Showing papers on "Free algebra published in 2005"
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
TL;DR: A computer algebra library for basic operations in an arbitrary Ore ring can be used as a basis for various algorithms in Ore rings, in particular, in differential, shift, and q-shift rings.
Abstract: We present some algorithms related to rings of Ore polynomials (or, briefly, Ore rings) and describe a computer algebra library for basic operations in an arbitrary Ore ring. The library can be used as a basis for various algorithms in Ore rings, in particular, in differential, shift, and q-shift rings.
53 citations
TL;DR: In this paper, the authors introduced M-Armendariz rings, which are generalizations of Armendariz ring, and investigated their properties for a monoid M and showed that every reduced ring is M-armendariz for any unique product monoid, where N is a unique monoid.
Abstract: For a monoid M, we introduce M-Armendariz rings, which are generalizations of Armendariz rings; and we investigate their properties. Every reduced ring is M-Armendariz for any unique product monoid M. We show that if R is a reduced and M-Armendariz ring, then R is M × N-Armendariz, where N is a unique product monoid. It is also shown that a finitely generated Abelian group G is torsion free if and only if there exists a ring R such that R is G-Armendariz. Moreover, we study the relationship between the Baerness and the PP-property of a ring R and those of the monoid ring R[M] in case R is M-Armendariz.
48 citations
TL;DR: In this paper, the authors studied the combinatorial, algebraic and geometric properties of the free product operation on matroids and showed that free product is associative and respects matroid duality.
27 citations
TL;DR: This paper shows how Millen's approach can be extended to public key cryptography, giving conditions under which security for a free algebra version of the protocol implies security of the version using rewrite rules.
26 citations
TL;DR: In this paper, the authors extend the Shirshov-Witt theorem to free algebras with a braided coproduct and show that if a subalgebra is a right categorical right coideal, then it is free.
18 citations
TL;DR: In this paper, a non-negative stellensatz and a null-stellatz for hereditary polynomials in a free *-algebra were proved. But the nullness of the null stellatz was not satisfied.
Abstract: We prove a non-negative-stellensatz and a null-stellensatz for a class of polynomials called hereditary polynomials in a free *-algebra.
15 citations
01 Jan 2005
TL;DR: A survey of results on the structure of free algebraic systems obtained during the past 50 years is given in this article, where the focus is on ways free algebras can be decomposed into simpler components and how the number of components and the way the components interact with each other.
Abstract: This article is a survey of selected results on the structure of free algebraic systems obtained during the past 50 years. The focus is on ways free algebras can be decomposed into simpler components and how the number of components and the way the components interact with each other can be readily determined. A common thread running through the exposition is a concrete method of representing a free algebra as an array of elements.
6 citations
TL;DR: In this article, free and finitely presented n-distributive modular ortholattices are discussed and it is shown that the free algebra on 3 generators has an unsolvable word problem for n ≥ 14.
Abstract: Free and finitely presented n-distributive modular ortholattices are discussed. In particular, we show that the free algebra on 3 generators has an unsolvable word problem for n ≥ 14.
6 citations
4 citations
TL;DR: In this paper, the relations for the specialisation at of the generic composition algebra are given, as well as those for Reineke's composition monoid, which is a proper factor of the specialization of the composition algebra.
Abstract: The Kronecker quiver is considered, and the relations for the specialisation at of the generic composition algebra are given, as well as those for Reineke's composition monoid. As a corollary, it is deduced that the composition monoid is a proper factor of the specialisation of the composition algebra. A normal form is also obtained for the varieties occurring in the composition monoid in terms of Schur roots.
TL;DR: It is shown that those Σ -labeled n -posets that can be generated from the singletons by the binary and ω -ary product operations form the free algebra on Σ in a variety axiomatizable by an infinite collection of simple equations.
Abstract: A Σ -labeled n -poset is an (at most) countable set, labeled in the set Σ , equipped with n partial orders. The collection of all Σ -labeled n -posets is naturally equipped with n binary product operations and n ω -ary product operations. Moreover, the ω -ary product operations give rise to n ω -power operations. We show that those Σ -labeled n -posets that can be generated from the singletons by the binary and ω -ary product operations form the free algebra on Σ in a variety axiomatizable by an infinite collection of simple equations. When n = 1 , this variety coincides with the class of ω -semigroups of Perrin and Pin. Moreover, we show that those Σ -labeled n -posets that can be generated from the singletons by the binary product operations and the ω -power operations form the free algebra on Σ in a related variety that generalizes Wilke's algebras. We also give graph-theoretic characterizations of those n -posets contained in the above free algebras. Our results serve as a preliminary study to a development of a theory of higher dimensional automata and languages on infinitary associative structures.
TL;DR: The construction of free R-modules over a Cartesian closed topological category X is detailed, and it is shown that the insertion of generators is an embedding.
Abstract: The construction of free R-modules over a Cartesian closed topological category X is detailed (where R is a ring object in X), and it is shown that the insertion of generators is an embedding. This result extends the well-known construction of free groups, and more generally of free algebras over a Cartesian closed topological category.
TL;DR: In this article, an analogue of the Magnus theorem for associative algebras without unity over arbitrary fields has been proved, namely, if an algebra is given by n + k generators and k relations and has an n-element system of generators, then this algebra is a free algebra of rank n.
Abstract: We prove an analogue of the Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by n + k generators and k relations and has an n-element system of generators, then this algebra is a free algebra of rank n.
TL;DR: In this article, the fundamental algebraic approach to quantum logics is presented, and two quantum consequence operations defined in these models are given, i.e., structural consequence operation (SOP) and logical matrices.
Abstract: In this paper we present the fundamentals of the so-called algebraic approach to propositional quantum logics. We define the set of formulas describing quantum reality as a free algebra freely generated by the set of quantum propositional variables. We define the general notion of logic as a structural consequence operation. Next we introduce the concept of logical matrices understood as a models of quantum logics. We give the definitions of two quantum consequence operations defined in these models.
Dissertation•
01 Jan 2005
TL;DR: In this article, the problem of finding the structure of all maximal sum-free subsets of a finite abelian group was studied, provided all the divisors of order of G are congruent to 1 modulo 3.
Abstract: In this thesis we discuss some problems relating the properties of a set A and those of A + A, when A is a subset of an abelian group. Given a finite abelian group G and A ⊂ G, we say A is sum-free if the sets 2A and A are disjoint. In chapter 2 we discuss the problem of finding the structure of all large sum-free subsets of G. We obtain the complete structure of all largest sum-free subsets of G, provided all the divisors of order of G are congruent to 1 modulo 3. In the same chapter we also give partial results regarding structure of all large maximal sum-free subsets of G. We say a sum-free set A is maximal if it is not a proper subset of any sum-free set. If there is a divisor of order of G which is not congruent to 1 modulo 3 then structure of all largest sum-free subsets of G was known before. Our results are based on a recent result of Ben Green and Imre Ruzsa [GR05]. Let SF (G) denote the set of all sum-free subsets of G and the symbol σ(G) denotes the number n(log2 |SF (G)|). In chapter 3 we improve the error term in asymptotic formula of σ(G) obtained by Ben Green and Imre Ruzsa [GR05]. The method used is a slight refinement of methods in [GR05]. In chapter 4 we discuss the following problem. Let A be an infinite subset of natural numbers. Suppose for all large natural numbers n the number of ways n can be written as a sum of two element (not necessarily distinct) of A is not equal to 1, then how “thin” can A be? The main result of this chapter is then an improvement of a result of Nicolas, Ruzsa, Sárközy [NRS98] and methods we use are refinement of methods developed in [NRS98]. A new ingredient used is an additive lemma proven by means of graph theory.