A thin-tall Boolean algebra which is isomorphic to each of its uncountable subalgebras

作者：Bonnet Robert^*; Rubin Matatyahu

来源：Topology and Its Applications, 2011, 158(13): 1503-1525.

DOI：10.1016/j.topol.2011.05.003

• Ostaszewski space
• Thin-tall
• Scattered space
• Superatomic Boolean algebra
• Rigid Boolean algebra

Theorem A (lozenge N(1)). There is a Boolean algebra B with the following properties:
(1) B is thin-tall, and
(2) B is downward-categorical.
That is, every uncountable subalgebra of B is isomorphic to B.
The algebra B from Theorem A has some additional properties. For an ideal K of B, set cmpl(B)(K) := {a is an element of B vertical bar a.b = 0 for all b is an element of K}. We say that K is almost principal if K boolean OR cmpl(B) (K) generates B.
(3) B is rigid in the following sense. Suppose that I, J are ideals in B and f : B/I -> B/J is a homomorphism with an uncountable range. Then there is an almost principal ideal K of B such that vertical bar cmpl(K)vertical bar No, I boolean AND K subset of J boolean AND K, and for every a is an element of K, f(a/I) = a/J.
(4) The Stone space of B is sub-Ostaszewski. Boolean-algebraically, this means that: if 1 is an uncountable ideal in B, then B/I has cardinality <= No.
(5) Every uncountable subalgebra of B contains an uncountable ideal of B.
(6) Every subset of B consisting of pairwise incomparable elements has cardinality <= No.
(7) Every uncountable quotient of B has properties (1)-(6).
Assuming lozenge N(1) we also construct a Boolean algebra C such that:
(1) C has properties (1) and (4)-(6) from Theorem A, and every uncountable quotient of C has properties (1) and (4)-(6).
(2) C is rigid in the following stronger sense. Suppose that I. J are ideals in C and f : C/I -> C/J is a homomorphism with an uncountable range. Then there is a principal ideal K of C such that vertical bar cmpl(K)vertical bar <= No. I boolean AND K subset of J boolean AND K, and for every a is an element of K, f (a/I) = a/J.

• 出版日期2011-8-15