On Presolid Varieties of Semirings
by
Hippolyte Hounnon
University of Potsdam, Germany
Coauthors: Prof. K. Denecke

Prehyperidentities in a given variety are identities which have the property that substituting the operation symbols which occur in those identities by any terms ( different from the variables ) of the appropriate arity the resulting identities are still satisfied in that variety. It is very natural to ask for the varieties such that every identity is satisfied as a prehyperidentity. Such varieties are called presolid. It turns out that the set of all presolid varieties of a given type forms a complete lattice, and this lattice is fully described by K.Denecke and J. Koppitz in the case of semigroups. Since semirings are important algebraic structures we will study this lattice in case of semirings. To do this we will give first some necessary conditions for semirings to be presolid. Secondly we will determine all presolid varieties of idempotent semirings and prove that in this case presolidity means solidity. Finally we will determine some presolid varieties of semirings which are not solid in particular the least one and the greatest one.

Date received: January 17, 2001