Fix a partial orderP=(X,<). We first show that bipartite orders are sufficient to study structural properties of the lattice of maximal antichains. We show that all orders having the same lattice of maximal antichains can be reduced to one representative order (called the poset of irreducibles by Markowsky 14). We then define thestrong simplicial elimination schemeto characterize orders which have distributive lattice of maximal antichains. The notion of simplicial elimination corresponds to the decomposition process described in 14 for extremal lattices. This notion leads to simple greedy algorithms for distributivity checking, lattice recognition and jump number computation. In the last section, we give several algorithms for lattices and orders.
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