Regular elements and Green's relations in Menger algebras of terms

Klaus Denecke; Prakit Jampachon

Discussiones Mathematicae - General Algebra and Applications (2006)

  • Volume: 26, Issue: 1, page 85-109
  • ISSN: 1509-9415

Abstract

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Defining an (n+1)-ary superposition operation S n on the set W τ ( X n ) of all n-ary terms of type τ, one obtains an algebra n - c l o n e τ : = ( W τ ( X n ) ; S n , x 1 , . . . , x n ) of type (n+1,0,...,0). The algebra n-clone τ is free in the variety of all Menger algebras ([9]). Using the operation S n there are different possibilities to define binary associative operations on the set W τ ( X n ) and on the cartesian power W τ ( X n ) n . In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative operations as fundamental operations.

How to cite

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Klaus Denecke, and Prakit Jampachon. "Regular elements and Green's relations in Menger algebras of terms." Discussiones Mathematicae - General Algebra and Applications 26.1 (2006): 85-109. <http://eudml.org/doc/276840>.

@article{KlausDenecke2006,
abstract = {Defining an (n+1)-ary superposition operation $S^n$ on the set $W_\{τ\}(X_n)$ of all n-ary terms of type τ, one obtains an algebra $n-clone τ := (W_\{τ\}(X_n); S^n, x_1, ..., x_n)$ of type (n+1,0,...,0). The algebra n-clone τ is free in the variety of all Menger algebras ([9]). Using the operation $S^n$ there are different possibilities to define binary associative operations on the set $W_\{τ\}(X_n)$ and on the cartesian power $W_\{τ\}(X_n)^n$. In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative operations as fundamental operations.},
author = {Klaus Denecke, Prakit Jampachon},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {term; superposition of terms; Menger algebra; regular element; Green's relations},
language = {eng},
number = {1},
pages = {85-109},
title = {Regular elements and Green's relations in Menger algebras of terms},
url = {http://eudml.org/doc/276840},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Klaus Denecke
AU - Prakit Jampachon
TI - Regular elements and Green's relations in Menger algebras of terms
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2006
VL - 26
IS - 1
SP - 85
EP - 109
AB - Defining an (n+1)-ary superposition operation $S^n$ on the set $W_{τ}(X_n)$ of all n-ary terms of type τ, one obtains an algebra $n-clone τ := (W_{τ}(X_n); S^n, x_1, ..., x_n)$ of type (n+1,0,...,0). The algebra n-clone τ is free in the variety of all Menger algebras ([9]). Using the operation $S^n$ there are different possibilities to define binary associative operations on the set $W_{τ}(X_n)$ and on the cartesian power $W_{τ}(X_n)^n$. In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative operations as fundamental operations.
LA - eng
KW - term; superposition of terms; Menger algebra; regular element; Green's relations
UR - http://eudml.org/doc/276840
ER -

References

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  1. [1] K. Denecke, Stongly Solid Varieties and Free Generalized Clones, Kyungpook Math. J. 45 (2005), 33-43. Zbl1156.08002
  2. [2] 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. 
  3. [3] K. Denecke and S.L. Wismath, Complexity of Terms, Composition and Hypersubstitution, Int. J. Math. Math. Sci. 15 (2003), 959-969. Zbl1015.08005
  4. [4] K. Denecke and P. Jampachon, N-solid varieties and free Menger algebras of rank n, East-West Journal of Mathematics 5 (1) (2003), 81-88. Zbl1083.08005
  5. [5] K. Denecke and P. Jampachon, Clones of Full Terms, Algebra Discrete Math. 4 (2004), 1-11. Zbl1091.08003
  6. [6] K. Denecke and J. Koppitz, M-solid Varieties of Algebras, Advances in Mathematics, Springer Science+Business Media, Inc., 2006. Zbl1094.08001
  7. [7] J.M. Howie, Fundamenntals of Semigroup Theory, Oxford Science Publications, Clarendon Press, Oxford 1995. 
  8. [8] K. Menger, The algebra of functions: past, present, future, Rend. Mat. 20 (1961), 409-430. Zbl0113.03904
  9. [9] B.M. Schein and V.S. Trohimenko, Algebras of multiplace functions, Semigroup Forum 17 (1979), 1-64. 
  10. [10] V.S. Trohimenko, v-regular Menger algebras, Algebra Univers. 38 (1997), 150-164. 

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