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<p>The research work carried out in this thesis is aimed&nbsp;&nbsp; at fuzzifying algebraic and relational structures in the framework of Ω-sets, where Ω is a complete lattice.<br />Therefore we attempt to synthesis universal algebra and fuzzy set theory. Our&nbsp; investigations of Ω-algebraic structures are based on Ω-valued equality, satisability of identities and cut techniques. We introduce Ω-algebras, Ω-valued congruences,&nbsp; corresponding quotient&nbsp; Ω-valued-algebras and&nbsp; 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<br />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&nbsp; subalgebras. In addition, an&nbsp; 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&nbsp; Ω-valued order which is&nbsp; 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.<br />We proved results that characterizes Ω-structures and many other interesting results.<br />Also the connection between Ω-algebra and the notion of weak congruences is presented.<br />We conclude with what we feel are most interesting areas for future work.</p> / <p>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. Na&scaron;a 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 jednako&scaron;ću 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 uop&scaron;tenje klasične potpune mreže. Dokazali smo jo&scaron; neke rezultate koji karakteri&scaron;u Ω-strukture.Data je i veza izmedju Ω-algebre i pojma slabih kongruencija.Na kraju je dat prikaz pravaca daljih istrazivanja.</p>

<p>In this work a comparative analysis of several approaches to fuzzy algebraic structures and comparison of previous approaches to the recent one developed at University of&nbsp; Novi Sad has been done. Special attention is paid to reducts and expansions of algebraic structures in fuzzy settings. Besides mentioning all the relevant algebras and properties developed in this setting, particular new algebras and properties are developed and investigated. Some new structures, in particular Omega Boolean algebras, Omega Boolean lattices and Omega Boolean rings are developed in the framework of omega structures. Equivalences among these structures are elaborated in details. Transfers from Omega groupoids to Omega groups and back are demonstrated. Moreover, normal subgroups are introduced in a particular way. Their connections to congruences are elaborated in this settings. Subgroups, congruences and normal subgroups are investigated for Ω-groups. These are latticevalued algebraic structures, defined on crisp algebras which are not necessarily groups, and in which the classical equality is replaced by a lattice-valued one. A normal Ω-subgroup is defined as a particular class in an Ω-congruence. Our main result is that the quotient groups over cuts of a normal Ω- subgroup of an Ω-group G, are classical normal subgroups of the corresponding quotient groups over G. We also describe the minimal normal Ω-subgroup of an Ω-group, and some other constructions related to Ω-valued congruences.Further results that are obtained are theorems that connect various approaches of fuzzy algebraic structures. A special notion of a generalized lattice valued Boolean algebra is introduced. The universe of this structure is an algebra with two binary, an unary and two nullary operations (as usual), but which is not a crisp Boolean algebra in general. A main element in our approach is a fuzzy&nbsp; quivalence relation such that the Boolean algebras identities are approximately satisfied related to the considered fuzzy equivalence. Main properties of the new introduced notions are proved, and a connection with the notion of a structure of a generalized fuzzy lattice is provided.</p> / <p>Ovaj rad bavi se komparativnom analizom različitih pristupa rasplinutim (fazi) algebarskim strukturama i odnosom tih struktura sa odgovarajućim klasičnim&nbsp;&nbsp; algebrama. Posebna pažnja posvećena je poredenju postojećih pristupa ovom&nbsp;&nbsp; problemu sa novim tehnikama i pojmovima nedavno razvijenim na Univerzitetu u Novom Sadu. U okviru ove analize, proučavana su i pro&scaron;irenja kao i redukti algebarskih struktura u kontekstu rasplinutih algebri. Brojne važne konkretne algebarske strukture istraživane su u ovom kontekstu, a neke nove uvedene su i ispitane. Bavili smo se detaljnim istrazivanjima Ω-grupa, sa stanovista kongruencija, normalnih podgrupa i veze sa klasicnim grupama. Nove strukture koje su u radu uvedene u posebnom delu, istrazene su sa aspekta svojstava i medusobne ekvivalentnosti. To su Ω-Bulove algebre, kao i odgo-varajuce mreže i Bulovi prsteni. Uspostavljena je uzajamna ekvivalentnost tih struktura analogno odnosima u klasičnoj algebri. U osnovi na&scaron;e konstrukcije su mrežno vrednosne algebarske strukture denisane na klasičnim algebrama koje ne zadovoljavaju nužno identitete ispunjene na odgovarajucim klasičnim strukturama (Bulove algebre, prsteni, grupe itd.), već su to samo algebre istog tipa. Klasična jednakost zamenjena je posebnom kompatibilnom rasplinutom (mrežno-vrednosnom) relacijom ekvivalencije. Na navedeni nacin i u cilju koji je u osnovi teze (poredenja sa postojecim pristupima u ovoj naucnoj oblasti) proucavane su (vec denisane)&nbsp; Ω-grupe. U nasim istraživanju uvedene su odgovarajuće normalne podgrupe. Uspostavljena je i istražena njihova veza sa Ω-kongruencijama. Normalna podgrupa&nbsp; Ω-grupe definisana je kao posebna&nbsp; klasa Ω-kongruencije. Jedan od rezultata u ovom delu je da su količničke grupe definisane pomocu nivoa Ω-jednakosti klasične normalne podgrupe odgovarajućih količničkih podgrupa polazne&nbsp; -grupe. I u ovom slučaju osnovna&nbsp; struktura na kojoj je denisana Ω-grupa je grupoid, ne nužno grupa. Opisane su osobine najmanje normalne podgrupe u terminima Ω-kongruencija, a date su i neke konstrukcije&nbsp; Ω-kongruencija.</p><p>Rezultati koji su izloženi u nastavku povezuju različite pristupe nekim mrežno- vrednosnim strukturama. Ω-Bulova algebra je uvedena na strukturi sa dve binarne, unarnom i dve nularne operacije, ali za koju se ne zahteva ispunjenost klasičnih aksioma. Identiteti za Bulove algebre važe kao mrežno-teoretske formule u odnosu na mrežno-vrednosnu jednakost. Klasicne Bulove algebre ih zadovoljavaju, ali obratno ne vazi: iz tih formula ne slede standardne aksiome za Bulove algebre. Na analogan nacin uveden je i&nbsp; Ω-Bulov prsten. Glavna svojstva ovih struktura su opisana. Osnovna osobina je da se klasične Bulove algebre odnosno Bulovi prsteni javljaju kao količničke strukture na nivoima Ω -jednakosti. Veza ove strukture sa Ω-Bulovom mrežom je pokazana.</p><p>Kao ilustracija ovih istraživanja, u radu je navedeno vi&scaron;e primera.</p>

<p>Intuicionistički rasplinuti skupovi su već proučavani i definisani u kontekstu mrežnovrednosnih struktura, ali svaka od postojećih definicija imala je odgovarajuće nedostatke. U ovom radu razvijena je definicija intuicionističkog poset-vrednosnog rasplinutog skupa, kojom se poset predstavlja kao podskup distributivne mreže. Na ovaj način možemo ispitivati funkcije pripadanja i nepripadanja i njihove odnose bez upotrebe komplementiranja na posetu. Takođe, u ovako postavljenim okvirima, svaki poset (a samim tim i mreža) može biti kodomen intuicionističkog rasplinutog skupa (čime se isključuje uslov ograničenosti poseta). Primenom uvedene definicije razmatrane su IP-vrednosne rasplinute relacije, x-blokovi ovih relacija i familije<br />njihovih nivoa.Razvijene su jake poset vrednosne relacije reciprociteta koje&nbsp; predstavljaju uop&scaron;tenje relacija reciprociteta sa intervala [0,1]. Pokazano je da ovakve relacije imaju svojstva slična poset-vrednosnim relacijama preferencije. Međutim, postoje velika ograničenja za primenu ovakvih relacija jer su zahtevi dosta jaki.<br />Uvedene su IP-vrednosne relacije reciprociteta koje se mogu definisati za veliku klasu poseta.Ovakve relacije pogodne su za opisivanje preferencija. Posmatrana je intuicionistička poset-vrednosna relacija preferencije, koja je refleksivna rasplinuta relacija, nad skupom alternativa. U samom procesu vi&scaron;ekriterijumskog odlučivanja<br />može se pojaviti situacija kada alternative nisu međusobno uporedive u odnosu na relaciju preferencije, kao i nedovoljna određenost samih alternativa. Da bi se prevazi&scaron;li ovakvi problemi uvodi se intuicionistička poset-vrednosna relacija preferencije kao intuicionistička rasplinuta relacija na skupu alternativa sa vrednostima u uređenom skupu. Analizirana su neka njena svojstva. Ovakav model pogodan je za upoređivanje alternativa koje nisu, nužno, u linearnom poretku. Dato je nekoliko opravdanja za uvodjenje oba tipa definisanih relacija. Jedna od mogućnosti jeste preko mreže intervala elemenata iz konačnog lanca S, a koji predstavljaju ocene određene alternative. Relacije preferencije mogu uzimati vrednosti sa ove mreže i time se može prevazići nedostatak informacija ili neodlučnost donosioca odluke.</p> / <p>Intuitionistic fuzzy sets have already been explored in depth and defined in the context of lattice-valued intuitionistic fuzzy sets, however, every existing definition has certain drawbacks. In this thesis, a definition of poset-valued intuitionistic fuzzy sets is developed, which introduces a poset as a subset of a distributive lattice. In this manner, functions of membership and non-membership can be examined as well as&nbsp; their relations without using complement in the poset. Also, in such framework, each poset (and the lattice) can be a co-domain of an intuitionistic fuzzy set (which excludes the condition of the bounded poset). Introduced definition defines IP-valued fuzzy relations, x-blocks of these relations andfamilies of their levels. Strong IP-valued&nbsp; reciprocialy relations have been developed as a generalization of reciprocal relations from interval [0,1]. It has been shown that these relations have properties similar to the P-valued preferences relations. However, there are great constraints on the application of these relations because the requirements are quite strong.IP- valued reciprocial relations have been introduced, which can be defined for a large class of posets. Such relations are suitable for describing preferences.An intuitionistic poset-valued preference relation, which is a reflexive fuzzy relation, over the set of&nbsp; alternatives, has been examined. In the process of a multi-criteria decision making, a situation can occur that the alternatives cannot be compared by the preference relation, as well as insufficient determination of the mentioned alternatives. In order to overcome similar problems, we have introduced an intuitionistic poset-valued preference relation as an intuitionistic fuzzy set over the set of alternatives with values in a certain poset. We have analyzed some its performances. This model is suitable for comparing alternatives which are not necessarily linearly ordered. There are several justifications for the introduction of&nbsp; both types of defined relations. One of the possibilities is via the lattice of the intervals&nbsp; of elements from the finite chain S, which represent the preference of a particular alternative. Preferences relations can take values from this lattice and this can overcome the lack of informations or the decisiveness of the decision maker.</p>

Development of electric vehicles is motivated by global concerns over the need for environmental protection. In addition to its zero-emission characteristics, an electric propulsion system enables high performance torque control that may be used to maximize vehicle performance obtained from energy-efficient, low rolling resistance tires typically associated with degraded road-holding ability. A simultaneous plant/controller optimization is performed on an electric vehicle traction control system with respect to conflicting energy use and performance objectives. Due to system nonlinearities, an iterative simulation-based optimization approach is proposed using a system model and a genetic algorithm (GA) to guide search space exploration. The system model consists of: a drive cycle with a constant driver torque request and a step change in coefficient of friction, a single-wheel longitudinal vehicle model, a tire model described using the Magic Formula and a constant rolling resistance, and an adhesion gradient fuzzy logic traction controller. Optimization is defined in terms of the all at once variable selection of: either a performance oriented or low rolling resistance tire, the shape of a fuzzy logic controller membership function, and a set of fuzzy logic controller rule base conclusions. A mixed encoding, multi-chromosomal GA is implemented to represent the variables, respectively, as a binary string, a real-valued number, and a novel rule base encoding based on the definition of a partially ordered set (poset) by delta inclusion. Simultaneous optimization results indicate that, under straight-line acceleration and unless energy concerns are completely neglected, low rolling resistance tires should be incorporated in a traction control system design since the energy saving benefits outweigh the associated degradation in road-holding ability. The results also indicate that the proposed novel encoding enables the efficient representation of a fix-sized fuzzy logic rule base within a GA.

Development of electric vehicles is motivated by global concerns over the need for environmental protection. In addition to its zero-emission characteristics, an electric propulsion system enables high performance torque control that may be used to maximize vehicle performance obtained from energy-efficient, low rolling resistance tires typically associated with degraded road-holding ability. A simultaneous plant/controller optimization is performed on an electric vehicle traction control system with respect to conflicting energy use and performance objectives. Due to system nonlinearities, an iterative simulation-based optimization approach is proposed using a system model and a genetic algorithm (GA) to guide search space exploration. The system model consists of: a drive cycle with a constant driver torque request and a step change in coefficient of friction, a single-wheel longitudinal vehicle model, a tire model described using the Magic Formula and a constant rolling resistance, and an adhesion gradient fuzzy logic traction controller. Optimization is defined in terms of the all at once variable selection of: either a performance oriented or low rolling resistance tire, the shape of a fuzzy logic controller membership function, and a set of fuzzy logic controller rule base conclusions. A mixed encoding, multi-chromosomal GA is implemented to represent the variables, respectively, as a binary string, a real-valued number, and a novel rule base encoding based on the definition of a partially ordered set (poset) by delta inclusion. Simultaneous optimization results indicate that, under straight-line acceleration and unless energy concerns are completely neglected, low rolling resistance tires should be incorporated in a traction control system design since the energy saving benefits outweigh the associated degradation in road-holding ability. The results also indicate that the proposed novel encoding enables the efficient representation of a fix-sized fuzzy logic rule base within a GA.