Nd-solid varieties

Klaus Denecke; Prisana Glubudom

Discussiones Mathematicae - General Algebra and Applications (2007)

  • Volume: 27, Issue: 2, page 245-262
  • ISSN: 1509-9415

Abstract

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A non-deterministic hypersubstitution maps any operation symbol of a tree language of type τ to a set of trees of the same type, i.e. to a tree language. Non-deterministic hypersubstitutions can be extended to mappings which map tree languages to tree languages preserving the arities. We define the application of a non-deterministic hypersubstitution to an algebra of type τ and obtain a class of derived algebras. Non-deterministic hypersubstitutions can also be applied to equations of type τ. Formally, we obtain two closure operators which turn out to form a conjugate pair of completely additive closure operators. This allows us to use the theory of conjugate pairs of additive closure operators for a characterization of M-solid non-deterministic varieties of algebras. As an application we consider M-solid non-deterministic varieties of semigroups.

How to cite

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Klaus Denecke, and Prisana Glubudom. "Nd-solid varieties." Discussiones Mathematicae - General Algebra and Applications 27.2 (2007): 245-262. <http://eudml.org/doc/276954>.

@article{KlausDenecke2007,
abstract = {A non-deterministic hypersubstitution maps any operation symbol of a tree language of type τ to a set of trees of the same type, i.e. to a tree language. Non-deterministic hypersubstitutions can be extended to mappings which map tree languages to tree languages preserving the arities. We define the application of a non-deterministic hypersubstitution to an algebra of type τ and obtain a class of derived algebras. Non-deterministic hypersubstitutions can also be applied to equations of type τ. Formally, we obtain two closure operators which turn out to form a conjugate pair of completely additive closure operators. This allows us to use the theory of conjugate pairs of additive closure operators for a characterization of M-solid non-deterministic varieties of algebras. As an application we consider M-solid non-deterministic varieties of semigroups.},
author = {Klaus Denecke, Prisana Glubudom},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {Non-deterministic hypersubstitution; conjugate pair of additive closure operators; M-solid non-deterministic variety; nondeterministic hypersubstitution; solid nondeterministic variety; term algebra},
language = {eng},
number = {2},
pages = {245-262},
title = {Nd-solid varieties},
url = {http://eudml.org/doc/276954},
volume = {27},
year = {2007},
}

TY - JOUR
AU - Klaus Denecke
AU - Prisana Glubudom
TI - Nd-solid varieties
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2007
VL - 27
IS - 2
SP - 245
EP - 262
AB - A non-deterministic hypersubstitution maps any operation symbol of a tree language of type τ to a set of trees of the same type, i.e. to a tree language. Non-deterministic hypersubstitutions can be extended to mappings which map tree languages to tree languages preserving the arities. We define the application of a non-deterministic hypersubstitution to an algebra of type τ and obtain a class of derived algebras. Non-deterministic hypersubstitutions can also be applied to equations of type τ. Formally, we obtain two closure operators which turn out to form a conjugate pair of completely additive closure operators. This allows us to use the theory of conjugate pairs of additive closure operators for a characterization of M-solid non-deterministic varieties of algebras. As an application we consider M-solid non-deterministic varieties of semigroups.
LA - eng
KW - Non-deterministic hypersubstitution; conjugate pair of additive closure operators; M-solid non-deterministic variety; nondeterministic hypersubstitution; solid nondeterministic variety; term algebra
UR - http://eudml.org/doc/276954
ER -

References

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  1. [1] K. Denecke and S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall/CRC, Boca Raton, London, New York, Washington, D.C. 2002. 
  2. [2] K. Denecke, P. Glubudom and J. Koppitz, Power Clones and Non-Deterministic Hypersubstitutions, preprint 2005. Zbl1221.08002
  3. [3] F. Gécseg and M. Steinby, Tree Languages, pp. 1-68 in: Handbook of Formal Languages, Vol. 3, Chapter 1, Tree Languages, Springer-Verlag 1997. 
  4. [4] K. Denecke and J. Koppitz, M-solid Varieties of Algebras, Advances in Mathematics, Vol. 10, Springer 2006. Zbl1094.08001
  5. [5] S. Leeratanavalee, Weak hypersubstitutions, Thesis, University of Potsdam 2002. 
  6. [6] K. Menger, The algebra of functions: past, present, future, Rend. Mat. 20 (1961), 409-430. Zbl0113.03904
  7. [7] B.M. Schein, and V.S. Trokhimenko, Algebras of multiplace functions, Semigroup Forum 17 (1979), 1-64. Zbl0397.08001
  8. [8] W. Taylor, Abstract Clone Theory, Algebras and Orders, Kluwer Academic Publishers, Dordrecht, Boston, London (1993), 507-530. Zbl0792.08005

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