Completeness results for metrized rings and lattices

TL;DR

This paper investigates the metric completeness of Boolean rings and lattices, providing examples and generalizations that clarify the conditions under which these algebraic structures are complete in their metrics.

Contribution

It establishes the completeness of certain metrized Boolean rings and lattices, generalizes this to broader classes, and explores conditions affecting completeness.

Findings

Boolean ring of measurable subsets is complete in its metric.

Lattice completeness depends on specific metric inequalities.

Weaker metric conditions may lead to incompleteness.

Abstract

The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (e.g., $\{0\})$ that are closed under the natural metric, but has no prime ideals closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Together, these facts answer a question posed by J.Gleason. From this example, rings of arbitrary characteristic with the corresponding properties are obtained. The result that $B$ is complete in its metric is generalized to show that if $L$ is a lattice given with a metric satisfying identically either the inequality $d(x\vee y,\,x\vee z)\leq d(y,z)$ or the inequality $d(x\wedge y,\,x\wedge z)\leq d(y,z),$ and if in $L$ every increasing Cauchy sequence converges and every decreasing Cauchy sequence converges, then every Cauchy sequence in $L$ converges; i.e., $L$ is complete as a metric space. We show by example that if the above inequalities are replaced by the weaker conditions $d(x,\,x\vee y)\leq d(x,y),$ respectively $d(x,\,x\wedge y)\leq d(x,y),$ the completeness conclusion can fail. We end with two open questions.