Disjoint sum of partially ordered sets
disjoint sum of posets
Let and be two partially ordered sets (cf. Partially ordered set).
The disjoint sum of and is the disjoint union of the sets and with the original ordering on and and no other comparable pairs. A poset is disconnected if it is (isomorphic to) the disjoint sum of two sub-posets. Otherwise it is connected. The maximal connected sub-posets are called components.
The disjoint sum is the direct sum in the category of posets and order-preserving mappings. The direct product in this category is the Cartesian product with partial ordering
|[a1]||W.T. Trotter, "Partially ordered sets" R.L. Graham (ed.) M. Grötschel (ed.) L. Lovász (ed.) , Handbook of Combinatorics , I , North-Holland (1995) pp. 433–480|
Disjoint sum of partially ordered sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjoint_sum_of_partially_ordered_sets&oldid=15649