Nuclei and their uses

⊇ Basic ⊇ Congruence ⊇ Division of sets of intervals, those sets closed under taking (a) similar intervals, (b) sub-intervals, (c) abutting intervals, (d) suprema, respectively. We let B(Λ) and D(Λ) be the family of all basic sets and division sets, respectively. Abstract and basic sets are not particularly important, but the do help with the organization. However, observe that the family B(Λ) of all basic sets is closed under arbitrary unions and intersections, and so is a frame. Two intervals abut if the have the form [a, b], [b, c] for a common element b. A set of interval is a congruence set precisely when it is the family of intervals collapsed by some lattice congruence. Division sets are the ones in bijective correspondence with the nuclei. For each nucleus j the set Dj is a division set. Conversely, for each division set D and each a ∈ Λ we setand basic sets are not particularly important, but the do help with the organization. However, observe that the family B(Λ) of all basic sets is closed under arbitrary unions and intersections, and so is a frame. Two intervals abut if the have the form [a, b], [b, c] for a common element b. A set of interval is a congruence set precisely when it is the family of intervals collapsed by some lattice congruence. Division sets are the ones in bijective correspondence with the nuclei. For each nucleus j the set Dj is a division set. Conversely, for each division set D and each a ∈ Λ we set j(a) = ∨ X where x ∈ X ⇐⇒ [a, x] ∈ D to obtain the corresponding nucleus. To generate a nucleus we select a set of intervals that we want to collapse, and then generate a division set.