A semiring  is an algebra (S, +, *, 0) such that (S, +, 0) is a commutative monoid
, multiplication is associative and distributes over addition from both sides, and 0 is a zero element with respect to multiplication.
[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.
If (S, [cross product], 1) is a commutative monoid
then S is called a commutative semiring.
Define an [F.sub.1]-ring to be a commutative monoid
with an absorbing element 0.
Definition 2.1. A BL-algebra is an algebra (A, [disjunction], [and], [??], [right arrow], 0,1) of type (2, 2, 2, 2,0, 0) such that (A, [disjunction], [and], 0,1) is a bounded lattice, (A, [??], 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
. Further, we have characterized a left regular ordered LA -semigroup by using the properties of fuzzy interior ideals, and give some equivalent statements for an ordered LA-semigroup to become a left regular ordered LA-semigroup.
A monoid ([direct sum], [Z.sub.[direct sum]]) may be a commutative monoid
(i.e., when [direct sum] is commutative) or an idempotent monoid (i.e., when [inverted] A x : x [direct sum] x = x), or both.
Definition 5 Let R be an abelian Hopf monoid, such that each [R.sup.I] is also a commutative monoid
with multiplication [*.sub.I] and identity [1.sub.I].
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.