On interval decomposition lattices

Abstract

Intervals in binary or n-ary relations or other discrete structures generalize the concept of interval in an ordered set. They are defined abstractly as closed sets of a closure system on a set V, satisfying certain axioms. Decompositions are partitions of V whose blocks are intervals, and they form an algebraic semimodular lattice. Latticetheoretical properties of decompositions are explored, and connections with particular types of intervals are established.

Interval Decomposition Lattices are Balanced

“…Since DpV, Qq Ď PartpV q, it is ordered by refinement, where for any π 1 , π 2 P DpV, Qq, π 1 ≤ π 2 holds if and only if every block of π 2 is the union of some blocks of π 1 . In [5] we proved the following. Proposition 1.2.…”

“…The following result was proved in full generality in [17] and [5]: Proposition 1.4. If pV, Qq is an algebraic closure system satisfying condition pI 1 q, then DpV, Qq is an algebraic semimodular lattice.…”

“…An interval system pV, Iq was defined in [5] as an algebraic closure system with the following properties:…”

“…Examples of interval systems given in [5] include modules of graphs and relational intervals. These latter include order intervals in linearly ordered sets.…”

Classification lattices are geometric for complete atomistic lattices

“…Actually, the problem is also related with the study of CD-bases of a lattice (see [9,10]), or to the investigation of the decomposition systems of a closure system (cf. [11,12]). In other words, the solution of this problem may have useful applications in several related fields.…”