Details
| Original language | English |
|---|---|
| Pages (from-to) | 295-314 |
| Number of pages | 20 |
| Journal | ORDER |
| Volume | 7 |
| Issue number | 3 |
| Publication status | Published - Sept 1990 |
Abstract
The category BPC of bounded posets and so-called cut continuous maps has concrete products, and the Dedekind-MacNeille completion gives rise to a reflector from BPC to the full subcategory CLJ of complete lattices and join-preserving maps. Like CLJ, the category BPC has a functional internal hom-functor in the sense of Banaschewski and Nelson. But, in contrast to CLJ, arbitrary universal bimorphisms do not exist in BPC. However, a natural tensor product is defined in terms of so-called G-ideals, such that the desired universal property holds at least for BPC-morphisms into complete lattices. Moreover, this tensor product is associative and distributes over (cartesian) products. The tensor product of an arbitrary family of bounded posets is isomorphic to that of their normal completions; hence, restricted to the subcategory CLJ, it agrees with the usual one.
Keywords
- (universal) bimorphism, AMS subject classifications (1980): 06A15, 06A23, 18A40, 18D10, complete lattice, cut continuous map, internal hom-functor, left adjoint map, poset, Tensor product
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
- Mathematics(all)
- Geometry and Topology
- Computer Science(all)
- Computational Theory and Mathematics
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In: ORDER, Vol. 7, No. 3, 09.1990, p. 295-314.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Tensor products for bounded posets revisited - To the memory of Evelyn Nelson
AU - Erné, Marcel
PY - 1990/9
Y1 - 1990/9
N2 - The category BPC of bounded posets and so-called cut continuous maps has concrete products, and the Dedekind-MacNeille completion gives rise to a reflector from BPC to the full subcategory CLJ of complete lattices and join-preserving maps. Like CLJ, the category BPC has a functional internal hom-functor in the sense of Banaschewski and Nelson. But, in contrast to CLJ, arbitrary universal bimorphisms do not exist in BPC. However, a natural tensor product is defined in terms of so-called G-ideals, such that the desired universal property holds at least for BPC-morphisms into complete lattices. Moreover, this tensor product is associative and distributes over (cartesian) products. The tensor product of an arbitrary family of bounded posets is isomorphic to that of their normal completions; hence, restricted to the subcategory CLJ, it agrees with the usual one.
AB - The category BPC of bounded posets and so-called cut continuous maps has concrete products, and the Dedekind-MacNeille completion gives rise to a reflector from BPC to the full subcategory CLJ of complete lattices and join-preserving maps. Like CLJ, the category BPC has a functional internal hom-functor in the sense of Banaschewski and Nelson. But, in contrast to CLJ, arbitrary universal bimorphisms do not exist in BPC. However, a natural tensor product is defined in terms of so-called G-ideals, such that the desired universal property holds at least for BPC-morphisms into complete lattices. Moreover, this tensor product is associative and distributes over (cartesian) products. The tensor product of an arbitrary family of bounded posets is isomorphic to that of their normal completions; hence, restricted to the subcategory CLJ, it agrees with the usual one.
KW - (universal) bimorphism
KW - AMS subject classifications (1980): 06A15, 06A23, 18A40, 18D10
KW - complete lattice
KW - cut continuous map
KW - internal hom-functor
KW - left adjoint map
KW - poset
KW - Tensor product
UR - http://www.scopus.com/inward/record.url?scp=34249955694&partnerID=8YFLogxK
U2 - 10.1007/BF00418657
DO - 10.1007/BF00418657
M3 - Article
AN - SCOPUS:34249955694
VL - 7
SP - 295
EP - 314
JO - ORDER
JF - ORDER
SN - 0167-8094
IS - 3
ER -