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All pre-solid varieties of semirings

100%

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.

Binary relations on the monoid of V-proper hypersubstitutions

100%

Denecke K. , Srithus R.

Discussiones Mathematicae - General Algebra and Applications

2006

tom 26

nr 2

233-251

In this paper we consider different relations on the set P(V) of all proper hypersubstitutions with respect to a given variety V and their properties. Using these relations we introduce the cardinalities of the corresponding quotient sets as degrees and determine the properties of solid varieties having given degrees. Finally, for all varieties of bands we determine their degrees.

The greatest regular-solid variety of semigroups % Dedicated to R. McKenzie's 60$^th$ birthday %

88%

Denecke K. , Koppitz J. , Pabhapote N.

Discussiones Mathematicae - General Algebra and Applications

2008

tom 28

nr 1

91-119

A regular hypersubstitution is a mapping which takes every $n_i$-ary operation symbol to an $n_i$-ary term. A variety is called regular-solid if it contains all algebras derived by regular hypersubstitutions. We determine the greatest regular-solid variety of semigroups. This result will be used to give a new proof for the equational description of the greatest solid variety of semigroups. We show that every variety of semigroups which is finitely based by hyperidentities is also finitely based by identities.