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Sequential convergences in Boolean algebras. (English) Zbl 0668.54002

Let B be a Boolean algebra. The notion of sequential convergence in B which is compatible with all operations of B is introduced. It is proved that such convergence is uniquely determined by the system of all sequences which converge to the zero element of B (denoted in the next by \(0_ B)\). Let Conv B denote all convergences on B. The author characterizes those subsets S of \(B^ N\) for which there exists \(\alpha\in Conv B\) such that every sequence of S \(\alpha\)-converges to \(0_ B\). The partially ordered set Conv B ordered in the natural way is investigated. It has the least element, no atom and it is closed under taking arbitrary infima. The existence of the greatest element of Conv B is equivalent to any of the following conditions: (i) Conv B is a lattice; (ii) Conv B is a complete lattice; (iii) the subset of these convergences which set of sequences converging to \(0_ B\) is generated by one sequence is join-semilattice. If B is completely distributive then Conv B has the greatest element. Also there are proved two inequalities concerning the cardinals \(D(P)=\sup \{card A:\) A is a subset of an ordered set P with pairwise disjoint elements\(\}\) and \(L(P)=\sup \{card C:\) C is a subchain of an ordered set \(P\}\). If B is infinite then D(B)\(\leq D(Conv B)\) and D(B)\(\leq L(Conv B)\).
Reviewer: M.Harminc

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
06E99 Boolean algebras (Boolean rings)
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References:

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