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All pre-solid varieties of semirings

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Denecke K. , Hounnon H.

2004

tom Vol. 37, nr 1

13--34

A semiring is an algebra with two binary associative operations 4- and o which satisfy two distributive laws. Single semirings as well as classes of semirings are important structures in Automata Theory. Nevertheless, not so much is known about varieties of semirings. An identity t w t' is called a pre-hyperidentity of a variety V of semirings if whenever the operation symbols occurring in t and in t' are replaced by binary terms different from variables, the identity which results, holds in V. A variety V of semirings is called pre-solid if every identity holds as a pre-hyperidentity in V. The set of all pre-solid varieties of semirings forms a complete sublattice of the lattice of all varieties of semirings. To get more insight into the lattice of all varieties of semirings we will give a complete characterization of the lattice of all pre-solid varieties of semirings.