Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Completeness results for metrized rings and lattices   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.

23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat