Limit systems and chain condition
Katsuya Eda · Tsukuba Journal of Mathematics · 1980
There is an interesting theorem about the countable chain condition of complete Boolean algebras in [11].Roughly speaking, the wellordered injective direct limit of complete Boolean algebras which satisfy the countable chain condition also satisfies the countable chain condition.In this paper, we shall investigate relations among limit systems of topolo- gical spaces, complete Boolean algebras and complete pseudo-Boolean algebras, and state the application of the above theorem.\S 1 is devoted to the basic definitions and preliminary results.Limit systems and their relations are described in \S 2. And in \S 3, we shall discuss about the chain conditions of the wellordered direct limits.\S 1. Basic definitions and preliminary results.We use the lattice theoretic symbols and the set theoretic ones.They are usual ones, but we shall give a few remarks about the symbols concerning a pseudo-Boolean algebra.A pseudo-Boolean algebra is a lattice with the least element $0$ and the operation $\Rightarrow$ , where $a\Rightarrow b$ is the maximal element $x$ such that $a\wedge x\leqq b$ .A Boolean algebra or a pseudo-Boolean algebra is complete if every subset of it has the least upper bound and the greatest lower bound.Many informations about the relationship between a Boolean algebra and a pseudo-Boolean algebras are in [10].Let $A$ be a pseudo-Boolean algebra and $R(A)$ be the set $\{a\Rightarrow 0;a\in A\}$ .An element of $R(A)$ is called a regular element.PROPOSITION 1 [10] $R(A)$ is a Boolean algebra the ordering of which is the restriction of the