In this thesis we describe linear, idempotent, strong Mal’cev conditions for congruence ^-semidistributivity in locally finite varieties. In [40] authors presented one such Mal’cev condition and technique they used is generalization of one result from [46]. Main difference is that simple application of Pigeonhole prinicple from [46] becomes much more complicated Ramsey style argument in [40]. Here we dditionaly generalize that argument and we present the proof of above mentioned characterization. All these papers use deep result [4] by L. Barto on solvability of Constraint Satisfaction Problem by local consistency checking methods, so third chapter of this thesis is dedicated to detailed presentation of [4]. Also, we prove that some strong Mal’cev condition characterizes congruence ^-semidistributivity in locally finite varieties if and only if it is realized in certain four element algebra. Finally, we work on the problem of finding optimal strong Mal’cev condition for existence of Taylor term in general case. In [53] M. Olsak presented suprising result that exisstence of Taylor term is strong Malcev property. The term from first version of [53] has arity 12, but here we prove that arity can be reduced to 9.

Let S be a locally small category, and fix two (not necessarily distinct) objectsi, j in S. Let Sij and Sji denote the set of all morphisms i → j and j → i, respectively. Fixa ∈ 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 tocertain 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 allof 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 .

Usmereni stepeni graf ~G(G) grupe G uveli su Kelarev i Quinn [37] kao digraf sa skupom cvorova G u kome je x ! y ako je y stepen elementa x, a stepeni graf G(G) je odgovarajuci prost graf, i njega su prvi proucavali Chakrabarty, Ghosh i Sen [17]. Obogaceni stepeni graf G e(G) od G, koji je uveden u [1], je prost graf sa istim skupom cvorova u kome su dva cvora susedna ako su oba stepeni nekog elementa te grupe.U disertaciji su predstavljeni dokazi iz [12] i [73] da se, za konacnu stepenoasocijativnu lupu G sa inverzima, ~ G(G), G(G) i G e(G) medusobno odreduju. Ovo povlaci da sva tri navedena grafa pridruzena konacnoj grupi u istoj meri odreduju razne osobine te grupe, kao sto su broj elemenata bilo kog reda i nilpotentnost te grupe. Dokazano je da, u slucaju torziono slobodne grupe u kojoj je svaki nejedinicni element sadrzan u jedinstvenoj maksimalnoj ciklicnoj podgrupi, stepeni graf odreduje usmereni stepeni graf, sto je rezultat rada [14], i analogno je dokazano i za torziono slobodne grupe klase nilpotentnosti klase 2. Pruzen je dokaz da je svaki automorzam stepenog grafa stepeno-asocijativne lupe sa inverzima automorzam obogacenog grafa. Dat je opis obogacenih stepenih grafova konacnih Abelovih grupa. Prezentirano je nekoliko potrebnih uslova da graf bude obogaceni stepeni graf neke konacne grupe, kao i algoritam koji za obogaceni stepeni graf konacne nilpotentne grupe daje obogaceni stepeni graf njene podgrupe Sylowa.Komutirajuci graf grupe je prost graf ciji je skup cvorova nosac grupe, i u kome su dva elementa susedna ako komutiraju. U disertaciji je predstavljen dokaz Bernharda Neumanna [54] da, ako komutirajuci graf grupe nema beskonacan nezavisan skup, onda on nema ni proizvoljno velike konacne nezavisne skupove. Okarakterisane su nilpotentne grupe ciji stepeni graf nema beskonacni nezavisni skup, sto je rezultat rada [1]. Prezentovan je dokaz Shitova [69] da je hromatski broj stepenog grafa stepeno-asocijativnog grupoida najvise prebrojiv, i dokazano je da je hromatski broj obogacenog stepenog grafa stepeno-asocijativne lupe sa inverzima takodenajvise prebrojiv. Izlozen je dokaz iz [1] da je stepeni graf svake grupe ogranicenog eksponenta perfektan, i data je karakterizacija konacnih nilpotentnih grupa ciji je obogaceni stepeni graf perfektan., The directed power graph ~G(G) of a group G was introduced by Kelarev and Quinn [37] as the digraph with its vertex set G in which x ! y if y is a power of x.The power graph G(G) is the underlying simple graph, and it was rst studied by Chakrabarty, Ghosh and Sen [17]. The enhanced power graph G e(G) of G, which was introduced in [1], is the simple graph with the same vertex set in which two vertices are adjacent if they are powers of one element.In this thesis are presented the proofs from [12] and [73] that, for any powerassociative loop G with inverses,~ G(G), G(G) and G e(G) determine each other. It follows that each of these three graphs associated to a nite group provides the same amount of information about the group, such as the number of elements of any order and nilpotency of the group. It is also proved that, in the case of a torsionfree group in which every non-identity element is contained in a unique maximal cyclic subgroup, the power graph determines the directed power graph, which is a result from [14], and the same is proved for torsion-free groups of nilpotency class 2.It is proved that an automorphism of the power graph of a power-associative loop with inverses is an automorphism of the enhanced power graph. A description of enhanced power graphs of abelian groups is given. Several necessary conditions for a graph to be the enhanced power graph of a nite group are presented, as well as an algorithm which, given the enhanced power graph of a nite nilpotent group, constructs the enhanced power graph of the Sylow subgroup. The commuting graph of a group is the simple graph whose vertex set is the universe of the group, and in which two elements are adjacent if they commute. In the thesis is presented the proof by Bernhard Neumann [54] that, if the commuting graph of a group doesn't have any innite independent set, then there is a nite bound on cardinality of its independent sets. Nilpotent groups whose power graphs don't have any innite independent set are characterized, which is a result from [1]. The proof of Shitov [69] that the chromatic number of the power graph of a power-associative groupoid is at most countable is presented, and it is proved that the chromatic number of the enhanced power graph of power-associative loops with inverses are at most countable too. The proof from [1] that the power graph of any group of nite exponent is presented, and nite nilpotent groups whose enhanced power graph is perfect are characterized.

In this thesis we generalize the well-known connections between closure operators, closure systems and complete lattices. We introduce a special kind of a partial closure operator, named sharp partial closure operator, and show that each sharp partial closure operator uniquely corresponds to a partial closure system. We further introduce a special kind of a partial clo-sure system, called principal partial closure system, and then prove the representation theorem for ordered sets with respect to the introduced partial closure operators and partial closure systems.Further, motivated by a well-known connection between matroids and geometric lattices, given that the notion of matroids can be naturally generalized to partial matroids (by dening them with respect to a partial closure operator instead of with respect to a closure operator), we dene geometric poset, and show that there is a same kind of connection between partial matroids and geometric posets as there is between matroids and geometric lattices. Furthermore, we then dene semimod-ular poset, and show that it is indeed a generalization of semi-modular lattices, and that there is a same kind of connection between semimodular and geometric posets as there is betweensemimodular and geometric lattices.Finally, we note that the dened notions can be applied to im-plicational systems, that have many applications in real world,particularly in big data analysis., U ovoj tezi uopštavamo dobro poznate veze između operatora zatvaranja, sistema zatvaranja i potpunih mreža. Uvodimo posebnu vrstu parcijalnog operatora zatvaranja, koji nazivamo oštar parcijalni operator zatvaranja, i pokazujemo da svaki oštar parcijalni operator zatvaranja jedinstveno korespondira parcijalnom sistemu zatvaranja. Dalje uvodimo posebnu vrstu parcijalnog sistema zatvaranja, nazvan glavni parcijalni sistem zatvaranja, a zatim dokazujemo teoremu reprezentacije za posete u odnosu na uvedene parcijalne operatore zatvaranja i parcijalne sisteme zatvaranja. Dalje, s obzirom na dobro poznatu vezu između matroida i geometrijskih mreža, a budući da se pojam matroida može na prirodan nacin uopštiti na parcijalne matroide (definišući ih preko parcijalnih operatora zatvaranja umesto preko operatora zatvaranja), definišemo geometrijske uređene skupove i pokazujemo da su povezani sa parcijalnim matroidima na isti način kao što su povezani i matroidi i geometrijske mreže. Osim toga, definišemo polumodularne uređene skupove i pokazujemo da su oni zaista uopštenje polumodularnih mreža i da ista veza postoji između polumodularnih i geometrijskih poseta kao što imamo između polumodularnih i geometrijskih mreža. Konačno, konstatujemo da definisani pojmovi mogu biti primenjeni na implikacione sisteme, koji imaju veliku primenu u realnom svetu, posebno u analizi velikih podataka.

In this thesis we generalize the well-known connections between closure operators, closure systems and complete lattices. We introduce a special kind of a partial closure operator, named sharp partial closure operator, and show that each sharp partial closure operator uniquely corresponds to a partial closure system. We further introduce a special kind of a partial clo-sure system, called principal partial closure system, and then prove the representation theorem for ordered sets with respect to the introduced partial closure operators and partial closure systems. Further, motivated by a well-known connection between matroids and geometric lattices, given that the notion of matroids can be naturally generalized to partial matroids (by dening them with respect to a partial closure operator instead of with respect to a closure operator), we dene geometric poset, and show that there is a same kind of connection between partial matroids and geometric posets as there is between matroids and geometric lattices. Furthermore, we then dene semimod-ular poset, and show that it is indeed a generalization of semi-modular lattices, and that there is a same kind of connection between semimodular and geometric posets as there is between semimodular and geometric lattices. Finally, we note that the dened notions can be applied to im-plicational systems, that have many applications in real world,particularly in big data analysis. U ovoj tezi uopštavamo dobro poznate veze između operatora zatvaranja, sistema zatvaranja i potpunih mreža. Uvodimo posebnu vrstu parcijalnog operatora zatvaranja, koji nazivamo oštar parcijalni operator zatvaranja, i pokazujemo da svaki oštar parcijalni operator zatvaranja jedinstveno korespondira parcijalnom sistemu zatvaranja. Dalje uvodimo posebnu vrstu parcijalnog sistema zatvaranja, nazvan glavni parcijalni sistem zatvaranja, a zatim dokazujemo teoremu reprezentacije za posete u odnosu na uvedene parcijalne operatore zatvaranja i parcijalne sisteme zatvaranja. Dalje, s obzirom na dobro poznatu vezu između matroida i geometrijskih mreža, a budući da se pojam matroida može na prirodan nacin uopštiti na parcijalne matroide (definišući ih preko parcijalnih operatora zatvaranja umesto preko operatora zatvaranja), definišemo geometrijske uređene skupove i pokazujemo da su povezani sa parcijalnim matroidima na isti način kao što su povezani i matroidi i geometrijske mreže. Osim toga, definišemo polumodularne uređene skupove i pokazujemo da su oni zaista uopštenje polumodularnih mreža i da ista veza postoji između polumodularnih i geometrijskih poseta kao što imamo između polumodularnih i geometrijskih mreža. Konačno, konstatujemo da definisani pojmovi mogu biti primenjeni na implikacione sisteme, koji imaju veliku primenu u realnom svetu, posebno u analizi velikih podataka.

The research work carried out in this thesis is aimed at fuzzifying algebraic and relational structures in the framework of Ω-sets, where Ω is a complete lattice.Therefore we attempt to synthesis universal algebra and fuzzy set theory. Our investigations of Ω-algebraic structures are based on Ω-valued equality, satisability of identities and cut techniques. We introduce Ω-algebras, Ω-valued congruences, corresponding quotient Ω-valued-algebras and Ω-valued homomorphisms and we investigate connections among these notions. We prove that there is an Ω-valued homomorphism from an Ω-algebra to the corresponding quotient Ω-algebra. The kernelof an Ω-valued homomorphism is an Ω-valued congruence. When dealing with cut structures, we prove that an Ω-valued homomorphism determines classical homomorphisms among the corresponding quotient structures over cut subalgebras. In addition, an Ω-valued congruence determines a closure system of classical congruences on cut subalgebras. In addition, identities are preserved under Ω-valued homomorphisms. Therefore in the framework of Ω-sets we were able to introduce Ω-attice both as an ordered and algebraic structures. By this Ω-poset is defined as an Ω-set equipped with Ω-valued order which is antisymmetric with respect to the corresponding Ω-valued equality. Thus defining the notion of pseudo-infimum and pseudo-supremum we obtained the definition of Ω-lattice as an ordered structure. It is also defined that the an Ω-lattice as an algebra is a bi-groupoid equipped with an Ω-valued equality fulfilling some particular lattice Ω-theoretical formulas. Thus using axiom of choice we proved that the two approaches are equivalent. Then we also introduced the notion of complete Ω-lattice based on Ω-lattice. It was defined as a generalization of the classical complete lattice.We proved results that characterizes Ω-structures and many other interesting results.Also the connection between Ω-algebra and the notion of weak congruences is presented.We conclude with what we feel are most interesting areas for future work., Tema ovog rada je fazifikovanje algebarskih i relacijskih struktura u okviru omega- skupova, gdeje Ω kompletna mreza. U radu se bavimo sintezom oblasti univerzalne algebre i teorije rasplinutih (fazi) skupova. Naša istraživanja omega-algebarskih struktura bazirana su na omega-vrednosnoj jednakosti,zadovoljivosti identiteta i tehnici rada sa nivoima. U radu uvodimo omega-algebre,omega-vrednosne kongruencije, odgovarajuće omega-strukture, i omega-vrednosne homomorfizme i istražujemo veze izmedju ovih pojmova. Dokazujemo da postoji Ω -vrednosni homomorfizam iz Ω -algebre na odgovarajuću količničku Ω -algebru. Jezgro Ω -vrednosnog homomorfizma je Ω- vrednosna kongruencija. U vezi sa nivoima struktura, dokazujemo da Ω -vrednosni homomorfizam odredjuje klasične homomorfizme na odgovarajućim količničkim strukturama preko nivoa podalgebri. Osim toga, Ω-vrednosna kongruencija odredjuje sistem zatvaranja klasične kongruencije na nivo podalgebrama. Dalje, identiteti su očuvani u Ω- vrednosnim homomorfnim slikama.U nastavku smo u okviru Ω-skupova uveli Ω-mreže kao uredjene skupove i kao algebre i dokazali ekvivalenciju ovih pojmova. Ω-poset je definisan kao Ω -relacija koja je antisimetrična i tranzitivna u odnosu na odgovarajuću Ω-vrednosnu jednakost. Definisani su pojmovi pseudo-infimuma i pseudo-supremuma i tako smo dobili definiciju Ω-mreže kao uredjene strukture. Takodje je definisana Ω-mreža kao algebra, u ovim kontekstu nosač te strukture je bi-grupoid koji je saglasan sa Ω-vrednosnom jednakošću i ispunjava neke mrežno-teorijske formule. Koristeći aksiom izbora dokazali smo da su dva pristupa ekvivalentna. Dalje smo uveli i pojam potpune Ω-mreže kao uopštenje klasične potpune mreže. Dokazali smo još neke rezultate koji karakterišu Ω-strukture.Data je i veza izmedju Ω-algebre i pojma slabih kongruencija.Na kraju je dat prikaz pravaca daljih istrazivanja.

The research work carried out in this thesis is aimed at fuzzifying algebraic and relational structures in the framework of Ω-sets, where Ω is a complete lattice. Therefore we attempt to synthesis universal algebra and fuzzy set theory. Our investigations of Ω-algebraic structures are based on Ω-valued equality, satisability of identities and cut techniques. We introduce Ω-algebras, Ω-valued congruences, corresponding quotient Ω-valued-algebras and Ω-valued homomorphisms and we investigate connections among these notions. We prove that there is an Ω-valued homomorphism from an Ω-algebra to the corresponding quotient Ω-algebra. The kernel of an Ω-valued homomorphism is an Ω-valued congruence. When dealing with cut structures, we prove that an Ω-valued homomorphism determines classical homomorphisms among the corresponding quotient structures over cut subalgebras. In addition, an Ω-valued congruence determines a closure system of classical congruences on cut subalgebras. In addition, identities are preserved under Ω-valued homomorphisms. Therefore in the framework of Ω-sets we were able to introduce Ω-attice both as an ordered and algebraic structures. By this Ω-poset is defined as an Ω-set equipped with Ω-valued order which is antisymmetric with respect to the corresponding Ω-valued equality. Thus defining the notion of pseudo-infimum and pseudo-supremum we obtained the definition of Ω-lattice as an ordered structure. It is also defined that the an Ω-lattice as an algebra is a bi-groupoid equipped with an Ω-valued equality fulfilling some particular lattice Ω-theoretical formulas. Thus using axiom of choice we proved that the two approaches are equivalent. Then we also introduced the notion of complete Ω-lattice based on Ω-lattice. It was defined as a generalization of the classical complete lattice. We proved results that characterizes Ω-structures and many other interesting results. Also the connection between Ω-algebra and the notion of weak congruences is presented. We conclude with what we feel are most interesting areas for future work. Tema ovog rada je fazifikovanje algebarskih i relacijskih struktura u okviru omega- skupova, gdeje Ω kompletna mreza. U radu se bavimo sintezom oblasti univerzalne algebre i teorije rasplinutih (fazi) skupova. Naša istraživanja omega-algebarskih struktura bazirana su na omega-vrednosnoj jednakosti,zadovoljivosti identiteta i tehnici rada sa nivoima. U radu uvodimo omega-algebre,omega-vrednosne kongruencije, odgovarajuće omega-strukture, i omega-vrednosne homomorfizme i istražujemo veze izmedju ovih pojmova. Dokazujemo da postoji Ω -vrednosni homomorfizam iz Ω -algebre na odgovarajuću količničku Ω -algebru. Jezgro Ω -vrednosnog homomorfizma je Ω- vrednosna kongruencija. U vezi sa nivoima struktura, dokazujemo da Ω -vrednosni homomorfizam odredjuje klasične homomorfizme na odgovarajućim količničkim strukturama preko nivoa podalgebri. Osim toga, Ω-vrednosna kongruencija odredjuje sistem zatvaranja klasične kongruencije na nivo podalgebrama. Dalje, identiteti su očuvani u Ω- vrednosnim homomorfnim slikama.U nastavku smo u okviru Ω-skupova uveli Ω-mreže kao uredjene skupove i kao algebre i dokazali ekvivalenciju ovih pojmova. Ω-poset je definisan kao Ω -relacija koja je antisimetrična i tranzitivna u odnosu na odgovarajuću Ω-vrednosnu jednakost. Definisani su pojmovi pseudo-infimuma i pseudo-supremuma i tako smo dobili definiciju Ω-mreže kao uredjene strukture. Takodje je definisana Ω-mreža kao algebra, u ovim kontekstu nosač te strukture je bi-grupoid koji je saglasan sa Ω-vrednosnom jednakošću i ispunjava neke mrežno-teorijske formule. Koristeći aksiom izbora dokazali smo da su dva pristupa ekvivalentna. Dalje smo uveli i pojam potpune Ω-mreže kao uopštenje klasične potpune mreže. Dokazali smo još neke rezultate koji karakterišu Ω-strukture.Data je i veza izmedju Ω-algebre i pojma slabih kongruencija.Na kraju je dat prikaz pravaca daljih istrazivanja.

У дисертацији се разматра утицај примене рачунара на предавање и учење аналитичкегеометрије. Израђени су одговарајући динамични радни листови којима је пропраћено градивоизложено у званичном уџбенику. Приказана је примена рачунара и софтвера GeoGebra уизграђивању рачунарског окружења погодног за изучавање аналитичке геометрије у равни., U disertaciji se razmatra uticaj primene računara na predavanje i učenje analitičkegeometrije. Izrađeni su odgovarajući dinamični radni listovi kojima je propraćeno gradivoizloženo u zvaničnom udžbeniku. Prikazana je primena računara i softvera GeoGebra uizgrađivanju računarskog okruženja pogodnog za izučavanje analitičke geometrije u ravni., The thesis considers the impact that computer usage makes on analytic instructionprocess. Appropriate dynamic worksheets are developed in order to ease students’ understanding of the subject material included in the official textbook. Computers and GeoGebra were used to build a computer empowered learning environment suitable for plane analytic geometry instruction.

У дисертацији се разматра утицај примене рачунара на предавање и учење аналитичке геометрије. Израђени су одговарајући динамични радни листови којима је пропраћено градиво изложено у званичном уџбенику. Приказана је примена рачунара и софтвера GeoGebra у изграђивању рачунарског окружења погодног за изучавање аналитичке геометрије у равни. U disertaciji se razmatra uticaj primene računara na predavanje i učenje analitičke geometrije. Izrađeni su odgovarajući dinamični radni listovi kojima je propraćeno gradivo izloženo u zvaničnom udžbeniku. Prikazana je primena računara i softvera GeoGebra u izgrađivanju računarskog okruženja pogodnog za izučavanje analitičke geometrije u ravni. The thesis considers the impact that computer usage makes on analytic instruction process. Appropriate dynamic worksheets are developed in order to ease students’ understanding of the subject material included in the official textbook. Computers and GeoGebra were used to build a computer empowered learning environment suitable for plane analytic geometry instruction.

Predmet ove disertacije je sintaksna karakterizacija kongruencijske polu{distributiv- nosti (u odnosu na inmum) lokalno konacnih varijeteta Maljcevljevim uslovima (posmatramo varijetete idempotentnih algebri). Dokazujemo da takva karakteri- zacija nije moguca sistemom identiteta koji koriste jedan ternarni i proizvoljan broj binarnih operacijskih simbola. Prvu karakterizaciju dobijamo jakim Maljcevljevim uslovom koji ukljucuje dva ternarna simbola: Lokalno konacan varijetet V zadovo- ljava uslov kongruencijske polu{distributivnosti (u odnosu na inmum) ako i samo ako postoje ternarni termi p i q (koji indukuju idempotentne term operacije) takvi da V zadovoljava: p(x; x; y) p(x; y; y) p(x; y; x) q(x; y; x) q(x; x; y) q(y; x; x). Ovaj uslov je optimalan u smislu da su broj terma, njihove visestrukosti i broj identiteta najmanji moguci. Druga karakterizacija koju dobijamo koristi jedan 4- arni simbol i data je jakim Maljcevljevim uslovom t(y; x; x; x) t(x; y; x; x) t(x; x; y; x) t(x; x; x; y) t(y; y; x; x) t(y; x; y; x) t(x; y; y; x) : Treca karakterizacija je data kompletnim Maljcevljevim uslovom: Postoje binarni term t(x; y) i wnu-termi !n(x1; : : : ; xn) varijeteta V tako za sve n > 3 vazi sledece: V j= !n(x; x; : : : ; x; y) t(x; y). The subject of this dissertation is a syntactic characterization of congruence ^{ semidistributivity in locally nite varieties by Mal'cev conditions (we consider va- rieties of idempotent algebras). We prove that no such characterization is possible by a system of identities including one ternary and any number of binary opera- tion symbols. The rst characterization is obtained by a strong Mal'cev condition involving two ternary term symbols: A locally nite variety V satises congruence meet{semidistributivity if and only if there exist ternary terms p and q (inducing idempotent term operations) such that V satises p(x; x; y) p(x; y; y) p(x; y; x) q(x; y; x) q(x; x; y) q(y; x; x). This condition is optimal in the sense that the number of terms, their arities and the number of identities are the least possible. The second characterization that we nd uses a single 4-ary term symbol and is given by the following strong Mal'cev condition t(y; x; x; x) t(x; y; x; x) t(x; x; y; x) t(x; x; x; y) t(y; y; x; x) t(y; x; y; x) t(x; y; y; x) : The third characterization is given by a complete Mal'cev condition: There exist a binary term t(x; y) and wnu-terms !n(x1; : : : ; xn) of variety V such that for all n > 3 the following holds: V j= !n(x; x; : : : ; x; y) t(x; y).

Disertacija je posvećena rešavanju jednačina distributivnosti gdenepoznate funkcije pripadaju nekim poznatim klasama operacija agregacijei primeni dobijenih rešenja u teoriji korisnosti. Dobijeni rezultati se generalno mogu podeliti u tri grupe. Prvu grupu čine rezultati iz Glave 2 dobijeni rešavanjem jednačina distributivnosti između GM-operacija agregacije ioslabljenih uninormi, GM-operacija agregacije i oslabljenih nulanormi, kao iGM-operacija agregacije i operacija agregacije bez neutralnog i absorbujućegelementa. Druga grupa rezultata, takođe iz Glave 2, je dobijena rešavanjemjednačina uslovne (oslabljene) distributivnosi neprekidne nulanorme u odnosuna neprekidnu t-konormu, i neprekidne nulanonorme u odnosu na uninorme izklasa Umin ∪Umax. Treća grupa rezultata (Glava 3) je proistekla iz primene dobijenih rezultata o uslovoj distributivnosti nulanorme u odnosu na t-konormuu teoriji korisnosti., This dissertation is devoted to solving distributivity equations involving some well-known classes of aggregation operators, and applicationthe obtained results to utility theory. In general, the obtained results canbe divided into three groups. The first group are results from Chapter 2 obtained by solving distributivity equations between GM-aggregation operatorsand relaxed nullnorm, GM-aggregation operators and relaxed uninorms, aswell as GM-aggregation operators and aggregation operators without neutraland absorbing element. The second group are results, also from Chapter 2,obtained by solving conditional (relaxed) distributivity of continuous nullnorm with respect to continuous t-conorm, as well as continuous nullnormwith respect to uninorms from the classes Umin ∪ Umax. The third group areresults (Chapter 3) arising from the application results on conditional distributivity of nullnorm with respect to t-conorm in utility theory.

In the thesis we are concerned with actual problems on palindromic subwords and palindromic factors of finite and infinite words. The main course of the research are the ways of determining which of two given words is “more palindromic” than the other one, that is, defining a measure for the degree of “palindromicity” of a word. Particularly, we pay attention to two actual approaches: the so-called MP-ratio and the so-called palindromic defect, and answer several open questions about them.Namely, concerning the MP-ratio, a few plausible-looking question have been asked in the literature, which would have, if answered positively, made computations of MP-ratios significantly simpler. We add one more related question to these ones, and then show that, rather unexpectedly, all these questions have negative answer.Concerning the palindromic defect, the main result of this work is a construction of an infinite class of infinite words that have several properties that were sought after in some recent works in this area. Among the most interesting facts is that that all these words are aperiodic words of a finite positive defect, having the set of factors closed under reversal---in some recent works, the construction of even a single word having these properties turned out to be quite hard. Using these words, which we are calling highly potential words, we check the validity of several open conjectures, and for several of them we find out that they are false., U tezi razmatramo aktuelne probleme u vezi s palindromskim podrečima i palindromskim faktorima konačnih i beskonačnih reči. Glavni pravac istraživanja jesu kriterijumi za određivanje koja od dve date reči je „palindromičnija“ od druge, tj. određivanje stepena „palindromičnosti“ date reči. Akcenat stavljamo na dva aktuelna pristupa: tzv. MP-razmeru i tzv. palindromski defekt, i odgovaramo na više otvorenih pitanja u vezi s njima. Naime, u vezi sa MP-razmerom u literaturi je postavljeno više pitanja, intuitivno uverljivih, koja bi, u slučaju pozitivnog razrešenja, znatno pojednostavila izračunavanje MP-razmere. Ovim pitanjima dodajemo još jedno srodno, a zatim pokazujemo da, prilično neočekivano, sva ova pitanja imaju negativan odgovor. U vezi s palindromskim defektom, glavni rezultat rada je konstrukcija beskonačne klase beskonačnih reči koje imaju više osobina za kojima je iskazana potreba u skorašnjim radovima iz ove oblasti. Među najzanimljivije spada činjenica da su sve aperiodične reči konačnog pozitivnog defekta, i da im je skup faktora zatvoren za preokretanje – u nekim skorašnjim radovima konstrukcija makar jedne reči s ovim osobinama pokazala se kao prilično teška. Pomoću ovih reči, koje nazivamo visokopotencijalne reči, ispitujemo validnost više otvorenih hipoteza, i za više njih ustanovljavamo da nisu validne.

In the thesis we are concerned with actual problems on palindromic subwords and palindromic factors of finite and infinite words. The main course of the research are the ways of determining which of two given words is “more palindromic” than the other one, that is, defining a measure for the degree of “palindromicity” of a word. Particularly, we pay attention to two actual approaches: the so-called MP-ratio and the so-called palindromic defect, and answer several open questions about them. Namely, concerning the MP-ratio, a few plausible-looking question have been asked in the literature, which would have, if answered positively, made computations of MP-ratios significantly simpler. We add one more related question to these ones, and then show that, rather unexpectedly, all these questions have negative answer. Concerning the palindromic defect, the main result of this work is a construction of an infinite class of infinite words that have several properties that were sought after in some recent works in this area. Among the most interesting facts is that that all these words are aperiodic words of a finite positive defect, having the set of factors closed under reversal---in some recent works, the construction of even a single word having these properties turned out to be quite hard. Using these words, which we are calling highly potential words, we check the validity of several open conjectures, and for several of them we find out that they are false. U tezi razmatramo aktuelne probleme u vezi s palindromskim podrečima i palindromskim faktorima konačnih i beskonačnih reči. Glavni pravac istraživanja jesu kriterijumi za određivanje koja od dve date reči je „palindromičnija“ od druge, tj. određivanje stepena „palindromičnosti“ date reči. Akcenat stavljamo na dva aktuelna pristupa: tzv. MP-razmeru i tzv. palindromski defekt, i odgovaramo na više otvorenih pitanja u vezi s njima. Naime, u vezi sa MP-razmerom u literaturi je postavljeno više pitanja, intuitivno uverljivih, koja bi, u slučaju pozitivnog razrešenja, znatno pojednostavila izračunavanje MP-razmere. Ovim pitanjima dodajemo još jedno srodno, a zatim pokazujemo da, prilično neočekivano, sva ova pitanja imaju negativan odgovor. U vezi s palindromskim defektom, glavni rezultat rada je konstrukcija beskonačne klase beskonačnih reči koje imaju više osobina za kojima je iskazana potreba u skorašnjim radovima iz ove oblasti. Među najzanimljivije spada činjenica da su sve aperiodične reči konačnog pozitivnog defekta, i da im je skup faktora zatvoren za preokretanje – u nekim skorašnjim radovima konstrukcija makar jedne reči s ovim osobinama pokazala se kao prilično teška. Pomoću ovih reči, koje nazivamo visokopotencijalne reči, ispitujemo validnost više otvorenih hipoteza, i za više njih ustanovljavamo da nisu validne.

We establish several properties of higher commutators, which wereintroduced by A. Bulatov, in congruence permutable varieties. We use thesecommutators to prove that the clone of polynomial functions of a finite Mal’cevalgebra whose congruence lattice is of height at most 2, can be described by afinite set of relations. For a finite nilpotent algebra of finite type that is a productof algebras of prime power order and generates congruence modular variety, weare able to show that the property of affine completeness is decidable. Moreover,polynomial equivalence problem has polynomial complexity in the length of theinput polynomials., Ustanovljavamo osobine viˇsih komutatora, koje je uveo A. Bulatov,u kongruencijki permutabilnim varijetetima. Te komutatore koristimo da bidokazali da se klon polinomijalnih funkcija konaˇcne Maljcevljeve algebre ˇcija jemreˇza kongruencija visine najviˇse dva moˇze opisati konaˇcnim skupom relacija. Zakonaˇcne nilpotentne algebre konaˇcnog tipa koje su proizvod algebri koje imaju redstepena prostog broja i koje generiˇsu kongruencijki modularan varijetet pokazu-jemo da je osobina afine kompletnosti odluˇciva. Takod¯e, pokazujemo za istu klasuda problem polinomijalne ekvivalencije ima polinomnu sloˇzenost u zavisnosti odduˇzine unetih polinomijalnih terma.

Ova teza se bavi ¤-kvazilinearnim varijetetima grupoida. Pokazano je da postoji ta·cno dvadeset osam idempotentnih ¤-kvazilinearnih varijeteta grupoida, od kojih dvadeset ·sest varijeteta imaju kona·cnu bazu i te baze su i navedene, dok preostala dva varijeteta imaju inherentno beskona·cnu bazu. U tezi je opisano ured enje svih idempotentnih ¤-kvazilinearnih varijeteta grupoida i nalazimo male grupoide koji generi·su svaki od navedenih varijeteta. Na kraju je pokazano da postoji kontinum mnogo ¤-kvazilinearnih variejeteta grupoida., The topic of this thesis are ¤-quasilinear varieties of groupoids.We show that there exist exactly twenty-eight idempotent ¤-quasilinear varieties of groupoids, twenty-six of which are ¯nitely based (and we explicitlygive ¯nite bases for each of them), while two are inherently non¯nitely based.We describe the ordering of these twenty-eight idempotent ¤-quasilinear varieties of groupoids and ¯nd small generating algebras for each of them. Inthe end we show that there exist continuum many ¤-quasilinear varieties ofgroupoids, not all of which are even locally ¯nite.