Details
| Original language | English |
|---|---|
| Pages (from-to) | 149-206 |
| Number of pages | 58 |
| Journal | Quaestiones mathematicae |
| Volume | 9 |
| Issue number | 1-4 |
| Publication status | Published - 1 Jan 1986 |
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.
Keywords
- Adjunction, Completion, Duality, Extension, Poset, Subset system, Z -continuous map, Z -embedding
ASJC Scopus subject areas
- Mathematics(all)
- Mathematics (miscellaneous)
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In: Quaestiones mathematicae, Vol. 9, No. 1-4, 01.01.1986, p. 149-206.
Research output: Contribution to journal › Article › Research › peer review
}
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 -