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Continuous tree-like scales

100%
Cummings J.
Open Mathematics
|
2010
|
tom 8
|
nr 2
314-318
EN
Answering a question raised by Luis Pereira, we show that a continuous tree-like scale can exist above a supercompact cardinal. We also show that the existence of a continuous tree-like scale at ℵω is consistent with Martin’s Maximum.
2
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Singular cardinals and strong extenders

76%
Apter A., Cummings J., Hamkins J.
Open Mathematics
|
2013
|
tom 11
|
nr 9
1628-1634
EN
We investigate the circumstances under which there exist a singular cardinal µ and a short (κ,µ)-extender E witnessing “κ is µ-strong”, such that µ is singular in Ult(V, E).
3
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The free one-generated left distributive algebra: basics and a simplified proof of the division algorithm

64%
Laver R., Miller S.
Open Mathematics
|
2013
|
tom 11
|
nr 12
2150-2175
EN
The left distributive law is the law a· (b· c) = (a·b) · (a· c). Left distributive algebras have been classically used in the study of knots and braids, and more recently free left distributive algebras have been studied in connection with large cardinal axioms in set theory. We provide a survey of results on the free left distributive algebra on one generator, A, and a new, simplified proof of the existence of a normal form for terms in A. Topics included are: the confluence of A, the linearity of the iterated left division ordering
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