Order extensions as adjoint functors

verfasst von

Marcel Erné

Abstract

A standard extension (resp. standard completion) is a function Z assigning to each poset P a (closure) system ZP of subsets such that x ⋚ y iff x belongs to every Z ε ZP with y ε Z. A poset P is Z -complete if each Z ε 2P has a join in P. A map f: P → P′ is Z—continuous if f⁻¹[Z′] ε ZP for all Z′ ε ZP′, and a Z—morphism if, in addition, for all Z ε ZP there is a least Z′ ε ZP′ with f[Z] ⊆ Z′. The standard extension Z is compositive if every map f: P → P′ with (x ε P: f(x) ⋚ y′) ε ZP for all y′ ε P′ is Z -continuous. We show that any compositive standard extension Z is the object part of a reflector from IPZ, the category of posets and Z -morphisms, to IRZ, the category of Z -complete posets and residuated maps. In case of a standard completion Z, every Z -continuous map is a Z -morphism, and IR2 is simply the category of complete lattices and join—preserving maps. Defining in a suitable way so-called Z -embeddings and morphisms between them, we obtain for arbitrary standard extensions Z an adjunction between IPZ and the category of Z -embeddings. Many related adjunctions, equivalences and dualities are studied and compared with each other. Suitable specializations of the function 2 provide a broad spectrum of old and new applications. AMS (MOS) subject classification (1980). 06A10/15, 18 A 40.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Typ

Artikel

Journal

Quaestiones mathematicae

Band

9

Seiten

149-206

Anzahl der Seiten

58

ISSN

1607-3606

Publikationsdatum

01.01.1986

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Mathematik (sonstige)

Elektronische Version(en)

https://doi.org/10.1080/16073606.1986.9632112 (Zugang: Geschlossen)