Discussiones Mathematicae General Algebra and Applications 21(2) (2001)175-200
doi: 10.7151/dmgaa.1036

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THE SEMANTICAL HYPERUNIFICATION PROBLEM

Klaus Denecke, Jörg Koppitz

University of Potsdam, Institute of Mathematics,
Am Neuen Palais,
14415 Potsdam, Germany
e-mail:
kdenecke@rz.uni-potsdam.de

Shelly Wismath

Department of Mathematics and C.S., University of Lethbridge,
Lethbridge, Ab., Canada T1K-3M4
e-mail:
wismaths@cs.uleth.ca

Abstract

A hypersubstitution of a fixed type t 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 Ft(X) of the considered type ([2]). If V is a variety of type t, 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.

Keywords: hypersubstitution, fully invariant congruence relation, hyperunification problem.

2000 Mathematics Subject Classification:  08B15, 08B25.

References

[1]K. Denecke, J. Hyndman and S.L. Wismath, The Galois correspondence between subvariety lattices and monoids of hypersubstitutions, Discuss.Math. - Gen. Algebra Appl. 20 (2000), 21-36.
[2]K. Denecke, J. Koppitz and St. Niwczyk, Equational Theories generated by Hypersubstitutions of Type (n), Internat. J. Algebra Comput., in print.
[3]K. Denecke and S.L. Wismath, The monoid of hypersubstitutions of type (2), Contributions to General Algebra 10 (1998), 109-126.
[4]K. Denecke and S.L. Wismath, Hyperidentities and Clones, Gordon and Breach Sci. Publ., Amsterdam 2000.
[5]J. Płonka, Proper and inner hypersubstitutions of varieties, p. 106-116 in: ``Proccedings of the International Conference `Summer School on Algebra and Ordered Sets', Olomouc 1994," Palacký University, Olomouc 1994.
[6]J.H. Siekmann, Universal Unification, p. 1-42 in: Lecture Notes in Computer Science, no. 170 (``International Conference on Automated Deductions (Napa, CA, 1984)"), Springer-Verlag, Berlin 1984.

Received 16 March 2001
Revised 8 August 2001