If (M, +,0) is a

commutative monoid, then its group completion Gp(M) can be described as the quotient M x M/~, where (m, n) ~ (p, q) when there exists k [member of]m such that

If (S, [cross product], 1) is a

commutative monoid then S is called a commutative semiring.

1]-ring to be a

commutative monoid with an absorbing element 0.

1) is a

commutative monoid and the following conditions hold for all x, y, z [member of] A,

We have shown that the set of all fuzzy interior ideals of a left regular ordered LA -semigroup with left identity forms a

commutative monoid.

I] is also a

commutative monoid with multiplication [*.

That is, JX is the free k-module on the

commutative monoid under coproduct of isomorphism classes of objects of [epsilon](X).

a) (A, *, [less than or equal to]) is a partially ordered

commutative monoid with a greatest element 1 where x [less than or equal to] y if and only if x [right arrow] y =1.

A rig (or semiring) is a ring without negatives: a set equipped with a

commutative monoid structure (+, 0) and a monoid structure (x, 1), the latter distributing over the former.

This is the left regular band analogue of free partially

commutative monoids (also called trace monoids or graph monoids [18, 14]) and of free partially commutative groups (also called right-angled Artin groups or graph groups [31]).

Trace monoids are obtained from free monoids by allowing certain pairs of generators to commute, which is the reason why they are also known as free partially

commutative monoids.

Garcia-Sanchez, Finitely generated

commutative monoids.