A lattice-ordered set is a poset (L, <=) in which each two-element subset {a, b} has an infimum, denoted inf{a, b}, and a supremum, denoted sup{a, b}. There is a natural relationship between lattice-ordered sets and lattices. In fact, a lattice (L, ⋀, ⋁) is obtained from a lattice-ordered poset (L, <=) by defining a⋀b = inf{a, b} and a⋁b = sup{a, b} for any a, b element L. Also, from a lattice (L, ⋀, ⋁), one may obtain a lattice-ordered set (L, <=) by setting a<=b in L if and only if a = a⋀b. One obtains the same lattice-ordered set (L, <=) from the given lattice by setting a<=b in L if and only if a⋁b = b. (In other words, one may prove that for any lattice, (L, ⋀, ⋁), and for any two members a and b of L, a⋀b = b if and only if a = a⋁b.)

lattice | partially ordered set