Analogical Proportions in a Lattice of Sets of Alignments Built on the Common Subwords in a Finite Language

We define the locally maximal subwords and locally minimal superwords common to a finite set of words. We also define the corresponding sets of alignments. We give a partial order relation between such sets of alignments, as well as two operations between them. We show that the constructed family of sets of alignments has the lattice structure. The study of analogical proportion in lattices gives hints to use this structure as a machine learning basis, aiming at inducing a generalization of the set of words.