  Alignments, i.e., position-wise comparisons of two or more strings or ordered
lists are of utmost practical importance in computational biology and a host of
other fields, including historical linguistics and emerging areas of research
in the Digital Humanities. The problem is well-known to be computationally hard
as soon as the number of input strings is not bounded. Due to its prac- tical
importance, a huge number of heuristics have been devised, which have proved
very successful in a wide range of applications. Alignments nevertheless have
received hardly any attention as formal, mathematical structures. Here, we
focus on the compositional aspects of alignments, which underlie most algo-
rithmic approaches to computing alignments. We also show that the concepts
naturally generalize to finite partially ordered sets and partial maps between
them that in some sense preserve the partial orders.
