# On sets related to maximal clones

• Volume: 32, Issue: 1, page 101-114
• ISSN: 1509-9415

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## Abstract

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For an arbitrary h-ary relation ρ we are interested to express n-clone Polⁿρ in terms of some subsets of the set of all n-ary operations Oⁿ(A) on a finite set A, which are in general not clones but we can obtain Polⁿρ from these sets by using intersection and union. Therefore we specify the concept a function preserves a relation and moreover, we study the properties of this new concept and the connection between these sets and Polⁿρ. Particularly we study ${R}_{a̲,b}^{n,k}$ for arbitrary partial order relations, equivalence relations and central relations.

## How to cite

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Yeni Susanti, and Klaus Denecke. "On sets related to maximal clones." Discussiones Mathematicae - General Algebra and Applications 32.1 (2012): 101-114. <http://eudml.org/doc/270196>.

@article{YeniSusanti2012,
abstract = {For an arbitrary h-ary relation ρ we are interested to express n-clone Polⁿρ in terms of some subsets of the set of all n-ary operations Oⁿ(A) on a finite set A, which are in general not clones but we can obtain Polⁿρ from these sets by using intersection and union. Therefore we specify the concept a function preserves a relation and moreover, we study the properties of this new concept and the connection between these sets and Polⁿρ. Particularly we study $R_\{a̲,b\}^\{n,k\}$ for arbitrary partial order relations, equivalence relations and central relations.},
author = {Yeni Susanti, Klaus Denecke},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {operations preserving relations; clones; semigroups; -ary operations},
language = {eng},
number = {1},
pages = {101-114},
title = {On sets related to maximal clones},
url = {http://eudml.org/doc/270196},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Yeni Susanti
AU - Klaus Denecke
TI - On sets related to maximal clones
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2012
VL - 32
IS - 1
SP - 101
EP - 114
AB - For an arbitrary h-ary relation ρ we are interested to express n-clone Polⁿρ in terms of some subsets of the set of all n-ary operations Oⁿ(A) on a finite set A, which are in general not clones but we can obtain Polⁿρ from these sets by using intersection and union. Therefore we specify the concept a function preserves a relation and moreover, we study the properties of this new concept and the connection between these sets and Polⁿρ. Particularly we study $R_{a̲,b}^{n,k}$ for arbitrary partial order relations, equivalence relations and central relations.
LA - eng
KW - operations preserving relations; clones; semigroups; -ary operations
UR - http://eudml.org/doc/270196
ER -

## References

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1. [1] A. Fearnley, Clones on Three Elements Preserving a Binary Relation, Algebra Universalis 56 (2007) 165-177. doi: 10.1007/s00012-007-1985-5 Zbl1115.08004
2. [2] Á. Szendrei, Clones in Universal Algebra (Les Presses de L' Université de Montréal, 1986). Zbl0603.08004
3. [3] I.G. Rosenberg, Über die Funktionale Vollständigkeit in den Mehrwertigen Logiken, Rozpravy Ćeskoslovenské Akad. véd, Ser. Math. Nat. Sci. 80(1970) 3-93 Zbl0199.30201
4. [4] K. Denecke, D. Lau, R. Pöschel and D. Schweigert, Hyperidentities, Hyperequational Classes and Clone Congruences, Contributions to General Algebra 7, Verlag Hölder-Pichler-Tempsky, Wien (1991) 97-118 Zbl0759.08005
5. [5] K. Denecke and S.L. Wismath, Hyperidentities and Clones (Gordon and Breach Science Publisher, 2000). Zbl0960.08001
6. [6] K. Denecke and S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science (Chapman and Hall, 2002).
7. [7] K. Denecke and Y. Susanti, Semigroups of n-ary Operations on Finite Sets, in: Proceedings of International Conference on Algebra on Algebra 2010 Advances in Algebraic Structures, W. Hemakul, S. Wahyuni and P.W. Sy(Ed(s)), (World Scientific, 2012) 157-176. doi: 10.1142/9789814366311_0011 Zbl1264.20070
8. [8] K. Denecke and Y. Susanti, On Sets Related to Clones of Quasilinear Operations, in: Proceedings of the 6th SEAMS-GMU International Conference on Mathematics and Its Application 2011, S. Wahyuni, I.E. Wijayanti and D. Rosadi (Ed(s)), (University of Gadjah Mada, 2012) 145-158
9. [9] R. Butkote and K. Denecke, Semigroup Properties of Boolean Operations, Asian-Eur. J. Math. 1(2008) 157-176 Zbl1176.20060
10. [10] R. Butkote, Universal-algebraic and Semigroup-theoretical Properties of Boolean Operations (Dissertation Universität Potsdam, 2009).

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