Related papers: Valuative dimension, constructive points of view
The main result from this note provides a constructive characterization of the valuative dimension, which bears a strong analogy to Lombardi's constructive characterization of the Krull dimension. While Lombardi's characterization uses the…
The first part of the present article consists in a survey about the dynamical constructive method designed using dynamical theories and dynamical algebraic structures. Dynamical methods uncovers a hidden computational content for numerous…
We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espan\~ol and the authors. We show that this gives a constructive…
We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espa\~nol and the authors. We show that the notion of Krull…
A valuation theory for superrings is developed, extending classical constructions from commutative algebra to the $\mathbb Z_2$-graded and supercommutative setting. We define valuations on superrings, investigate their fundamental…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
Classical logic is embedded into constructive logic, through a definition of the classical connectives and quantifiers in terms of the constructive ones.
Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically…
The classical work of Feferman Vaught gives a powerful, constructive analysis of definability in (generalized) product structures, and certain associated enriched Boolean structures. %structures in terms of definability in the component…
An age-old controversy in mathematics concerns the necessity and the possibility of constructive proofs. The controversy has been rekindled by recent advances which demonstrate the feasibility of a fully constructive mathematics. This…
We study different notions of connected constructive metric spaces. They differ the types of connected components and how different components relate to each other. These notions are equivalent in classical point set topology but they give…
It is shown that every commutative arithmetic ring $R$ has $lambda$-dimension $ leq 3$. An example of a commutative Kaplansky ring with $ lambda$-dimension 3 is given. If $R$ satisfies an additional condition then $ lambda$-dim($R$)
We give an explicit description of cubic rings over a discrete valuation ring, as well as a description of all ideals of such rings.
The strong global dimension of a ring is the supremum of the length of perfect complexes that are indecomposable in the derived category. In this note we characterize the noetherian commutative rings that have finite strong global…
Discussion of the necessity to use the constructive mathematics as the formalism of quantum theory for systems with many particles.
Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on…
The central structure in various versions of noncommutative geometry is a differential calculus on an associative algebra. This is an analogue of the calculus of differential forms on a manifold. In this short review we collect examples of…
A review is given of some 2-dimensional metrics for which noncommutative versions have been found. They serve partially to illustrate a noncommutative extension of the moving-frame formalism. All of these models suggest that there is an…
We give an example of a commutative coherent ring of infinite global dimension such that the category of perfect complexes has finite Rouquier dimension.
Extra dimensions are introduced: 3 in Classical Mechanics and 6 in Relativistic Mechanics, which represent orientations, resulting from rotations, of a particle, described by quaternions, and leading to a 7-dimensional, respectively…