| Title:
|
Relatively additive states on quantum logics (English) |
| Author:
|
Pták, Pavel |
| Author:
|
Weber, Hans |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
2 |
| Year:
|
2005 |
| Pages:
|
327-338 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous quantum logic which does not possess any two-valued 1-additive state with respect to a given Boolean subalgebra. This result strengthens Theorem 2 of [5] and presents a rather advanced example in the orthomodular combinatorics (see also [9], [13], [4], [6], [16], etc.). In the rest we show that Greechie logics allow for $2$-additive three-valued states, and in case of Greechie lattices we show that one can even construct many $2$-additive two-valued states. Some open questions are posed, too. (English) |
| Keyword:
|
(weak) state on quantum logic |
| Keyword:
|
Greechie paste job |
| Keyword:
|
Boolean algebra |
| MSC:
|
03G12 |
| MSC:
|
46C05 |
| MSC:
|
81P10 |
| idZBL:
|
Zbl 1121.03085 |
| idMR:
|
MR2176895 |
| . |
| Date available:
|
2009-05-05T16:51:17Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119527 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
