Norms of basic elementary operators on algebras of regular operators

We show that if E is an atomic Banach lattice with an ordercontinuous norm, A, B ∈ L^r(E) and M_A,_B is the operator on L^r(E) defined by M_A,_B(T) = AT B then ||M_A,_B||r = ||A||_r||B||_r but that there is no real α > 0 such that ||MA,B || ≥ α ||A||r||B ||r.

Two dimensional unital Riesz algebras, their representations and norms

We describe all two dimensional unital Riesz algebras and study representations of them in Riesz algebras of regular operators. Although our results are not complete, we do demonstrate that very varied behaviour can occur even though all these algebras can be given a Banach lattice algebra norm.

Reflections of universal algebras into semilattices, their Galois theories, and related factorization systems

Estabelecemos uma condição suficiente para a preservação dos produtos finitos, pelo reflector de uma variedade de álgebras universais numa subvariedade, que é, também, condição necessária se a subvariedade for idempotente. Esta condição é estabelecida, seguidamente, num contexto mais geral e caracteriza reflexões para as quais a propriedade de ser semi-exacta à esquerda e a propriedade, mais forte, de ter unidades estáveis, coincidem. Prova-se que reflexões simples e semi-exactas à esquerda coincidem, no contexto das variedades de álgebras universais e caracterizam-se as classes do sistema de factorização derivado da reflexão. Estabelecem-se resultados que ajudam a caracterizar morfismos de cobertura e verticais-estáveis em álgebras universais e no contexto mais geral já referido. Caracterizam-se as classes de morfismos separáveis, puramente inseparáveis e normais. O estudo dos morfismos de descida de Galois conduz a condições suficientes para que o seu par kernel seja preservado pelo reflector.

Generating finite integral relation algebras

Relation algebras and categories of relations in particular have proven to be extremely useful as a fundamental tool in mathematics and computer science. Since relation algebras are Boolean algebras with some well-behaved operations, every such algebra provides an atom structure, i.e., a relational structure on its set of atoms. In the case of complete and atomic structure (e.g. finite algebras), the original algebra can be recovered from its atom structure by using the complex algebra construction. This gives a representation of relation algebras as the complex algebra of a certain relational structure. This property is of particular interest because storing the atom structure requires less space than the entire algebra. In this thesis I want to introduce and implement three structures representing atom structures of integral heterogeneous relation algebras, i.e., categorical versions of relation algebras. The first structure will simply embed a homogeneous atom structure of a relation algebra into the heterogeneous context. The second structure is obtained by splitting all symmetric idempotent relations. This new algebra is in almost all cases an heterogeneous structure having more objects than the original one. Finally, I will define two different union operations to combine two algebras into a single one.

Generating Relation Algebras for Qualitative Spatial Reasoning

Basic relationships between certain regions of space are formulated in natural language in everyday situations. For example, a customer specifies the outline of his future home to the architect by indicating which rooms should be close to each other. Qualitative spatial reasoning as an area of artificial intelligence tries to develop a theory of space based on similar notions. In formal ontology and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts. We shall introduce abstract relation algebras and present their structural properties as well as their connection to algebras of binary relations. This will be followed by details of the expressiveness of algebras of relations for region based models. Mereotopology has been the main basis for most region based theories of space. Since its earliest inception many theories have been proposed for mereotopology in artificial intelligence among which Region Connection Calculus is most prominent. The expressiveness of the region connection calculus in relational logic is far greater than its original eight base relations might suggest. In the thesis we formulate ways to automatically generate representable relation algebras using spatial data based on region connection calculus. The generation of new algebras is a two pronged approach involving splitting of existing relations to form new algebras and refinement of such newly generated algebras. We present an implementation of a system for automating aforementioned steps and provide an effective and convenient interface to define new spatial relations and generate representable relational algebras.

International Conference on Semigroups, Algebras and Applications (ICSA 2015)

Mathematicians who make significant contributions towards development of mathematical science are not getting the recognition they deserve, according to Cusat Vice Chancellor Dr. J. Letha. She was delivering the inaugural address at the International Conference on Semigroups, Algebras and Applications (ICSA 2015) organized by Dept. of Mathematics, Cochin university of Science and Technology on Thursday. Mathematics plays an important role in the development of basic science. The academic community should not delay in accepting and appreciating this, Dr. Letha added. Dr. Godfrey Louis, Dean, Faculty of Science presided over the inaugural function. Prof. P. G. Romeo, Head, Dept. of Mathematics, Prof. John C. Meakin, University of Nebraska-Lincoln, USA, Prof. A. N. Balchand, Syndicate Member, Prof. K. A. Zakkariya, Syndicate Member, Prof. A. R. Rajan, Emeritus Professor, University of Kerala and Prof. A. Vijayakumar, Dept. of Mathematics, Cusat addressed the gathering. Around 50 research papers will be presented at the Conference.Prof. K. S. S. Nambooripad, the internationally famous mathematician with enormous contributions in the field of semigroup theory, who has attained eighty years of age will be felicitated on 18th at 5.00 pm during a function presided over by Dr. K. Poulose Jacob, Pro-Vice Chancellor. Dr. Suresh Das, Executive President, KSCSTE, Dr. A. M. Mathai, Director, CMSS and President, Indian Mathematical Society, Dr. P. G. Romeo, Head, Dept. of Mathematics and Dr. B. Lakshmi, Dept. of Mathematics will speak on the occasion.

Computing Generators of Free Modules over Orders in Group Algebras

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition E[G] \cong \oplus_xM_x is explicitly computable and each M_x is in fact a matrix ring over a field, this leads to an algorithm that either gives elements \alpha_1,...,\alpha_d \in X such that X = A\alpha_1 \oplus ... \oplusA\alpha_d or determines that no such elements exist. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [K : E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A(E[G];O_L) \subseteq E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = \IQ.

Elimination in Operator Algebras

A large class of special functions are solutions of systems of linear difference and differential equations with polynomial coefficients. For a given function, these equations considered as operator polynomials generate a left ideal in a noncommutative algebra called Ore algebra. This ideal with finitely many conditions characterizes the function uniquely so that Gröbner basis techniques can be applied. Many problems related to special functions which can be described by such ideals can be solved by performing elimination of appropriate noncommutative variables in these ideals. In this work, we mainly achieve the following: 1. We give an overview of the theoretical algebraic background as well as the algorithmic aspects of different methods using noncommutative Gröbner elimination techniques in Ore algebras in order to solve problems related to special functions. 2. We describe in detail algorithms which are based on Gröbner elimination techniques and perform the creative telescoping method for sums and integrals of special functions. 3. We investigate and compare these algorithms by illustrative examples which are performed by the computer algebra system Maple. This investigation has the objective to test how far noncommutative Gröbner elimination techniques may be efficiently applied to perform creative telescoping.