Partially Ordered Set

A partially ordered set (poset) is a set with a relation that is reflexive, antisymmetric, and transitive. One element covers another, written , when with nothing strictly between them. The Hasse diagram draws the covering relations, and is recovered from them by transitivity. A poset is a thin category, with one arrow for each relation .

Definition of grading

A poset is graded by a rank function with whenever . Every maximal chain between two elements then has the same length, and the rank sorts the Hasse diagram into levels.

Definition of meet, join, (semi)lattice

The meet is the greatest lower bound and the join the least upper bound, when these exist. A meet-semilattice has all binary meets (equivalently a top element and all meets of finite non-empty sets), whereas a lattice has all binary meets and joins. Adjoining a bottom element to a meet-semilattice with a top makes it a lattice.

References

• Davey & Priestley, Introduction to Lattices and Order, 2nd ed. (Cambridge, 2002)