We define and study classification systems in an arbitrary CJgenerated complete lattice L. Introducing a partial order among the classification systems of L, we obtain a complete lattice denoted by Cls(L). By using the elements of the classification systems, another lattice is also constructed: the box lattice B(L) of L. We show that B(L) is an atomistic complete lattice, moreover Cls(L)=Cls(B(...

The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, ${0})$ that are closed under the natural metric, but has no prime ideal 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. Togethe...

If (L,∨,∧, 0, 1, ) is a complete ortholattice, f : Ln → L any partial function, then there is a complete ortholattice L containing L as a subortholattice, and a ortholattice polynomial p with coefficients in L such that p(a1, . . . , an) = f(a1, . . . , an) for all a1, . . . , an ∈ L. Iterating this construction long enough yields a complete ortholattice in which every function can be interpola...