Invariant Computation in a Poset

Djamel Talem; Bachir Sadi

doi:10.1007/s11083-021-09559-2

Abstract

Kong and Ribemboim (1994) define for every poset P a sequence P = D0(P), D(P), D2(P), D3(P)… of posets, where Di(P) = D(Di− 1(P)) consists of all maximal antichains of Di− 1(P). They prove that for a finite poset P, there exists an integer i ≥ 0 such that Di(P) is a chain. In this paper, for every finite poset P, we show how to calculate the smallest integer i for which Di(P) is a chain.

Research Article
23
10.1007/bf02115811

Computing on-line the lattice of maximal antichains of posets

Sep 01, 1994
Order
Guy-Vincent Jourdan + 2 more +2

We consider the on-line computation of the lattice of maximal antichains of a finite poset\(\tilde P\). This on-line computation satisfies what we call the “linear extension hypothesis”: the new incoming vertex is always maximal in the current subposet of\(\tilde P\). In addition to its theoretical interest, this abstraction of the lattice of antichains of a poset has structural properties which give it interesting practical behavior. In particular, the lattice of maximal antichains may be useful for testing distributed computations, for which purpose the lattice of antichains is already widely used. Our on-line algorithm has a run time complexity of\(\mathcal{O}((\left| P \right| + \omega ^2 (P))\omega (P)\left| {MA(P)} \right|),\), where |P| is the number of elements of the poset,\(\tilde P\), |MA(P)| is the number of maximal antichains of\(\tilde P\) and ω(P) is the width of\(\tilde P\). This is more efficient than the best off-line algorithms known so far.