Glossary of Order Theory - L

L

• Lattice. A lattice is a poset in which all non-empty finite joins (suprema) and meets (infima) exist.

• Least element. For a subset X of a poset P, an element a of X is called the least element of X, if a ≤ x for every element x in X. The dual notion is called greatest element.

• The length of a chain is the number of elements less one. A chain with 1 element has length 0, one with 2 elements has length 1, etc.

• Linear. See total order.

• Linear extension. A linear extension of a partial order is an extension that is a linear order, or total order.

• Locale. A locale is a complete Heyting algebra. Locales are also called frames and appear in Stone duality and pointless topology.

• Locally finite poset. A partially ordered set P is locally finite if every interval = {x in P | a ≤ x ≤ b} is a finite set.

• Lower bound. A lower bound of a subset X of a poset P is an element b of P, such that b ≤ x, for all x in X. The dual notion is called upper bound.

• Lower set. A subset X of a poset P is called a lower set if, for all elements x in X and p in P, p ≤ x implies that p is contained in X. The dual notion is called upper set.