281 research outputs found
Algebraic geometry over algebraic structures II: Foundations
In this paper we introduce elements of algebraic geometry over an arbitrary
algebraic structure. We prove Unification Theorems which gather the description
of coordinate algebras by several ways.Comment: 55 page
Opening the AC-Unification Race
This note reports about the implementation of AC-unification algorithms, based on the variable-abstraction method of Stickel and on the constant-abstraction method of Livesey, Siekmann, and Herold. We give a set of 105 benchmark examples and compare execution times for implementations of the two approaches. This documents for other researchers what we consider to be the state-of-the-art performance for elementary AC-unification problems
Brane and string field structure of elementary particles
The two quantizations of QFT,as well as the attempt of unifying it with
general relativity,lead us to consider that the internal structure of an
elementary fermion must be twofold and composed of three embedded internal
(bi)structures which are vacuum and mass (physical) bosonic fields decomposing
into packets of pairs of strings behaving like harmonic oscillators
characterized by integers mu corresponding to normal modes at mu (algebraic)
quanta.Comment: 50 page
A perspective on non-commutative frame theory
This paper extends the fundamental results of frame theory to a
non-commutative setting where the role of locales is taken over by \'etale
localic categories. This involves ideas from quantale theory and from semigroup
theory, specifically Ehresmann semigroups, restriction semigroups and inverse
semigroups. We establish a duality between the category of complete restriction
monoids and the category of \'etale localic categories. The relationship
between monoids and categories is mediated by a class of quantales called
restriction quantal frames. This result builds on the work of Pedro Resende on
the connection between pseudogroups and \'etale localic groupoids but in the
process we both generalize and simplify: for example, we do not require
involutions and, in addition, we render his result functorial. We also project
down to topological spaces and, as a result, extend the classical adjunction
between locales and topological spaces to an adjunction between \'etale localic
categories and \'etale topological categories. In fact, varying morphisms, we
obtain several adjunctions. Just as in the commutative case, we restrict these
adjunctions to spatial-sober and coherent-spectral equivalences. The classical
equivalence between coherent frames and distributive lattices is extended to an
equivalence between coherent complete restriction monoids and distributive
restriction semigroups. Consequently, we deduce several dualities between
distributive restriction semigroups and spectral \'etale topological
categories. We also specialize these dualities for the setting where the
topological categories are cancellative or are groupoids. Our approach thus
links, unifies and extends the approaches taken in the work by Lawson and Lenz
and by Resende.Comment: 69 page
Unification theory
The purpose of this paper is not to give an overview of the state of art in unification theory. It is intended to be a short introduction into the area of equational unification which should give the reader a feeling for what unification theory might be about. The basic notions such as complete and minimal complete sets of unifiers, and unification types of equational theories are introduced and illustrated by examples. Then we shall describe the original motivations for considering unification (in the empty theory) in resolution theorem proving and term rewriting. Starting with Robinson\u27s first unification algorithm it will be sketched how more efficient unification algorithms can be derived.
We shall then explain the reasons which lead to the introduction of unification in non-empty theories into the above mentioned areas theorem proving and term rewriting. For theory unification it makes a difference whether single equations or systems of equations are considered. In addition, one has to be careful with regard to the signature over which the terms of the unification problems can be built. This leads to the distinction between elementary unification, unification with constants, and general unification (where arbitrary free function symbols may occur). Going from elementary unification to general unification is an instance of the so-called combination problem for equational theories which can be formulated as follows: Let E, F be equational theories over disjoint signatures. How can unification algorithms for E, F be combined to a unification algorithm for the theory E cup F
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