Rho-solid varieties of semigroups
by
Jörg Koppitz
Universität Potsdam

A hypersubstitution is a mapping \sigma from the set of operation symbols into the set of terms preserving the arity. Any hypersubstitution \sigma can be extended to a mapping ^\sigma from the set of terms into the set of terms. A variety V is said to be solid iff the application of ^\sigma for any hypersubstitution \sigma to any identity in V gives again an identity in V.

Sometimes one is interested in other mappings from the set of terms in the set of terms, but these mappings should be related to the hypesustitutions. In particular, this is the case in the computer sciences. Here we often substitute the operation symbol by terms in different way. Following this idea we define mappings \rho, which assigns to each hypersubstitution \sigma a mapping \rho(\sigma) from the set of terms into the set of terms. A variety V is said to be \rho-solid iff the application of \rho(\sigma) to any identity in V gives again an identity in the variety.
If \rho is the mapping, which assigns to any hypersubstitution \sigma its extension ^\sigma, so we get the usual solidity. Besides this example we will present particular such mappings \rho which play an important role in the computer sciences. For these mappings \rho we characterize the class of all \rho-solid varieties of semigroups.

Date received: April 30, 2004