Details
| Original language | English |
|---|---|
| Pages (from-to) | 267-284 |
| Number of pages | 18 |
| Journal | Topology and its applications |
| Volume | 73 |
| Issue number | 3 |
| Publication status | Published - 11 Nov 1996 |
Abstract
By a (fine) scale on a lattice L, we mean a (strictly) isotone real-valued function μ on L. The coarsest topology on L such that μ composed with the unary lattice operations becomes continuous is denoted by τμ. The following convergence structures on L are compared with each other: (i) order convergence, (ii) convergence in the order topology, (iii) τμ-convergence, (iv) ρμ-convergence, where ρμ(x, y) = μ(x V y) -μ,(x Λ y). We show that for any scale μ on an arbitrary complete lattice, order convergence agrees with ρμ-convergence and with τμ-convergence iff μ is a fine continuous scale such that join and meet operations of arbitrary arity are continuous with respect to ρμ-convergence. Furthermore, for any fine continuous scale μ on a bi-algebraic lattice, order convergence agrees with τμ-convergence. From these and related results, we derive various applications to the theory of measures and valuations on orthomodular lattices. For example, if μ is a fine scale on a complete orthomodular lattice then order convergence agrees with τμ, -convergence iff μ, is continuous and L is algebraic (or atomic and meet-continuous).
Keywords
- (Bi-)algebraic lattice, Content, Continuous, Measure, Order convergence, Order topology, Orthomodular lattice, Scale, Valuation
ASJC Scopus subject areas
- Mathematics(all)
- Geometry and Topology
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In: Topology and its applications, Vol. 73, No. 3, 11.11.1996, p. 267-284.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Convergence structures induced by scales
AU - Erné, Marcel
PY - 1996/11/11
Y1 - 1996/11/11
N2 - By a (fine) scale on a lattice L, we mean a (strictly) isotone real-valued function μ on L. The coarsest topology on L such that μ composed with the unary lattice operations becomes continuous is denoted by τμ. The following convergence structures on L are compared with each other: (i) order convergence, (ii) convergence in the order topology, (iii) τμ-convergence, (iv) ρμ-convergence, where ρμ(x, y) = μ(x V y) -μ,(x Λ y). We show that for any scale μ on an arbitrary complete lattice, order convergence agrees with ρμ-convergence and with τμ-convergence iff μ is a fine continuous scale such that join and meet operations of arbitrary arity are continuous with respect to ρμ-convergence. Furthermore, for any fine continuous scale μ on a bi-algebraic lattice, order convergence agrees with τμ-convergence. From these and related results, we derive various applications to the theory of measures and valuations on orthomodular lattices. For example, if μ is a fine scale on a complete orthomodular lattice then order convergence agrees with τμ, -convergence iff μ, is continuous and L is algebraic (or atomic and meet-continuous).
AB - By a (fine) scale on a lattice L, we mean a (strictly) isotone real-valued function μ on L. The coarsest topology on L such that μ composed with the unary lattice operations becomes continuous is denoted by τμ. The following convergence structures on L are compared with each other: (i) order convergence, (ii) convergence in the order topology, (iii) τμ-convergence, (iv) ρμ-convergence, where ρμ(x, y) = μ(x V y) -μ,(x Λ y). We show that for any scale μ on an arbitrary complete lattice, order convergence agrees with ρμ-convergence and with τμ-convergence iff μ is a fine continuous scale such that join and meet operations of arbitrary arity are continuous with respect to ρμ-convergence. Furthermore, for any fine continuous scale μ on a bi-algebraic lattice, order convergence agrees with τμ-convergence. From these and related results, we derive various applications to the theory of measures and valuations on orthomodular lattices. For example, if μ is a fine scale on a complete orthomodular lattice then order convergence agrees with τμ, -convergence iff μ, is continuous and L is algebraic (or atomic and meet-continuous).
KW - (Bi-)algebraic lattice
KW - Content
KW - Continuous
KW - Measure
KW - Order convergence
KW - Order topology
KW - Orthomodular lattice
KW - Scale
KW - Valuation
UR - http://www.scopus.com/inward/record.url?scp=15844406215&partnerID=8YFLogxK
U2 - 10.1016/s0166-8641(96)00074-0
DO - 10.1016/s0166-8641(96)00074-0
M3 - Article
AN - SCOPUS:15844406215
VL - 73
SP - 267
EP - 284
JO - Topology and its applications
JF - Topology and its applications
SN - 0016-660X
IS - 3
ER -