One candidate for e that exists in every semiring is the multiplicative identity 1.
Clearly all the properties of a semiring are satisfied where 0 is the additive identity and + 1 is multiplicative identity. Here + 1 and -1 are the units of S.
In mathematics, a multiplicative inverse for a number X is a number that when multiplied by X yields the multiplicative identity
By (1) and (2) [u.sub.e] is a multiplicative identity of A [[??].sup.[sigma].sub.[alpha]] G and by (3) the multiplication on A [[??].sup.[sigma].sub.[alpha]] G is associative.
By (7), (8), and (9) it follows that A [[??].sup.[sigma].sub.[alpha]] G has a multiplicative identity if and only if ob(G) is finite; in that case the multiplicative identity is [[summation].sub.e[member of]ob(G)] [u.sub.e].
The multiplicative identity
property keeps quant ities the same: 1 x 10) = 10.