*-Continuous Idempotent Left Semirings and Their Ideal Completion

In this paper, we introduce two notions of continuity for idempotent left semirings, which are called *-continuity and D-continuity. Also, for a *-continuous idempotent left semiring, we introduce a notion of *-ideals. Then, we show that the set of *-ideals of a *-continuous idempotent left semiring forms a D-continuous idempotent left serniring and the construction satisfies a universal property.