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A general extension theorem for directed-complete partial orders

Schuster P., Wessel D.

Reports on Mathematical Logic

2018

Vol. 53

79--96

The typical indirect proof of an abstract extension theorem, by the Kuratowski-Zorn lemma, is based on a onestep extension argument. While Bell has observed this in case of the axiom of choice, for subfunctions of a given relation, we now consider such extension patterns on arbitrary directed-complete partial orders. By postulating the existence of so-called total elements rather than maximal ones, we can single out an immediate consequence of the Kuratowski-Zorn lemma from which quite a few abstract extension theorems can be deduced more directly, apart from certain definitions by cases. Applications include Baer’s criterion for a module to be injective. Last but not least, our general extension theorem is equivalent to a suitable form of the Kuratowski-Zorn lemma over constructive set theory.

The equality case in some recent convexity inequalities

Maksa G., Pales Z.

Opuscula Mathematica

2011

Vol. 31, no. 2

269-277

In this paper, we investigate a functional equation related to some recently introduced and investigated convexity type inequalities.

On extension of solutions of a simultaneous system of iterative functional equations

Matkowski J.

Opuscula Mathematica

2009

Vol. 29, no. 4

415-421

Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form φ(x)=h(x, φ[ƒ1(x)],…, φ[ƒm(x)]) φ(x)=H(x,φ[F1(x),…, [Fn(x)]) to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [1-3]).

Extensions of solutions of a functional equation in two variables

Matkowski J.

Opuscula Mathematica

2009

Vol. 29, no. 4

423-425

An extension theorem for the functional equation of several variables ƒ (M(x,y))=N(ƒ(x), ƒ (y)), where the given functions M and N are left-side autodistributive, is presented.

Extension theorem for a functional equation

Burai P.

Journal of Applied Analysis

2006

Vol. 12, nr 2

293-299

We are doing to prove an extension theorem on a functional equation for special means studied by Domsta and Matkowski