# monoid

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## monoid

[′mä‚nȯid]
(mathematics)
A semigroup which has an identity element.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## monoid

An operator * and a value x form a monoid if * is associative and x is its left and right identity.
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References in periodicals archive ?
* (R, *) is an monoid, with the identity element noted 1
A semiring is an algebraic structure where the additive structure in the definition of a ring has been changed from an Abelian group to a monoid. The analogues for modules of rings are called semimodules.
[R.sub.a] = (Sa [union] [Sa.sup.2]] is the smallest right ideal of an ordered commutative monoid S containing a, for all a [member of] S.
Let [summation]* be the free monoid over the alphabet set [summation] and a set of strings S [subset or equal to] [summation]*.
In [1] Iqbal gave a linear system for the reducible and irreducible words of the braid monoid [MB.sub.n], which leads to compute the Hilbert series of [MB.sub.n].
Margolis, "Semigroup identities in the monoid of two-by-two tropical matrices," Semigroup Forum, vol.
This product induces a monoid structure on the set SL(k) of (ambient) isotopy classes of k-string links.
Theorem 2.8 The structure {GI, x) is a monoid under the operation(aI)(bI) = abI for all a, b in the group (G, x) and [I.sup.2] = I.
elements of a ring A constitute a multiplicative monoid. If it is a group, A is called the division ring.
For noncommutative monoid, [mathematical expression not reproducible], it first computes [a.sup.t] and then its inversion ([a.sup.t]) and finally takes two multiplications in the underlying implementation.

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