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Introduction to Lattice Theory with Computer Science Applications by Vijay K. Garg

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Chapter 15Lattice of Maximal Antichains

15.1 Introduction

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 c15-math-0001, its completion by normal cuts is the smallest lattice that has c15-math-0002 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 c15-math-0003 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 c15-math-0004 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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