REMARKS ON IDEAL COMPLETION OF *-CONTINUOUS IDEMPOTENT LEFT SEMIRINGS

It has been reported that the set of ∗-ideals of a ∗-continuous idempotent left semiring forms a D-continuous idempotent left semiring and the construction satisfies a universal property. However, we have found a ∗-continuous idempotent left semiring that the set of its ∗-ideals never forms a D-continuous idempotent left semiring. In this paper, we give this example and show that the reported properties hold if a ∗-continuous idempotent left semiring satisfies left distributivity.