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The ordering of commutative terms

Jaroslav Ježek (2006)

Czechoslovak Mathematical Journal

By a commutative term we mean an element of the free commutative groupoid F of infinite rank. For two commutative terms a , b write a ≤ b if b contains a subterm that is a substitution instance of a . With respect to this relation, F is a quasiordered set which becomes an ordered set after the appropriate factorization. We study definability in this ordered set. Among other things, we prove that every commutative term (or its block in the factor) is a definable element. Consequently, the ordered set has...

The Orthogonal Projection and the Riesz Representation Theorem

Keiko Narita, Noboru Endou, Yasunari Shidama (2015)

Formalized Mathematics

In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals...

The partially pre-ordered set of compactifications of Cp(X, Y)

A. Dorantes-Aldama, R. Rojas-Hernández, Á. Tamariz-Mascarúa (2015)

Topological Algebra and its Applications

In the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where...

The Point of Continuity Property: Descriptive Complexity and Ordinal Index

Bossard, Benoit, López, Ginés (1998)

Serdica Mathematical Journal

∗ Supported by D.G.I.C.Y.T. Project No. PB93-1142Let X be a separable Banach space without the Point of Continuity Property. When the set of closed subsets of its closed unit ball is equipped with the standard Effros-Borel structure, the set of those which have the Point of Continuity Property is non-Borel. We also prove that, for any separable Banach space X, the oscillation rank of the identity on X (an ordinal index which quantifies the Point of Continuity Property) is determined by the subspaces...

The point of continuity property, neighbourhood assignments and filter convergences

Ahmed Bouziad (2012)

Fundamenta Mathematicae

We show that for some large classes of topological spaces X and any metric space (Z,d), the point of continuity property of any function f: X → (Z,d) is equivalent to the following condition: (*) For every ε > 0, there is a neighbourhood assignment ( V x ) x ∈ X of X such that d(f(x),f(y)) < ε whenever ( x , y ) ∈ V y × V x . We also give various descriptions of the filters ℱ on the integers ℕ for which (*) is satisfied by the ℱ-limit of any sequence of continuous functions from a topological space into a metric space.

The positive and generalized discriminators don't exist

A.G. Pinus (2000)

Discussiones Mathematicae - General Algebra and Applications

In this paper it is proved that there does not exist a function for the language of positive and generalized conditional terms that behaves the same as the discriminator for the language of conditional terms.

The positivity problem for fourth order linear recurrence sequences is decidable

Pinthira Tangsupphathawat, Narong Punnim, Vichian Laohakosol (2012)

Colloquium Mathematicae

The problem whether each element of a sequence satisfying a fourth order linear recurrence with integer coefficients is nonnegative, referred to as the Positivity Problem for fourth order linear recurrence sequence, is shown to be decidable.

The power set of ω Elementary submodels and weakenings of CH

István Juhász, Kenneth Kunen (2001)

Fundamenta Mathematicae

We define a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles. SEP is implied by each of CH*, the weak Freeze-Nation property of (ω), and the (ℵ₁,ℵ₀)-ideal property. SEP implies the principle C ₂ s ( ω ₂ ) , but does not follow from C ₂ s ( ω ₂ ) , or even C s ( ω ₂ ) .

The prime and maximal spectra and the reticulation of BL-algebras

Laurenťiu Leuštean (2003)

Open Mathematics

In this paper we study the prime and maximal spectra of a BL-algebra, proving that the prime spectrum is a compact T 0 topological space and that the maximal spectrum is a compact Hausdorff topological space. We also define and study the reticulation of a BL-algebra.

The problems of definition

Jean-Pierre Ginisti (1991)

Mathématiques et Sciences Humaines

The aim of this paper is to present the great kinds of definitions known in mathematical logic, their goals and their means, from their historical and philosophical background (notably thanks to the proof of two theorems), and in order to situate, within this field, the others contributions which make up this number.

The product of two ordinals is hereditarily dually discrete

M.Á. Gaspar-Arreola, F. Hernández-Hernández (2012)

Commentationes Mathematicae Universitatis Carolinae

In Dually discrete spaces, Topology Appl. 155 (2008), 1420–1425, Alas et. al. proved that ordinals are hereditarily dually discrete and asked whether the product of two ordinals has the same property. In Products of certain dually discrete spaces, Topology Appl. 156 (2009), 2832–2837, Peng proved a number of partial results and left open the question of whether the product of two stationary subsets of ω 1 is dually discrete. We answer the first question affirmatively and as a consequence also give...

The Properties of Sets of Temporal Logic Subformulas

Mariusz Giero (2012)

Formalized Mathematics

This is a second preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in [17]. We introduce two modified definitions of a subformula. In the former one we treat until-formula as indivisible. In the latter one, we extend the set of subformulas of until-formulas by a special disjunctive formula. This is needed to construct a temporal model. We also define an...

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