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

Theorem A – ⋄ℵ1There 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 cmplB(K):={a∈B|a⋅b=0for allb∈K}. We say that K is almost principal if K∪cmplB(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 |cmpl(K)|⩽ℵ0, I∩K⊆J∩K, and for every a∈K, f(a/I)=a/J.(4)The Stone space of B is sub-Ostaszewski. Boolean-algebraically, this means that: if I is an uncountable ideal in B, then B/I has cardinality ⩽ℵ0.(5)Every uncountable subalgebra of B contains an uncountable ideal of B.(6)Every subset of B consisting of pairwise incomparable elements has cardinality ⩽ℵ0.(7)Every uncountable quotient of B has properties (1)–(6).Assuming ⋄ℵ1 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 |cmpl(K)|⩽ℵ0, I∩K⊆J∩K, and for every a∈K, f(a/I)=a/J.