Glossary of Order Theory - M

M

• Maximal chain. A chain in a poset to which no element can be added without losing the property of being totally ordered. This is stronger than being a saturated chain, as it also excludes the existence of elements either less than all elements of the chain or greater than all its elements. A finite saturated chain is maximal if and only if it contains both a minimal and a maximal element of the poset.

• Maximal element. A maximal element of a subset X of a poset P is an element m of X, such that m ≤ x implies m = x, for all x in X. The dual notion is called minimal element.

• Meet. See infimum.

• Minimal element. A minimal element of a subset X of a poset P is an element m of X, such that x ≤ m implies m = x, for all x in X. The dual notion is called maximal element.

• Monotone. A function f between posets P and Q is monotone if, for all elements x, y of P, x ≤ y (in P) implies f(x) ≤ f(y) (in Q). Other names for this property are isotone and order-preserving. In analysis, in the presence of total orders, such functions are often called monotonically increasing, but this is not a very convenient description when dealing with non-total orders. The dual notion is called antitone or order reversing.