All presolid varieties of semirings
by
Hippolyte Hounnon

A semirings is an algebra with two binary operations + and · which satisfy the associative laws and two distributive laws. Prehyperidentities in a variety V are identities in V 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. Semirings are important structures in the Theory of Automata but there was no result concerning prehyperidentities for semirings. The application of the concept of prehyhyperidentities to varieties of semirings is a new method to study varieties of semirings. It is very natural to ask for 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. Now, we are going to prove that this lattice in the case of semirings is finite with 13 elements.

Date received: January 25, 2002