Previous |  Up |  Next


Boolean algebra; sequential convergence; disjoint sequence; Brouwerian lattice
We deal with the system ${\operatorname{Conv}} B$ of all sequential convergences on a Boolean algebra $B$. We prove that if $\alpha$ is a sequential convergence on $B$ which is generated by a set of disjoint sequences and if $\beta$ is any element of ${\operatorname{Conv}} B$, then the join $\alpha\vee\beta$ exists in the partially ordered set ${\operatorname{Conv}} B$. Further we show that each interval of ${\operatorname{Conv}} B$ is a Brouwerian lattice.
[1] J. Jakubík: Sequential convergences in Boolean algebras. Czechoslovak Math. J. 38 (1988), 520-530. MR 0950306
[2] J. Jakubík: Convergences and higher degrees of distributivity of lattice ordered groups and of Boolean algebras. Czechoslovak Math. J. 40 (1990), 453-458. MR 1065024
[3] H. Löwig: Intrinsic topology and completion of Boolean rings. Ann. Math. 13 (1941), 1138-1196. DOI 10.2307/1970464 | MR 0006494
[4] J. Novák M. Novotný: On the convergence in $\sigma$-algebras of point-sets. Czechoslovak Math. J. 3 (1953), 291-296. MR 0060572
[5] F. Papangelou: Order convergence and topological completion of commutative lattice-groups. Math. Ann. 155 (1964), 81-107. DOI 10.1007/BF01344076 | MR 0174498 | Zbl 0131.02601
Partner of
EuDML logo