Details
| Original language | English |
|---|---|
| Pages (from-to) | 69-83 |
| Number of pages | 15 |
| Journal | Journal of the Australian Mathematical Society |
| Volume | 42 |
| Issue number | 1 |
| Publication status | Published - 9 Apr 2009 |
Abstract
Several` “classical” results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own “up-completion” i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes with several applications of a generalized version of the Birkhoff Frink decomposition theorem for algebraic lattices.
Keywords
- algebraic, and phrases, compactly generated, irreducible element, poset, Scott completion, up-complete, weakly atomic
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Journal of the Australian Mathematical Society, Vol. 42, No. 1, 09.04.2009, p. 69-83.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Compact generation in partially ordered sets
AU - Erné, Marcel
PY - 2009/4/9
Y1 - 2009/4/9
N2 - Several` “classical” results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own “up-completion” i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes with several applications of a generalized version of the Birkhoff Frink decomposition theorem for algebraic lattices.
AB - Several` “classical” results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own “up-completion” i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes with several applications of a generalized version of the Birkhoff Frink decomposition theorem for algebraic lattices.
KW - algebraic
KW - and phrases
KW - compactly generated
KW - irreducible element
KW - poset
KW - Scott completion
KW - up-complete
KW - weakly atomic
UR - http://www.scopus.com/inward/record.url?scp=84974288961&partnerID=8YFLogxK
U2 - 10.1017/S1446788700033966
DO - 10.1017/S1446788700033966
M3 - Article
AN - SCOPUS:84974288961
VL - 42
SP - 69
EP - 83
JO - Journal of the Australian Mathematical Society
JF - Journal of the Australian Mathematical Society
SN - 1446-7887
IS - 1
ER -