On some small cardinals for Boolean algebras

Abstract

Assume that all algebras are atomless. (1) Spind(A× B)=Spind(A)∪ Spind(B). (2) Spind(∏^w_{i∈ I}A_i)={ω}∪⋃_{i∈ I} Spind (A_i). Now suppose that κ and λ are infinite cardinals, with κ uncountable and regular and with κ<λ. (3) There is an atomless Boolean algebra A such that 𝔲(A)=κ and 𝔦(A)=λ. (4) If λ is also regular, then there is an atomless Boolean algebra A such that 𝔰(A)=𝔰(A)=κ and 𝔞(A)=λ. All results are in ZFC, and answer some problems posed in Monk [Mon01] and Monk [MonInf].