Details
| Original language | English |
|---|---|
| Pages (from-to) | 215-227 |
| Number of pages | 13 |
| Journal | Topology and its applications |
| Volume | 61 |
| Issue number | 3 |
| Publication status | Published - 24 Feb 1995 |
Abstract
A lattice is order-topological iff its order convergence is topological and makes the lattice operations continuous. We show that the following properties are equivalent for any complete orthomodular lattice L: 1. (i) L is order-topological, 2. (ii) L is continuous (in the sense of Scott), 3. (iii) L is algebraic, 4. (iv) L is compactly atomistic, 5. (v) L is a totally order-disconnected topological lattice in the order topology. A special class of complete order-topological orthomodular lattices, namely the compact topological orthomodular lattices, are characterized by various algebraic conditions, for example, by the existence of a join-dense subset of so-called hypercompact elements.
Keywords
- Atomistic, Compact, Compactly generated, Continuous, Order convergence, Order topology, Order-topological, Ortho(modular) lattice, Totally order-disconnected
ASJC Scopus subject areas
- Mathematics(all)
- Geometry and Topology
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In: Topology and its applications, Vol. 61, No. 3, 24.02.1995, p. 215-227.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Order-topological complete orthomodular lattices
AU - Erné, Marcel
AU - Riečanová, Zdenka
PY - 1995/2/24
Y1 - 1995/2/24
N2 - A lattice is order-topological iff its order convergence is topological and makes the lattice operations continuous. We show that the following properties are equivalent for any complete orthomodular lattice L: 1. (i) L is order-topological, 2. (ii) L is continuous (in the sense of Scott), 3. (iii) L is algebraic, 4. (iv) L is compactly atomistic, 5. (v) L is a totally order-disconnected topological lattice in the order topology. A special class of complete order-topological orthomodular lattices, namely the compact topological orthomodular lattices, are characterized by various algebraic conditions, for example, by the existence of a join-dense subset of so-called hypercompact elements.
AB - A lattice is order-topological iff its order convergence is topological and makes the lattice operations continuous. We show that the following properties are equivalent for any complete orthomodular lattice L: 1. (i) L is order-topological, 2. (ii) L is continuous (in the sense of Scott), 3. (iii) L is algebraic, 4. (iv) L is compactly atomistic, 5. (v) L is a totally order-disconnected topological lattice in the order topology. A special class of complete order-topological orthomodular lattices, namely the compact topological orthomodular lattices, are characterized by various algebraic conditions, for example, by the existence of a join-dense subset of so-called hypercompact elements.
KW - Atomistic
KW - Compact
KW - Compactly generated
KW - Continuous
KW - Order convergence
KW - Order topology
KW - Order-topological
KW - Ortho(modular) lattice
KW - Totally order-disconnected
UR - http://www.scopus.com/inward/record.url?scp=0013153199&partnerID=8YFLogxK
U2 - 10.1016/0166-8641(94)00040-A
DO - 10.1016/0166-8641(94)00040-A
M3 - Article
AN - SCOPUS:0013153199
VL - 61
SP - 215
EP - 227
JO - Topology and its applications
JF - Topology and its applications
SN - 0166-8641
IS - 3
ER -