Semigroup with Bi-ideals

This paper introduces strongly prime, semi prime, strongly irreducible and irreducible bi-ideals of semigroups. The space of strongly prime bi-ideals is studied topologically. We introduce the notation of (m,n) ideals of semigroups, (m,n) bi-ideals and generalized (m,n) bi-ideals and generalized (m,n) *-bi-ideals. We also characterize those semigroups for which each bi-ideal is strongly prime.