OrderedMonoid -- the class of all ordered monoids

OrderedMonoid -- the class of all ordered monoids

The type OrderedMonoid is a member of the class Type. Each object of class OrderedMonoid is called an ordered monoid. Each ordered monoid is also a member of class Monoid.

More general types (whose methods may also apply) :

Monoid -- the class of all monoids

Type -- the class of all types

MutableHashTable -- the class of all mutable hash tables

HashTable -- the class of all hash tables

Types of ordered monoid :

GeneralOrderedMonoid -- the class of all ordered free commutative monoids

An ordered monoid is a multiplicative monoid together with an ordering of its elements. The ordering is required to be compatible with the multiplication in the sense that if x < y then x z < y z. The class of all ordered monomials is OrderedMonoid.

The reason for making a separate class for ordered monoids is that monoid rings can be implemented more efficiently for them - an element of the monoid ring can be stored as a sorted list, each element of which is a pair consisting of an element of the monoid and a coefficient. See PolynomialRing.

A free commutative ordered monoid can be created with monoid.

See also:

Monoid -- the class of all monoids

group -- make a group (monoid with inverses)

Methods for using an ordered monoid :

degreeLength OrderedMonoid

Ring OrderedMonoid -- make a polynomial ring