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SOLID VARIETIES OF SEMIRINGS

    https://doi.org/10.1142/9789812792310_0006Cited by:1 (Source: Crossref)
    Abstract:

    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.

    Keywords:
    AMSC: 08B05, 08B15, 08A50, 16Y60