Given any poset, there are three important distinct lattices associated with it: the lattice of ideals, the lattice of normal cuts, and the lattice of maximal antichains. The lattice of ideals captures the notion of consistent cuts in a distributed computation and is discussed in Chapter 14. For a poset , its completion by normal cuts is the smallest lattice that has as its suborder and is discussed in Chapter 6. In this chapter, we discuss the lattice of maximal antichains with applications to global predicate detection.

Recall that an antichain is *maximal* if there does not exist any event that can be added to the set without violating the antichain property. The *lattice of maximal antichains*, denoted by is the set of all maximal antichains under the order consistent with the order on the lattice of consistent cuts.

The lattice of maximal antichains captures all maximal sets of concurrent events and has applications in detection of global predicates because it is usually much smaller than the lattice of consistent cuts. ...

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