Class Of Algebraic Structures Research Articles

Computing presentations for subgroups of polycyclic groups and of context-free groups

Algorithms for solving uniform decision problems for algebraic structures crucially depend on the chosen finite presentations for the structures under consideration. Rewriting techniques have been used very successfully to solve uniform decision problems, when the presentations considered involve finite, noetherian, and (λ)-confluent rewriting systems. Whenever the class of algebraic structures considered is closed under the operation of taking finitely generated substructures, then the algorithms for solving the uniform decision problems can be applied to the substructures as well. However, since these algorithms depend on the form of the presentations, this involves the task of constructing a presentation of a certain form for a substructure given a presentation of this form for the structure itself and a finite set of generating elements for the substructure. This problem, which has received a lot of attention in algebra, is here investigated from an algorithmic point of view. The structures considered are the following two classes of groups, which have been studied extensively before: the polycyclic groups and the context-free groups. Finitely generated context-free groups can be presented by finite, monadic, and λ-confluent string-rewriting systems. Due to their nice algorithmic properties these systems provide a way to effectively solve many decision problems for context-free groups. Since finitely generated subgroups of context-free groups are again contextfree, they can be presented in the same way. Here we describe a process that, from a finite, monadic, and λ-confluent string-rewriting system presenting a context-free groupG and a finite subsetU ofG, determines a presentation of this form for the subgroup 〈U〉 ofG that is generated byU. For finitely presented polycyclic groups we obtain an analogous result, when we use finite confluent PCP2-presentations to describe these groups.

Quantifying over propositions in relevance logic: nonaxiomatisability of primary interpretations of ∀pand ∃p

A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take amodelto be one of these structures, α, together with some function or relation which associates with every formulaAa subset ofα. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and Meyer [1973], a formula can only be associated with subsets which are closed upwards. It is natural to take apropositionof α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers theprimaryinterpretation associated with the semantics.

On classical and quantum integrable field theories associated to Kac-Moody current algebras

We present classical and quantum algebraic structures for two-dimensional integrable field theories associated to Kac-Moody current algebras. We obtain in particular classical and quantum discretized versions of such current algebras. The corresponding monodromy matrix is shown to satisfy extended quantum group relations, leading to integrable properties of these theories. We apply our constructions to the lattice non-abelian Toda field theory.