Free complete extensions of Boolean algebras

Superatomic Boolean algebras

“…Conditions (c)-(e) are developed in [1]. Mostowski and Tarski also showed that all homomorphic images and subalgebras of superatomic Boolean algebras are superatomic.…”

“…In spite of the "naturalness" of the statement of the theorem, the similar statement, "If a Boolean algebra is generated by the union of finitely many atomic subalgebras, then it is atomic" is false. Consider the following: Let A denote the direct product of B t and J5 2 , and define A t to be the subalgebra of A with elements (0, 0), (0,1), (1,0), (1,1) and A 2 to be the subalgebra of A generated by all elements of the form (z, 0), where z is an atom of B, and the elements (x if #*), i = 1, 2, . It is easy to see that both A t and A 2 are atomic, and that their union generates the nonatomic Boolean algebra A.…”

“…It was proven by F. M. Yaqub ([8]) that, if a ^ 2 Ko , then the free aextension of a Boolean algebra B is α:-representable if and only if B is superatomic. In considering the question of the existence of free complete extensions of Boolean algebras, it was found ( [1]) that a Boolean algebra B has a free complete extension if and only if B is superatomic. Other results are summarized in the following theorem: THEOREM …”

Algumas aplicações de combinatória infinita a espaços de funções contínuas

“…Como principal referência, de maneira geral, para as Seções 1, 2 usamos (KOPPELBERG, 1989). Para a Seção 2, em particular usamos (DAY, 1965), (DAY, 1967) …”