by So Kwok Yu, Andy. / Bibliography: leaves 64-65 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1985

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

The purpose of this project was to study the classical result in this direction discovered by O. Ore in 1938, as well as related theorems and corollaries. Ore's Theorem and its corollaries provide us with several results relating distributive lattices with cyclic groups.

<p>Definisane su neke klase polugrupa koje su uop&scaron;tenja tzv. antiinverznih polugrupa. Za neke op&scaron;te klase polugrupa nađene su baze u smislu Ljapina. Dati su identiteti algebri koje se potapaju u polumreže.</p> / <p>Some new classes of semigroups are defined which are generalizations of so called antiinverse semigroups. The Lyapin bases of some well known classes of semigroups are determined. The identities of the variety of subalgebras of semilattices are determined.</p>

Les mathématiques tropicales sont une branche des mathématiques correspondant à l'étude d'une algèbre modifiée grâce à la redéfinition de l'addition et de la multiplication. Les mathématiques tropicales sont généralement définies grâce au minimum et à l'addition (algèbre min-plus) mais le terme est parfois utilisé pour désigner l'algèbre max-plus, définie grâce au maximum et à l'addition. Briec et Horvath ont introduit une notion de convexité très proche qui apparait comme un cas limite d’opérateurs utilisés en théorie de l’optimisation par Avriel (1972) et de Ben-Tal (1977). En suivant cette ligne d’investigation, nous allons proposer, dans le domaine de l’économie de la production et de l’optimisation de portefeuille, une certaine classe de modèles économiques à élaborés à partir de ces notions. Pour ce faire, nous introduisons une nouvelle classe de technologie de production permettant de prendre en compte les structures d’homothe´tie-translation dans la mesure de productivité au travers du concept de la Convexité Max-Plus. Ensuite, nous allons établir une relation topologique entre plusieurs classes de modèles convexes généralisés connus. Nous analysons pour cela la limite de Painlevé-Kuratowski des modèles CES-CET et des technologies non paramétriques satisfaisant une hypothèse de rendements d’échelle alpha. On montre que leurs limites topologiques convergent vers les modèles de production B-convexe et Cobb-Douglas. Enfin, nous allons montrer que l'amélioration de l'efficacité technique d’une coalition d’entreprises s'avère compatible avec les technologies de semi-treillis dans un jeu coopératif. Nous introduisons ensuite, le concept d’écart absolu moyen dans la sélection du portefeuille en utilisant le « Shortage Function » qui prend en compte simultanément la réduction des inputs et l’augmentation des outputs comme dans la théorie de la production. Enfin, nous allons étendre le concept de B-convexité et de l’inverse B-convexité en se concentrant sur le calcul des mesures d’efficacité technique dans le graphe. / Tropical algebra is the tropical analogue of linear algebra by redefining the usual operation addition by the maximization operation and the usual addition operation as multiplication. Briec and Horvath introduced a concept of convexity very close to this concept quoted above which appears as one of the limits of use of the theory of optimization by Avriel (1972) and Ben-Tal (1977). Following this line of investigation, we give an overview of contributions involving a semilattice structure of production technologies and an optimization portfolio. To do that, firstly, we propose a framework allowing to consider both semilattice structure and translation homothetic properties in productivity measurement. We introduce the concept of Max-Plus convexity which combine both an upper semilattice structure and an additivity assumption. We establish a topological relation between several classes of known generalized convex models using some basic algebraic convex structures. We analyze the Painlevé-Kuratowski limit of the CES-CET and Alpha-returns to scale models. It is shown that their topological limits yield the B-convex and Cobb-Douglas production models. Moreover, we show that the improvement of technical efficiency is compatible with semilattice technologies in a cooperative game. Then, we introduce a criterion to measure portfolio efficiency based upon the minimization of the maximum absolute deviation and minimum absolute deviation from the expected return using the Shortage function. Finally, we derive simple closed-form expressions to calculate the hyperbolic measure in the case of inverse and B-Convexity that evaluates technical efficiency in the full input-output space.