article · Kobe University Repository Kernel (Kobe University) · 1993
The stationarity preserving part of a complete Boolean algebra
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Abstract
An algebraic formula expressing the A-value of the statement that all stationary sets in V remain stationary in Vᴬ is obtained, where A is an arbitrary complete Boolean algebra. It is also shown how this formula leads to some of the known characterizations of A being proper. Forcing that preserves the stationarity of stationary sets in the ground model, named proper forcing by S. Shelah, has been a subject of considerable interest in set theory. Here, stationarity means that of subsets of P ≤ ω(X) = {M ⊆ X | Card(M) ≤ ω} with the base set X allowed to be arbitrary. In this paper, I will look at the Boolean-value || All stationary sets in V remain stationary || A for an arbitrary complete Boolean algebra A. Working with a complete Boolean algebra rather than with a poset, I will obtain an algebraic formula that expresses this value in terms of the infinite meet and join operations of A (Theorem 1.9). I will also show how this formula leads naturally to some of the known characterizations of A being proper (Theorems 2.1 and 2.3). I am grateful to the referee for suggesting a number of improvements on my original manuscript.
Cite this paper
Takahashi, J. (1993). The stationarity preserving part of a complete Boolean algebra. Kobe University Repository Kernel (Kobe University). https://doi.org/10.24546/e0003588
Takahashi, Joji. “The stationarity preserving part of a complete Boolean algebra.” Kobe University Repository Kernel (Kobe University), 1993, https://doi.org/10.24546/e0003588.
Takahashi, Joji. 1993. “The stationarity preserving part of a complete Boolean algebra.” Kobe University Repository Kernel (Kobe University). https://doi.org/10.24546/e0003588.
Takahashi, J. (1993) ‘The stationarity preserving part of a complete Boolean algebra’, Kobe University Repository Kernel (Kobe University). Available at: https://doi.org/10.24546/e0003588.
@article{takahashi1993stationarity,
title = {{The stationarity preserving part of a complete Boolean algebra}},
author = {Takahashi, Joji},
journal = {Kobe University Repository Kernel (Kobe University)},
year = {1993},
doi = {10.24546/e0003588}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.