A hypersubstitution of a fixed type τ maps n-ary operation symbols of the type to n-ary terms of the type. Such a mapping induces a unique mapping defined on the set of all terms of type t. The kernel of this induced mapping is called the kernel of the hypersubstitution, and it is a fully invariant congruence relation on the (absolutely free) term algebra of the considered type ([2]). If V is a variety of type τ, we consider the composition of the natural homomorphism with the mapping induced by a hypersubstitution. The kernel of this mapping is called the semantical kernel of the hypersubstitution with respect to the given variety. If the pair (s,t) of terms belongs to the semantical kernel of a hypersubstitution, then this hypersubstitution equalizes s and t with respect to the variety. Generalizing the concept of a unifier, we define a semantical hyperunifier for a pair of terms with respect to a variety. The problem of finding a semantical hyperunifier with respect to a given variety for any two terms is then called the semantical hyperunification problem. We prove that the semantical kernel of a hypersubstitution is a fully invariant congruence relation on the absolutely free algebra of the given type. Using this kernel, we define three relations between sets of hypersubstitutions and sets of varieties and introduce the Galois correspondences induced by these relations. Then we apply these general concepts to varieties of semigroups.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1036,
author = {Klaus Denecke and J\"org Koppitz and Shelly Wismath},
title = {The semantical hyperunification problem},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {21},
year = {2001},
pages = {175-200},
zbl = {1008.08002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1036}
}
Klaus Denecke; Jörg Koppitz; Shelly Wismath. The semantical hyperunification problem. Discussiones Mathematicae - General Algebra and Applications, Tome 21 (2001) pp. 175-200. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1036/
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