On additive semigroups of idempotent semirings with identity

The notion of an admissible semigroup is introduced. This is an (additive) semigroup on which a multiplication turning it into a multiplicatively idempotent semiring with identity can be defined. Admissible semilattices are studied. In particular, countable admissible and inadmissible “universal” semilattices are constructed, into which all finite semilattices are isomorphically embedded.