SOLID VARIETIES OF SEMIRINGS

In [5] the concept of an ID-semiring (S;+,.) as an algebra of type (2,2) where (S;+) and (S;.) are idempotent semigroups (bands) and where four distributive laws: xy + z ≈ (x + z)(y + z), x + yz ≈ (x + y)(x + z), x(y + z) ≈ xy + xz, (x + y)z ≈ xz + xz are satisfied, was introduced. An ID-semiring (S; +,.) is normal if x + y + u + v ≈ x + u + y + v and xyuv ≈ xuyv are identities. In this paper we want to determine the least and the greatest solid variety of normal ID-semirings, i.e. varieties in which every identity is satisfied as a hyperidentity. The result is that the variety of all normal ID-semirings is solid. The variety RA_(2,2) of rectangular algebras of type (2,2), i.e. the variety generated by all projection algebras of type (2,2) is the least non-trivial solid variety of normal ID-semirings.