Stacks and splits of partially ordered sets

The dimension of a partiruly ordered set (X, P) is the smallest positive integer i for which there exists a function f which assigns to each x E X a sequence {f(x)(i): I s i 41) of TeaI numbers so that x Sy in P i! and only if f(x)(i)Sf(y)(i) for each i = 1,2,. , , , t. TIE irltelval ~lmension of (X, P) is the sm;.llcst integer t for which there exists a function F which assigns to 2ach x E X a sequence {F(x)(i): 1 Q i s t) of closed intervals of the real line R so that x < y in P f and only if a -C b in R for every a E FQxNi), b E F(y)(i), and i = 1,2, . _ _ , z. For I 22, a martially ordered set (poset) is said to be t-irreducible (resp. t-interval irreducible) if it has Jimension I (resp. interval dimension t): and every proper subposet has dimension (resp. interval dimension) less than t. The only %rreduciile poset is a two element anti-chain, and the only 24ntcrva1 irreducible poset is the free sum of two chains each having two points. In sharp contrast, the coilection R of all %-reducible posets consists of 9 infinite families and 18 odd examples, and the collection %r of all 3-interval irreducible posets is suficiently complex to have avoided complete determination as of this date. Trotter and Moore determined R from Gallai’s forbidden subgraph characterization of comparability graphs. David Kelly independently determined % by a lattice theoretic argument combined wit$ the characterization of planar lattices Kelly and Ivan Rival had previously obtained. in this paper, we introduce a new operation called a stack which we wiil apply to posets of height one. In some ways the stack oneration is an inverse of the split operation on posets previously defined by Kimble. These operations behave predictably with respect to dimension and interval dimensiwl. In particular, the stack of a poset of height one plays a role in interval dimension theory which is analogous to the role played by the completion by cuts in dimension theory. As a consequence, we can exploit the similarities to Kelly’s approach to the determination of 3e to produce a relatively compact argument to determine the collection %(I. 1) of all 3-interval irredlncible posets of height one. This characterization problem has immediate combinatorial connections with a wide range of well-known forbidden subgraph problems including interval graph+ l&angle graphs, and circular arc graphs.