Colombia's telecommunication industry has changed drastically in the last decade. Among the most salient events, a series of mergers between some of the industry's largest providers which resulted in a reduced number of competitors. One would expect that this reduction in the number of competitors would translate into higher prices. However, competition is at such high level that the media even talk about a "price war". My dissertation aims to shed light on the causes of this apparent inconsistency between a smaller number of competitors and more competitive outcomes. I start by showing that, in effect, the latest of these mergers, the one between Comcel and Telmex, had a pro-competitive effects on the provision of broadband. Next I show that the services provided by Comcel and Telmex were complements and that the pro-competitive effects of the merger can be explained by this complementarity. Finally, I study the effects of price discrimination under oligopolistic competition. In chapter 1 I assess the ex-post short-run effect on broadband provision of the Comcel-Telmex merger. Employing administrative data about the universe of plans and firms providing wired telecom services, I use several difference-in-difference specifications to obtain estimates for the effect of the merger on price and download speeds of plans provided by the merging firm and its rivals. My estimates suggest that, in markets affected by the merger, download speeds rose by .6 Megabits per second on average. The average increase in the markets resulted from increases in the plans offered by the merging firm (1.2 Mbps) and increases in the speeds of plans provided by its rivals (.5 Mbps). In chapter 2 I study mergers of firms producing complementary goods. Mergers of firms producing complementary products have ambiguous effects on consumer welfare. The merged firm may lower prices because the merger internalizes the profits originated by the complementarity. But with the merger the firm gains the ability to bundle and with bundles the firm can exert price discrimination, increasing the prices of standalone products. I employ a comprehensive, administrative data set, which records prices, market shares, and plan attributes of the universe of Colombia’s telecom carriers, to assess which effect dominates. I estimate a random-coefficient discrete choice model of consumer demand model for bundled and standalone telecom products, in which the degree of substitutability or complementarity among products is an essential parameter of interest. I find that major telecom products display a mix of substitutability and complementarity, but in general hardwired and mobile services are perceived as complements by Colombian households. My counterfactual experiments using the estimated model, indicate positive net effects of mergers with complements: despite a small increase in the price of standalone goods, consumer surplus increases by around 7 million dollars per quarter. Finally, in chapter 3, I study price discrimination in an oligopolistic setting. Economic theory is not conclusive about the effects of banning third degree price discrimination under imperfect competition. Price discrimination can enhance competition if the firms practicing don't agree on the ranking of their markets. In this case, price discrimination can lead to lower prices in all markets. Thus, forcing the firms to charge uniform prices can increase their profits and reduce consumers' surplus. Using data on prices, market shares and characteristics of telecommunication services sold under price discrimination by Colombian telecom providers, I estimate a model of competition. The estimates allow me to simulate a counterfactual scenario in which firms lose their ability to exert price discrimination within a city. Simulating a ban on price discrimination has negligible effect on consumer surplus and increases profits slightly.

This dissertation concerns the study of canonical components of the SL(2, C) character varieties of hyperbolic 3-manifolds. Although character varieties have proven to be a useful tool in studying hyperbolic 3-manifolds, very little is known about their structure. Chapter 1 provides background on this subject. Chapter 2 is dedicated to the canonical component of the Whitehead link. We provide a projective model and show that this model is isomorphic to P^2 blown up at 10 points. The Whitehead link can be realized as 1/1 Dehn surgery on one cusp of both the Borromean rings and the 3-chain link. In Chapter 3 we examine the canonical components for the two families of hyperbolic link complements obtained by 1/n Dehn filling on one component of both the Borromean rings and the 3-chain link. These examples extend the work of Macasieb, Petersen and van Luijk who have studied the character varieties associated to the twist knot complements. We conjecture that the canonical components for the links obtained by 1/n Dehn filling on one component of the 3-chain link are all rational surfaces isomorphic to P^2 blown up at 9n + 1 points. A major goal is to understand how the algebro-geometric structure of these varieties reflects the topological structure of the associated manifolds. At the end of Chapter 3 we discuss common features of these examples and explain how our results lend insight into the affect Dehn surgery has on the character variety. We conclude, in Chapter 4, with a description of possible directions for future research. / text

Στην παρούσα διπλωματική εργασία, επιχειρήθηκε η μελέτη της εξέλιξης των γονιδίων C6, C7, C8α, C8β και C9 του συστήματος του Συμπληρώματος που ανήκουν στην οικογένεια MACPF. Για την συλλογή των δεδομένων και την ανάλυσή τους, χρησιμοποιήθηκαν εργαλεία της βιοπληροφορικής. Στο τελικό στάδιο επεξεργασίας, οικοδομήθηκαν φυλογενετικά δένδρα που αφορούσαν είτε τις πλήρεις αλληλουχίες των πρωτεϊνών της υπεροικογένειας MACPF, είτε ξεχωριστά κάθε μία περιοχή (domain) που εμφανίζεται στις πρωτεΐνες της οικογένειας MACPF του Συμπληρώματος. Τα συμπεράσματα από τη ανάλυση των φυλογενετικών δένδρων, πιστοποιούν ότι πιθανόν από ένα προγονικό γονίδιο που ομοιάζει με τα γονίδια C6/C7 και έλκει την καταγωγή του στην εμφάνιση των πρώτων σπονδυλωτών και αποδεδειγμένα στο εξελικτικό στάδιο των χονδροιχθύων, μετά από συνεχείς γονιδιακούς διπλασιασμούς και απώλειες δομών, προέκυψαν τα υπόλοιπα γονίδια της οικογένειας MACPF, C8α, C8β και C9. Η μη εύρεση μέχρι σήμερα γονιδίων της οικογένειας αυτής σε ασπόνδυλα, αλλά και στο γονιδίωμα των αγνάθων, καθώς και η μη ταυτοποίηση μέχρι σήμερα των γονιδίων C7 και C9 στους χονδροιχθύες, δεν επιτρέπει απόλυτα και τελικά συμπεράσματα για την εξελικτική τους πορεία. Αν η πρωταρχική εμφάνιση των γονιδίων C6/C7 συνδυαστεί και με την ανεξάρτητη εξέλιξη μιας άλλης οικογένειας γονιδίων C3/C4/C5, στο ίδιο εξελικτικό στάδιο, με τα οποία και αλληλεπιδρά στο πρωτεϊνικό επίπεδο, κατά τον καταρράκτη ενεργοποίησης της λυτικής οδού του Συμπληρώματος, τότε ενισχύεται η άποψη ότι προηγήθηκε κατά την εξέλιξη η εμφάνιση των γονιδίων C6/C7. / The aim of the present thesis is to study the evolution of the genes of the complement of the C6, C7, C8a, C8b and C9 proteins that belong to the MACPF superfamily. Collection and analysis of the data was conducted by means of bioinformatics’ tools. In the final evaluation stage phylogenetic trees were constructed concerning full protein sequences of the MACPF superfamily members, as well as distinct conserved domains isolated within the MACPF proteins of the complement. Conclusions provided from the analysis of the phylogenetic trees, certify the possibility of the presence of an ancestral gene resembling C6/C7 genes, whose origin is located in the era of primitive bony organisms appearance and more precisely, and recently proven, in the evolutionary stage of cartilaginous fishes. This gene after undergoing consecutive genetic duplications and loss of domains generated the rest of the MACPF family genes, C8a, C8b and C9. The fact that genes of this family are not yet isolated in non- bony organisms, nor in the genome of agnatha species, combined with the lack of identification up to now of C7 and C9 genes in the cartilaginous fishes, does not allow definite and certain conclusions concerning their evolutionary development. Nevertheless if the primary appearance of C6/C7 genes would be combined to the simultaneous evolution of a distinct family of C3/C4/C5 genes, in the same evolutionary period, between which interaction takes place in protein level, within the activation cascade of the lytic pathway of the complement, the theory that C6/C7 genes appearance took place fist in the evolution is fortified.

The general goal of this dissertation is to highlight the role of discourse pragmatics in the explanation of the use of the indicative and subjunctive moods in Spanish sentential complements. This dissertation examines mood selection in Spanish complements in order to illustrate the shortcomings of the traditional semantic/syntactic approach (Terrell & Hooper (1974), Hooper (1975), P. Klein (1974), Fukushima (1978-79), Bell (1980), and Takagaki (1984)) and to provide within the Information Structure framework (Lambrecht 1994; 2001) a detailed analysis of mood selection in Spanish complement clauses. Considering some existing pragmatic approaches to Spanish mood selection (e.g., Lavandera 1983, Guitart 1991, Mejías-Bikandi 1994, 1998), they are found to be inadequate because they are based on decontextualized sentences. This dissertation considers the context where sentences take place and contributes to our understanding of mood selection in Spanish complements as a formal reflection of the pragmatic properties and relations of the discourse referents that are denoted by noun complements, considering pragmatic notions of presupposition and assertion of propositional referents, their activation, and the pragmatic relations of topic/focus of these referents in the utterances. The notion of pragmatic assertion used in this dissertation is based on the notion of speaker intent, and it is equated with the notion of inactive discourse referents, which are in turn linked to the use of indicative mood in complements of assertive matrices. The notion of pragmatic presupposition is equated with the notion of active referents in the discourse, which are in turn linked to the use of subjunctive mood in complements of doubt/negation and comment matrices. However, this thesis argues that not all uses of subjunctive are motivated by the active status of propositional referents. Volitional and possibility uses of subjunctive are analyzed, similarly to assertive matrices, as activating a discourse referent. Yet, contrary to assertive matrices, and following Fauconnier’s (1985) theory of mental spaces, the referent activated belongs to the domain that represents an individual’s view of reality. This account of mood distribution in complement clauses is eventually extended to adjectival and adverbial subordinates and provides an explanation of mood distribution in all subordinate contexts in Spanish. / text

Esta tesis consta de tres capítulos en donde analizo mercados con fricciones. En los dos primeros capítulos, estas fricciones surgen debido a la carencia información completa acerca de los precios por parte de los consumidores. Puntualmente, en el primer capítulo, se estudia como se desarrolla la competición de precio multiproducto en este tipo de ambiente. Se encuentra que la búsqueda por precios bajos por parte de los consumidores conlleva a que las firmas fijen los precios de los productos complementarios de manera correlativamente inversa. De esta manera, las firmas buscan incrementar sus ganancias valiéndose de aquellos consumidores que no investigan lo suficiente. En el siguiente capítulo se analiza cuales son los efectos que genera la búsqueda de mejores precios en la determinación de los mismos cuando las firmas tienen diferentes costos marginales. Se demuestra que firmas con diferentes estructuras de costos no pueden fijar los mismos precios en equilibrio. Debido a esto, mayores costos conllevan a mayores precios promedios. Finalmente, en el tercer capítulo, las fricciones emergen debido a que las firmas no tienen acceso a todos los mercados. Se analiza la competición en cantidades que se desarrolla luego de la etapa de inversión en capacidad productiva. Se demuestra que la capacidad productiva es mayor que la generada en un modelo Cournot estándar debido los incentivos pro-competitivos presentes en los mercados fragmentados. / This thesis consists of three chapters analyzing markets with frictions. In the first two chapters frictions result from consumers not knowing all the prices and searching for them. The first chapter studies multiproduct price competition in this environment. It finds that consumer search induces firms to negatively correlate prices of complements in order to rip-off consumers who do not search enough. The second chapter studies the effects of consumer search on price competition when firms have different marginal costs. It demonstrates that firms with different costs cannot charge common prices in equilibrium. Due to this, the higher are the costs the higher are the average prices charged by firms. In the third chapter frictions emerge because firms do not have access to all the markets. It analyzes quantity competition following a capacity investment stage to show that equilibrium capacity is larger than in a standard Cournot model because of pro-competitive incentives in fragmented markets.

Research has focused on how the effects of different types of feedback vary as a function of the complexity of the linguistic targets and on the learning of inflectional features. However, few studies have investigated the learning of rule-based verb constructions. Grounded in the interactionist approach and usage-based theory, this study investigated the effects of corrective feedback on the acquisition of rule-based verb constructions among English-speaking learners of Chinese. Specifically, this study examined the effects of input-providing feedback and output-prompting feedback on the learning of two verb constructions. Data were drawn from 18 learners of Chinese from second-year Chinese classes in an American university. The participants were divided into two groups and took a pretest, treatment, and two posttests. Learners also filled out a questionnaire about their perception and preference of feedback types. Contrary to previous research, results indicated that both recasts and metalinguistic clues had positive effects on learners' learning of the target constructions. Moreover, learners of different proficiency preferred different types of feedback. The study results provided a categorization of verb constructions into four classes based on the rules that govern their formations and constraints that work on the constructions and identified stages learners moved through when learning verb constructions. The researcher proposed an instructional model of rule-based verb constructions. The model will help instructors recognize the stage the learners' are in and provide insight into how to help learners move to a higher stage by providing instruction, corrective feedback, and practice activities.

<p>This major project in Building Engineering has its focus on inquiring into what methods has been used in the planning process of the residential building</p><p>Ljusbojen 1. This project has been conducted in cooperation with Kadesjös Ingenjörsbyrå AB, an engineering contractor firm which also acts as the property developer. Initially several different general methods for planning residential buildings have been presented. After the methods from the process of building Ljusbojen 1 has been analyzed compared to the general methods presented in part 1. This shows what the determining factors are during the planning process; factors such as cost, quality, time management, know-how and environmental issues. Ljusbojen 1 has been a good building to study as it has a quite rare architecture and is meant to be perceived as a luxurious resident. The building has been divided into several different parts as this is how they are in the presented in offers, also the different parts are more easily analyzed; the parts are: foundation preparation, framework, exterior, rooming complements and installations. The basis of information has been extracted from relevant literature, planning project reports and in-person interviews. As the scope of this project covers a large field work has been concentrated on the most interesting methods. As a general result it can be determined that the building industry has been pressured to develop and try new methods mainly because of new and daring architectures. However, many existing and well known methods are still in use as the companies know that they work and can offer lower price from these.</p>

Το σύστημα του συμπληρώματος περιλαμβάνει περισσότερες από 30 πρωτεΐνες του ορού και της κυτταρικής επιφάνειας και αποτελεί έναν από τους αρχαιότερους φυλογενετικά μηχανισμούς ανοσίας του ξενιστή. Η ενεργοποίησή του, μέσω της κλασικής, της εναλλακτικής ή της λεκτινικής οδού, οδηγεί σε έναν καταρράκτη αντιδράσεων. Οι αντιδράσεις αυτές συγκλίνουν στην τελική λυτική οδό, οδηγώντας στη δημιουργία του συμπλόκου MAC, με αποτέλεσμα την ωσμωτική λύση του κυττάρου στόχου. Στην παρούσα εργασία, σχετικά με τη μελέτη της μοριακής εξέλιξης των γονιδίων της τελικής λυτικής οδού του συμπληρώματος, κλωνοποίηθηκε η αλληλουχία του cDNA του έκτου συστατικού του συμπληρώματος (C6). Στη συνέχεια, προκειμένου να εξαχθούν συμπεράσματα σχετικά με τη συντήρηση όλων των δομικών γονιδίων του συμπλόκου MAC κατά τη διάρκεια της φυλογενετικής εξέλιξης, δημιουργήθηκε φυλογενετικό δέντρο, στο οποίο συγκρίνονται τα παραπάνω γονίδια με τα αντίστοιχα ορθόλογα, σε διάφορους οργανισμούς. Από τη φυλογενετική ανάλυση φαίνεται ότι το C6 της όρνιθας ομαδοποιείται με τα αντίστοιχα ορθόλογα των θηλαστικών και του βατράχου, δημιουργώντας μια καλά καθορισμένη υποομάδα μέσα στην οικογένεια των πρωτεϊνών MACPF. Σύμφωνα με την υπάρχουσα βιβλιογραφία, εκτός από το ρόλο που διαδραματίζει το σύστημα του συμπληρώματος στην ανοσία, έχει δειχθεί η εμπλοκή του, μεταξύ άλλων, σε διαδικασίες αναγέννησης ιστών, ενεργοποίησης και πολλαπλασιασμού κυττάρων και ρύθμισης της απόπτωσης. Προκειμένου λοιπόν να διερευνηθεί περαιτέρω η πιθανότητα εμπλοκής του συστήματος του συμπληρώματος σε διαδικασίες που σχετίζονται με την ανάπτυξη, μελετήθηκε το αναπτυξιακό και ιστικό πρότυπο έκφρασης συστατικών του στην όρνιθα. Η ιστική μελέτη περιελάμβανε τη διερεύνηση και ημι-ποσοτική ανάλυση της έκφρασης, σε επίπεδο mRNA, σε οκτώ ιστούς της ενήλικης όρνιθας: εγκέφαλο, καρδιά, έντερο, νεφρό, ήπαρ, πνεύμονα, σπλήνα και στόμαχο. Η αναπτυξιακή μελέτη, περιελάμβανε τη διερεύνηση και ημι-ποσοτική ανάλυση της έκφρασης, σε επίπεδο mRNA, σε ολικό ομογενοποιημένο έμβρυο 4 και 6 ημερών, σε ήπαρ εμβρύου 12 και 17 ημερών και σε ήπαρ νεογένητου ατόμου 2 και 5 ημερών. Συγκεκριμένα, μελετήθηκε η έκφρασή του γονιδίου C3 σε διάφορα αναπτυξιακά στάδια, όπου και πιστοποιήθηκε η παρουσία του μορίου σε όλα τα στάδια. Επίσης, μελετήθηκε η ιστική και η αναπτυξιακή έκφραση των δομικών γονιδίων του συμπλόκου MAC (C6, C7, C8α, C8β, C8γ) και η αναπτυξιακή των ρυθμιστικών γονιδίων του (CD59, CLU,VTN). Η μελέτη του ιστοειδικού προτύπου έκφρασης για τα δομικά συστατικά του συμπλόκου, έδειξε ότι τα διάφορα συστατικά παρουσιάζουν ποικίλη έκφραση στους υπό μελέτη ιστούς, ενώ κύρια πηγή έκφρασής τους αποτελεί το ήπαρ. Κατά τη μελέτη του αναπτυξιακού προτύπου έκφρασης, δείχθηκε ότι τα δομικά συστατικά του συμπλόκου πρωτοεμφανίζονται τη 12η εμβρυική μέρα ενώ δεν παρατηρείται έκφρασή τους σε νεογέννητο άτομο 2 ημερών. Αντίθετα, η μελέτη του αναπτυξιακού προτύπου έκφρασης των ρυθμιστικών μορίων του συμπλόκου, έδειξε έκφραση όλων των συστατικών σε όλα τα υπό μελέτη στάδια. Το γεγονός αυτό αντανακλά πιθανώς την πλειοτροπική δράση των ρυθμιστικών αυτών πρωτεϊνών, καθώς εμπλέκονται σε πρόσθετες λειτουργίες οι οποίες δε σχετίζονται με τη δράση του συμπληρώματος, όπως είναι η ανάπτυξη και ωρίμανση οργάνων. Ανάλογα συμπεράσματα θα μπορούσαν να εξαχθούν και στην περίπτωση του μορίου C3, λόγω της έκφρασής του σε πρώιμα στάδια ανάπτυξης της όρνιθας, παρά την απουσία έκφρασης των γονιδίων του συμπλόκου MAC στα αντίστοιχα αναπτυξιακά στάδια. Συμπερασματικά, η παραπάνω εργασία αποτελεί την πρώτη αναφορά στη χωρική και χρονική μελέτη της έκφρασης γονιδίων του συμπληρώματος στην όρνιθα. Βάσει των παραπάνω, περαιτέρω διερεύνηση αναφορικά με τη συμμετοχή των συστατικών του συμπληρώματος σε αναπτυξιακές διεργασίες, παρουσιάζει πλέον ιδιαίτερο ενδιαφέρον. / The complement system includes more than 30 serum and membrane proteins and constitutes one of the most phylogenetically ancient immune mechanisms of the host. Its activation, via the classic, alternative or lectin pathway, leads to a cascade of reactions. All these reactions converge to the terminal lytic pathway, leading to the MAC complex creation, which results in osmotic lysis of the targeted cells. In the present work, concerning the study of the molecular evolution of the terminal complement pathway genes, the sixth complement component (C6) of the chicken (Gallus gallus) was cloned. Afterwards, a phylogenetic tree was created, in which the above genes are compared with the respective orthologs, in various organisms. As it seems from the phylogenetic analysis, chicken’s C6 clusters with the respective mammalian and frog orthologs, creating a well defined subgroup in the MACPF gene family. According to literature, besides the complement system’s role in immunity, it has been shown that it is also involved, in tissue regeneration, cell activation, proliferation and regulation of apoptosis. In order to further investigate the hypothesis of complement system’s involvement in developmental procedures, the developmental and tissue expression profile of complement components were studied in the chicken. The tissue study involved the investigation and semi-quantitative expression analysis, at the mRNA level, in eight tissues deriving from one adult chicken: brain, heart, intestine, kidney, liver, lung, spleen and stomach. The developmental study included the investigation and semi-quantitative expression analysis, at the mRNA level, of the whole homogenized 4 and 6 day embryo, in the liver from a 12 and 17 day embryo and the liver from a neonate chicken, 2 and 5 days old. In particular, the expression of C3 gene was studied in various developmental stages where the gene expression was certified in all stages. Moreover, the tissue and developmental expression of MAC complex structural genes (C6, C7, C8α, C8β, C8γ) was studied, as well as the developmental expression of the regulatory genes of MAC complex (CD59, CLU,VTN). The study of the tissue expression profile for the structural MAC components was shown that the various components exhibit a variable expression, with the liver being the major source of expression. In the developmental expression 117 profile study, it was shown that all structural MAC components initially appear in the 12th embryonic day, while no expression of them was detected in the 2 day old neonate chicken. On the contrary, the developmental expression profile study of the regulatory MAC genes was shown that they expressed throughout the developmental stages. This fact possibly reflects the regulatory proteins pleiotropic action, as they are involved in additional functions, which are not related to complement functions, such as development and organs maturation. Analogous conclusions could be drawn in the case of C3 gene, because of its expression in stages of chicken early development, despite the absence of expression of structural MAC genes in the respective developmental stages. Concluding, the above study is the first reference to the spatial and temporal expression of complement genes in the chicken. In accordance with the above mentioned, further investigation concerning the involvement of complement components in developmental procedures, appears to be of great interest.

This major project in Building Engineering has its focus on inquiring into what methods has been used in the planning process of the residential building Ljusbojen 1. This project has been conducted in cooperation with Kadesjös Ingenjörsbyrå AB, an engineering contractor firm which also acts as the property developer. Initially several different general methods for planning residential buildings have been presented. After the methods from the process of building Ljusbojen 1 has been analyzed compared to the general methods presented in part 1. This shows what the determining factors are during the planning process; factors such as cost, quality, time management, know-how and environmental issues. Ljusbojen 1 has been a good building to study as it has a quite rare architecture and is meant to be perceived as a luxurious resident. The building has been divided into several different parts as this is how they are in the presented in offers, also the different parts are more easily analyzed; the parts are: foundation preparation, framework, exterior, rooming complements and installations. The basis of information has been extracted from relevant literature, planning project reports and in-person interviews. As the scope of this project covers a large field work has been concentrated on the most interesting methods. As a general result it can be determined that the building industry has been pressured to develop and try new methods mainly because of new and daring architectures. However, many existing and well known methods are still in use as the companies know that they work and can offer lower price from these.

Στο κεφάλαιο 1 γίνεται μελέτη διατεταγμένων αλγεβρικών δομών. Δίνονται ορισμοί, αποδείξεις και στοιχειώδη αποτελέσματα, απαραίτητα σε όλη την πορεία της εργασίας. Ορίζουμε μερικώς διατεταγμένα σύνολα και μερική διάταξη σε αλγεβρικά συστήματα, βλέπουμε υπό ποίες προϋποθέσεις η μερική διάταξη επεκτείνεται σε ολική και άρα το σύνολο γίνεται ολικώς διατεταγμένο και στη συνέχεια τα διατεταγμένα σύνολα με μία εσωτερική πράξη ορίζουν μερικώς ή ολικώς διατεταγμένες ομάδες. Στο κεφάλαιο 2 παρουσιάζουμε συμπληρώσεις διατεταγμένων συνόλων και συγκεκριμένα, τα συμπληρώματα Dedekind, Kurepa και Krasner καθώς και ορισμένες ιδιότητες αυτών. Ο Dedekind (1831-1916) όρισε τις τομές Dedekind με τη βοήθεια των οποίων επέκτεινε τη διάταξη των φυσικών στο σύνολο των πραγματικών και θεμελίωσε με αυτόν τον τρόπο το σύνολο αυτό ως ένα διατεταγμένο σώμα. Η κατασκευή της δομής των πραγματικών εφοδιασμένη με τις πράξεις της πρόσθεσης και του πολλαπλασιασμού και τη δίαταξη, καθώς και η κατασκευή της δομής του επιπέδου με τις ίδιες πραξεις και διάταξη κατά Dedekind παρουσιάζεται εκτενέστερα στο κεφάλαιο 3. Η γενίκευση της έννοιας του συμπληρώματος Kurepa και η εισαγωγή του συμπληρώματος Krasner, οφείλονται στον καθηγητή Λ. Ντόκα (1963). Η μέθοδος του Dedekind της συμπλήρωσης με τομές δεν είναι η μόνη μέθοδος κατασκευής των πραγματικών αριθμών. Η μέθοδος του G. Cantor (1845-1918) της συμπλήρωσης με ακολουθίες, είναι η δεύτερη εξίσου σημαντική μέθοδος, την οποία θα παρουσιάσουμε στο κεφάλαιο 4. Η μελέτη μας ολοκληρώνεται στο κεφάλαιο 5, όπου παρουσιάζεται ένα ενδιαφέρον αποτέλεσμα για τις μερικώς διατεταγμένες ομάδες και τις συνθήκες κάτω από τις οποίες αυτές επεκτείνονται σε ολικώς διατεταγμένες ομάδες, στηριζόμενοι στην εργασία “embedding groups into linear or lattice structures” των Κοντολάτου-Σταμπάκη (1987), όπου πραγματοποιούν επέκταση μίας μερικώς διατεταγμένης ομάδας, χρησιμοποιώντας τα αποτελέσματα του Fuchs για ύπαρξη επέκτασης ενός μερικώς διατεταγμένου συνόλου σε ολικώς διατεταγμένο. / In chapter 1 ordered algebraic structures are considered and we present certain definitions, proofs and elementary results which are necessary in the whole project. Partially ordered sets and partial order in algebraic systems is defined. Then we analyze under which conditions partial order can be extended to full order. This leads to fully ordered sets and those sets, along with an internal operation, define partially or fully ordered groups. In chapter 2 we present specific ordered set complements and in particular those of Dedekind, Kurepa and Krasner and furthermore we mention some of their properties. Dedekind sections where introduced by Dedekind (1831-1916), who used them in order to extend the order of natural numbers to the set of real numbers, making this set an ordered field. The construction of the real numbers structure along with the internal operations of addition and multiplication and order and the construction of the plane structure with the same operations and order, using Dedekind theory, is analytically presented in chapter 3. Due to L. Docas (1963), Kurepa complement was generalized and Krasner complement was introduced. Dedekind’s sections is not the only way to construct the set of real numbers. Another important method is that of G. Cantor (1845-1918), who used sequences for completion. We present this method in chapter 4. Finally, in chapter 5, we consider a paper published by A. Kontolatou and J. Stabakis (1987) entitled “Embedding groups into linear or lattice structures”. Fuchs’s results on the extend existence of a partially ordered set to fully ordered set is used. Based on the Kontolatou-Stabakis paper, we present an interesting result for partially ordered groups and certain conditions of how to extend those groups to fully ordered ones.