Collection Of Algebras Research Articles

On a Hopf operad containing the Poisson operad

A new Hopf operad Ram is introduced, which contains both the well-known Poisson operad and the Bessel operad introduced previously by the author. Besides, a structure of cooperad R is introduced on a collection of algebras given by generators and relations which have some similarity with the Arnold relations for the cohomology of the type A hyperplane arrangement. A map from the operad Ram to the dual operad of R is defined which we conjecture to be a isomorphism.

Functional equations with distortion and certain q-special functions

We show how some well-known q-special functions arise as solutions to analogue functional and difference equations using a doubly-indexed collection of algebras of formal power series type and the notion of distortion.

Algebraic Kripke Sheaf Semantics for Non-Classical Predicate Logics

In so-called Kripke-type models, each sentence is assigned either to true or to false at each possible world. In this setting, every possible world has the two-valued Boolean algebra as the set of truth values. Instead, we take a collection of algebras each of which is attached to a world as the set of truth values at the world, and obtain an extended semantics based on the traditional Kripke-type semantics, which we call here the algebraic Kripke semantics. We introduce algebraic Kripke sheaf semantics for super-intuitionistic and modal predicate logics, and discuss some basic properties. We can state the Godel-McKinsey-Tarski translation theorem within this semantics. Further, we show new results on super-intuitionistic predicate logics. We prove that there exists a continuum of super-intuitionistic predicate logics each of which has both of the disjunction and existence properties and moreover the same propositional fragment as the intuitionistic logic.

Nilpotency of class 3 which is stable in a certain collection of algebras

Nilpotency of class 3 which is stable in a certain collection of algebras

The Algebraic Framework for Object-Oriented Systems

Category Theory is introduced as the mathematical model for object-oriented systems which are distributed, heterogeneous, real-time, embedded, and open-ended. Each object can be represented as an algebra. A collection of algebras with morphisms form a category if they satisfy some conditions. After a brief introduction of categorical concepts which are needed to formulate the framework for object-oriented systems, they are explicated in terms of objects. Then some system design methodologies such as SADT, JSD, MASCOT 3, OOD, HOOD, MOON, ADM 3, and Petri nets are examined in the categorical framework and classified into four groups: functional, process-based, object-oriented, and net-based. Combining theoretical and practical results, the interactive system design tool OBJ-NET is briefly introduced.

The metric of a space and quadratic algebras

A collection of algebras with product that preserves the metric of the underlying vector space is investigated. It is shown that this collection includes the generalized quaternions, the composition algebras, and the algebra of color. All the algebras in this collection that are either commutative or alternative are determined; the ramifications of nondegenerate metrics are detailed. If the algebra is alternative and not of dimension 1, 2, 4, or 8 over the base field, the metric must be degenerate.