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

Theorem A ⋄ ℵ 1 is a Boolean algebra B with the following properties:(1) hin–tall, and ownward-categorical. is, every uncountable subalgebra of B is isomorphic to B. gebra B from Theorem A has some additional properties. ideal K of B, set cmpl B ( K ) : = { a ∈ B | a ⋅ b = 0 for all b ∈ K } . We say that K is almost principal if K ∪ cmpl B ( K ) generates B.(3) igid 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 | cmpl ( K ) | ⩽ ℵ 0 , I ∩ K ⊆ J ∩ K , and for every a ∈ K , f ( a / I ) = a / J . one space of B is sub-Ostaszewski. Boolean-algebraically, this means that: if I is an uncountable ideal in B, then B / I has cardinality ⩽ ℵ 0 . uncountable subalgebra of B contains an uncountable ideal of B. subset of B consisting of pairwise incomparable elements has cardinality ⩽ ℵ 0 . uncountable quotient of B has properties (1)–(6). ng ⋄ ℵ 1 we also construct a Boolean algebra C such that:(1) properties (1) and (4)–(6) from Theorem A, and every uncountable quotient of C has properties (1) and (4)–(6). igid 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 | cmpl ( K ) | ⩽ ℵ 0 , I ∩ K ⊆ J ∩ K , and for every a ∈ K , f ( a / I ) = a / J .