On co-Filters in Semigroups with Apartness

The logical environment of this research is the Intuitionistic Logic and principled-philosophical orientation of the Bishop’s Constructive Mathematics. In this paper, basing our consideration on the sets with the apartness relation, we analyze the lattices of all co-filters of an ordered semigroup under a co-quasiorder as a continuation of our article [?]. We prove a number of results related to co-filters in a semigroup with apartness and the lattice of all co-filters of such semigroups.

Co-quasiordered residuated systems: An introduction

In this paper, we introduce and discuss a concept of co-quasiordered residuated relational systems. The setting of this research is Bishop’s constructive mathematics — a mathematics based on the Intuitionisic logic and particular principle-philosophical orientation of this attitude. Moreover, we introduce the concept of co-filters in such relational system. Additionally, some of the fundamentals properties of these substructures we have been shown.

BROUWER’S FAN THEOREM AND CONVEXITY

In the framework of Bishop’s constructive mathematics we introduce co-convexity as a property of subsets B of ${\left\{ {0,1} \right\}^{\rm{*}}}$, the set of finite binary sequences, and prove that co-convex bars are uniform. Moreover, we establish a canonical correspondence between detachable subsets B of ${\left\{ {0,1} \right\}^{\rm{*}}}$ and uniformly continuous functions f defined on the unit interval such that B is a bar if and only if the corresponding function f is positive-valued, B is a uniform bar if and only if f has positive infimum, and B is co-convex if and only if f satisfies a weak convexity condition.

A constructive look at the completeness of the space (ℝ)

We show, within the framework of Bishop's constructive mathematics, that (sequential) completeness of the locally convex space (ℝ) of test functions is equivalent to the principle BD-ℕ which holds in classical mathemtatics, Brouwer's intuitionism and Markov's constructive recursive mathematics, but does not hold in Bishop's constructivism.

On Weak Markov's Principle

We show that the so-called weak Markov's principle (WMP) which states that every pseudo-positive real number is positive is underivable in T^{omega}:= E-HA^{omega} + AC. Since T^{omega} allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved within the framework of Bishop-style mathematics (which has been open for about 20 years). The underivability even holds if the ineffective schema of full comprehension (in all types) for negated formulas (in particular for $\exists$-free formulas) is added which allows to derive the law of excluded middle for such formulas.

Toward a constructive theory of unbounded linear operators

We show that the following results in the classical theory of unbounded linear operators on Hilbert spaces can be proved within the framework of Bishop's constructive mathematics: the Kato-Rellich theorem, the spectral theorem. Stone's theorem, and the self-adjointness of the most common quantum mechanical operators, including the Hamiltonians of electro-magnetic fields with some general forms of potentials.