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Free algebra

About: Free algebra is a research topic. Over the lifetime, 803 publications have been published within this topic receiving 10909 citations.


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Book ChapterDOI
04 Apr 2005
TL;DR: A novel algebraic description for models of the @p-calculus is obtained, and an existing construction is validated as the universal such model, and it is generalised to prove that all free-algebra models are fully abstract.
Abstract: The finite π-calculus has an explicit set-theoretic functor-category model that is known to be fully abstract for strong late bisimulation congruence. We characterize this as the initial free algebra for an appropriate set of operations and equations in the enriched Lawvere theories of Plotkin and Power. Thus we obtain a novel algebraic description for models of the π-calculus, and validate an existing construction as the universal such model. The algebraic operations are intuitive, covering name creation, communication of names over channels, and nondeterminism; the equations then combine these features in a modular fashion. We work in an enriched setting, over a “possible worlds” category of sets indexed by available names. This expands significantly on the classical notion of algebraic theories, and in particular allows us to use nonstandard arities that vary as processes evolve. Based on our algebraic theory we describe a category of models for the π-calculus, and show that they all preserve bisimulation congruence. We develop a direct construction of free models in this category; and generalise previous results to prove that all free-algebra models are fully abstract.

623 citations

Book ChapterDOI
D. A. Turner1
01 Jan 1985
TL;DR: The use of abstract data types in the LCF system has been studied in this article, where Gordon et al. show that many of the data types which in other languages would have to be expressed as Abstract Data Types can be expressed in Miranda as algebraic data types with associated laws.
Abstract: data types Many of the data types which in other languages would have to be expressed as abstract data types can be represented in Miranda as algebraic data types with associated laws. Nevertheless there is still a need for abstract data types, as may be seen from the following example (which is based on a use of abstract data types in the LCF system [Gordon et al 79]).

623 citations

Journal ArticleDOI
TL;DR: The notion of quasi-free algebras was introduced in this article for non-commutative versions of manifolds, where the corresponding non-ingular affine varieties of a manifold are modeled as quasi-freeness.
Abstract: This paper is concerned with a notion of nonsingularity for noncommutative algebras, which arises naturally in connection with cyclic homology. Let us consider associative unital algebras over the complex numbers. We call an algebra A quasi-free, when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism A -+ R/I, where I is a nilpotent ideal in R, can be lifted to a homomorphism A -+ R. If we restrict to the category of finitely generated commutative algebras, then this lifting property characterizes smooth algebras, the ones corresponding to nonsingular affine varieties. In this way quasi-free algebras appear as noncommutative analogues of smooth algebras. Stretching the analogy, we can even regard quasi-free algebras as analogues of manifolds. One of the aims of this paper is to develop the analogy further by showing that quasi-free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds. To give an example, let us consider the analogue of an embedding: an extension A = R/I, where A and R are quasi-free algebras playing the role of the submanifold and ambient manifold respectively. In the manifold situation, I/I2 is the module of linear functions on the nor2 mal bundle, and the symmetric algebra SA(III ) is the algebra of polynomial functions. Now in passing from commutative to noncommutative algebras, the symmetric algebra of a module is replaced by the tensor algebra of a bimodule.

416 citations

Journal ArticleDOI
TL;DR: In this article, the free field representation for Wess-Zumino-Witten model with arbitrary Kac-Moody algebra and arbitrary central charge is discussed, and the special role of βγ systems is emphasized.
Abstract: The free field representation or "bosonization" rule1 for Wess-Zumino-Witten model (WZWM) with arbitrary Kac-Moody algebra and arbitrary central charge is discussed. Energy-momentum tensor, arising from Sugawara construction, is quadratic in the fields. In this way, all known formulae for conformal blocks and correlators may be easily reproduced as certain linear combinations of correlators of these free fields. Generalization to conformal blocks on arbitrary Riemann surfaces is straightforward. However, projection rules in the spirit of Ref. 2 are not specified. The special role of βγ systems is emphasized. From the mathematical point of view, the construction involved represents generators of Kac-Moody (KM) algebra in terms of generators of a Heisenberg one. If WZW Lagrangian is considered as d−1 of Kirillov form on an orbit of KM algebra,3 then the free fields of interest (i.e. generators of the Heisenberg algebra) diagonalize Kirillov form and the action. Reduction of KM algebra within the same construction should naturally lead to arbitrary coset models.

237 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20232
20226
202111
202018
201918
201822