Geodesic
The semantical hyperunification problem
Discussiones Mathematicae. General Algebra and Applications, Tome 21 (2001) no. 2, pp. 175-200

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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 F_τ(X) 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.
Keywords: hypersubstitution, fully invariant congruence relation, hyperunification problem 
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Denecke, Klaus; Koppitz, Jörg; Wismath, Shelly. The semantical hyperunification problem. Discussiones Mathematicae. General Algebra and Applications, Tome 21 (2001) no. 2, pp. 175-200. http://geodesic.mathdoc.fr/item/DMGAA_2001_21_2_a3/