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<head><title>OrderedMonoid -- the class of all ordered monoids</title>
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<div><h1>OrderedMonoid -- the class of all ordered monoids</h1>
<div class="single"><h2>Description</h2>
<div>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 <a href="___Ordered__Monoid.html" title="the class of all ordered monoids">OrderedMonoid</a>.<p/>
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 <a href="___Polynomial__Ring.html" title="the class of all ordered monoid rings">PolynomialRing</a>.<p/>
A free commutative ordered monoid can be created with <a href="_monoid.html" title="make or retrieve a monoid">monoid</a>.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Monoid.html" title="the class of all monoids">Monoid</a> -- the class of all monoids</span></li>
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<div class="waystouse"><h2>Types of ordered monoid :</h2>
<ul><li><span><a href="___General__Ordered__Monoid.html" title="the class of all ordered free commutative monoids">GeneralOrderedMonoid</a> -- the class of all ordered free commutative monoids</span></li>
</ul>
<h2>Methods that use an ordered monoid :</h2>
<ul><li><span>degreeLength(OrderedMonoid), see <span><a href="_degree__Length.html" title="the number of degrees">degreeLength</a> -- the number of degrees</span></span></li>
<li><span><a href="___Ring_sp__Ordered__Monoid.html" title="make a polynomial ring">Ring OrderedMonoid</a> -- make a polynomial ring</span></li>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Ordered__Monoid.html" title="the class of all ordered monoids">OrderedMonoid</a> is <span>a <a href="___Type.html">type</a></span>, with ancestor classes <a href="___Monoid.html" title="the class of all monoids">Monoid</a> < <a href="___Type.html" title="the class of all types">Type</a> < <a href="___Mutable__Hash__Table.html" title="the class of all mutable hash tables">MutableHashTable</a> < <a href="___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="___Thing.html" title="the class of all things">Thing</a>.</p>
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