## 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)

### Keywords:

sequential convergence
Full Text:

### References:

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