21.
Initial Algebra and Final Coalgebra Semantics for Concurrency
- J. J. M. M. Rutten; D. Turi; Jan Rutten; Daniele Turi
The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial semantics from final semantics, using the initiality and finality to ensure their equality. Moreover, many facts about congruences (on algebras) and (generalized) bisimulations (on coalgebras) are shown to be dual as well. AMS Subject Classification (1991): 68Q10, 68Q55 CR Subject Classification (1991): D.3.1, F.1.2, F.3.2 Keywords & Phrases: Algebra, coalgebra,...
22.
Graph algebras
- Raeburn, Iain
Graph algebras are a family of operator algebras which are associated to directed graphs. These algebras have an attractive structure theory in which algebraic properties of the algebra are related to the behaviour of paths in the underlying graph. In the past few years there has been a great deal of activity in this area, and graph algebras have cropped up in a surprising variety of situations, including non-abelian duality, non-commutative geometry, and the classification of simple C-algebras.
23.
Graph algebras
- Raeburn, Iain
Graph algebras are a family of operator algebras which are associated to directed graphs. These algebras have an attractive structure theory in which algebraic properties of the algebra are related to the behaviour of paths in the underlying graph. In the past few years there has been a great deal of activity in this area, and graph algebras have cropped up in a surprising variety of situations, including non-abelian duality, non-commutative geometry, and the classification of simple C-algebras.
24.
B-Star-Algebra -- from MathWorld
- Weisstein, Eric W.
See: C-Star-Algebra
25.
Algebras associated with blaschke products of type g
- Guillory, Carroll; Li, Kin-Yin
Let ? (resp. ?fi) be the set of all interpolating Blaschke products of (resp. finite) type G. Let E (resp. Efi) be the Douglas algebra generated by H? and the complex conjugates of elements of ? (resp. ?fi). Our main results are that the set of all invertible inner functions in E (resp. Efi) is the set of all finite products of elements of ? (resp. ?fi), which is also the closure of ? (resp. ?fi) among the Blaschke products. Consequently, finite convex combinations of finite products of elements of ? (resp. ?fi) are dense in the closed unit ball...
26.
Abstract Algebra -- from MathWorld
- Weisstein, Eric W.
Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these structures are groups, rings, and fields. Important branches of abstract algebra are commutative algebra, representation theory, and homological algebra. Linear algebra, elementary number theory, and discrete mathematics are sometimes considered branches of abstract algebra. Ash (1998) includes the following areas in his...
27.
Peirce Algebras
- Renate A. Schmidt; Im Stadtwald; Chris Brink; Chris Brink; Katarina Britz; Katarina Britz; Renate Schmidt
We present a two-sorted algebra, called a Peirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a relationforming operator on sets (the Peirce product of Boolean modules) and a set-forming operator on relations (a cylindrification operation). Two applications of Peirce algebras are given. The first points out that Peirce algebras provide a natural algebraic framework for modelling certain programming constructs. The second shows that the so-called terminological...
28.
Measure Algebra -- from MathWorld
- Weisstein, Eric W.
A Boolean sigma-algebra which possesses a measure.
29.
Nonassociative Algebra -- from MathWorld
- Weisstein, Eric W.
An algebra which does not satisfy a(bc)=(ab)c is called a nonassociative algebra. See also: Algebra, Cayley Number, Complex Number, Division Algebra, Quaternion, Real Number
30.
Preboolean MV-algebras as bipartite MV-algebras.
- Cella, Carmela; Lettieri, Ada
In this note we characterize bipartite MV-algebras by introducing the notion of preboolean MV-algebras.
31.
MV ∗ –Algebras
- Renato Lewin; Facultad De Matemáticas; Marta Sagastume; Pedro Massey
In this paper we make an algebraic study of the variety of MV ∗ –algebras introduced by C. C.Chang as an algebraic counterpart for a logic with positive and negative truth values. We build the algebraic theory of MV ∗ –algebras within its own limits using a concept of ideal and of prime ideal that are very naturally related to the corresponding concepts in ℓ–groups. The main results are a subdirect representation theorem, a completeness theorem, a study of simple and semisimple algebras, and a characterization of the ideal of infinitesimals as an ℓ–group. In the last section we develop...
32.
Wajsberg algebras.
- Font, Josep M.; Rodríguez Chía, Antonio Manuel; Torrens Torrell, Antonio
We present the basic theory of the most natural algebraic counterpart of the À0-valued Lukasiewicz calculus, strictly logically formulated. After showing its lattice structure and its relation to C. C. Chang's MV-algebras we study the implicative filters and prove its equivalence to congruence relations. We present some properties of the variety of all Wajsberg algebras, among which there is a representation theorem. Finally we give some characterizations of linear, simple and semisimple algebras.
33.
Algebras difusas
- Montero de Juan, Francisco Javier
En este trabajo se propone una estructura de álgebra difusa (borrosa) basada en la distinción entre difusidad extensiva y comprehensiva, desarrollando y conectando los trabajos de Nahmias sobre variables difusas, de Klement sobre medibilidad difusa y de Nowakowska sobre estructuras de conceptos.
34.
Exceptional Jordan Algebra -- from MathWorld
- Weisstein, Eric W.
A Jordan algebra which is not isomorphic to a subalgebra. See also: Jordan Algebra, Special Jordan Algebra
35.
Special Jordan Algebra -- from MathWorld
- Weisstein, Eric W.
A Jordan algebra which is isomorphic to a subalgebra. See also: Exceptional Jordan Algebra, Jordan Algebra
36.
DI Algebras
this article is to introduce and study a new notion of algebra which
gives, by a similar procedure, a Leibniz algebra. The idea is to start with two
distinct operations for the product xy and the product yx, so that the bracket
is not necessarily skew-symmetric any more. Explicitly, we define a dialgebra
37.
Homological Algebra -- from MathWorld
- Weisstein, Eric W.
An abstract algebra concerned with results valid for many different kinds of spaces. Modules are the basic tools used in homological algebra. See also: Commutative Diagram, Diagram Chasing, Diagram Lemma, Module
38.
Algebra -- from MathWorld
- Weisstein, Eric W.
The word "algebra" is a distortion of the Arabic title of a treatise by al-Khwarizmi about algebraic methods. In modern usage, algebra has several meanings. One use of the word "algebra" is the abstract study of number systems and operations within them, including such advanced topics as groups, Rings, invariant theory, and cohomology. This is the meaning mathematicians associate with the word "algebra." When there is the possibility of confusion, this field of...
39.
Distance Learning Classrooms, Casio 9850 Curriculum Units and Derive Software Used to Understand College Algebra
- Nancy J. Priselac; Key Words Multimedia; Casio Color
This paper outlines a plan for a newly developed college algebra course at Garrett Community College in western Maryland currently offered to a critical mass of under-prepared and minority students in a distance learning interactive video classroom. The Bell Atlantic Fiber Optic System links Garrett Community College to identical interactive video classrooms in county high schools and state colleges and universities. Teachers and administrators from sister community colleges and universities have worked together to assist in the change occurring within the college algebra classroom. We have now moved from a traditional course to one offering a technologically supported interactive video...
40.
Unroll-And-Jam Guided by A Linear-Algebra-Based Data-Reuse Model
- A Linear-algebra-based,Data-reuse Model,Yiping Guan,Dr. Austin Melton
Because of the existence of a memory bottleneck in modern microprocessors, idle computational
cycles in pipelined multiple functional units slow down the program performance. One
solution to this problem is applying loop unroll-and-jam to improve the ratio of memory operations
to floating-point operations for loops according to the target machine optimal ratio. In doing
so, both enough computation and memory accesses will keep the processor utilization high.
Previous work on loop unroll-and-jam implemented data reuse analysis on a data dependence
graph [7]. This reuse analysis is used to compute the memory requirement in a loop. Unfortunately
the results from this method are neither precise nor efficient. The...
