TY - JOUR

T1 - Order extensions as adjoint functors

AU - Erné, Marcel

PY - 1986/1/1

Y1 - 1986/1/1

N2 - 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.

AB - 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.

KW - Adjunction

KW - Completion

KW - Duality

KW - Extension

KW - Poset

KW - Subset system

KW - Z -continuous map

KW - Z -embedding

UR - http://www.scopus.com/inward/record.url?scp=0007470001&partnerID=8YFLogxK

U2 - 10.1080/16073606.1986.9632112

DO - 10.1080/16073606.1986.9632112

M3 - Article

AN - SCOPUS:0007470001

VL - 9

SP - 149

EP - 206

JO - Quaestiones mathematicae

JF - Quaestiones mathematicae

SN - 1607-3606

IS - 1-4

ER -