Some Forcing Related Convergence Structures on Complete Boolean Algebras

Let convergences λi : B → P (B), i ≤ 4, on a complete Boolean algebra B be defined in the following way. For a sequence x = 〈xn : n ∈ ω〉 in B and the corresponding B-name for a subset of ω, τx = {〈ň, xn〉 : n ∈ ω}, let λi(x) = { {‖τx is infinite‖} if bi(x) = 1B, ∅ otherwise, where b1(x) = ‖τx is finite or cofinite‖, b2(x) = ‖τx is not unsupported‖, b3(x) = ‖τx is not a splitting real‖ and b4(x) = 1B. Then λ1 is the algebraic convergence generating the sequential topology on B, while the convergences λ2, λ3 and λ4, although different on each Boolean algebra producing splitting reals, generate the same topological convergence a generalization of the convergence on the Aleksandrov cube, considered in [18]. AMS Mathematics Subject Classification (2000): 54A20, 03E40, 03E75, 06E10, 54D55, 54A10.