Commutation of operations and its relationship with Menger and Mann superpositions

Fedir M. Sokhatsky

Discussiones Mathematicae - General Algebra and Applications (2004)

  • Volume: 24, Issue: 2, page 153-176
  • ISSN: 1509-9415

Abstract

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The article considers a problem from Trokhimenko paper [13] concerning the study of abstract properties of commutations of operations and their connection with the Menger and Mann superpositions. Namely, abstract characterizations of some classes of operation algebras, whose signature consists of arbitrary families of commutations of operations, Menger and Mann superpositions and their various connections are found. Some unsolved problems are given at the end of the article.

How to cite

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Fedir M. Sokhatsky. "Commutation of operations and its relationship with Menger and Mann superpositions." Discussiones Mathematicae - General Algebra and Applications 24.2 (2004): 153-176. <http://eudml.org/doc/287629>.

@article{FedirM2004,
abstract = {The article considers a problem from Trokhimenko paper [13] concerning the study of abstract properties of commutations of operations and their connection with the Menger and Mann superpositions. Namely, abstract characterizations of some classes of operation algebras, whose signature consists of arbitrary families of commutations of operations, Menger and Mann superpositions and their various connections are found. Some unsolved problems are given at the end of the article.},
author = {Fedir M. Sokhatsky},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {Menger superposition; Superassociativity; (unitary) Menger algebra; selektor; n-ary groupoid; (extented) Menger multisemigroup (of operations); commutation of an operation; unar (of commutations); Mann superposition; abstract characterization of Menger algebras; function algebra},
language = {eng},
number = {2},
pages = {153-176},
title = {Commutation of operations and its relationship with Menger and Mann superpositions},
url = {http://eudml.org/doc/287629},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Fedir M. Sokhatsky
TI - Commutation of operations and its relationship with Menger and Mann superpositions
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2004
VL - 24
IS - 2
SP - 153
EP - 176
AB - The article considers a problem from Trokhimenko paper [13] concerning the study of abstract properties of commutations of operations and their connection with the Menger and Mann superpositions. Namely, abstract characterizations of some classes of operation algebras, whose signature consists of arbitrary families of commutations of operations, Menger and Mann superpositions and their various connections are found. Some unsolved problems are given at the end of the article.
LA - eng
KW - Menger superposition; Superassociativity; (unitary) Menger algebra; selektor; n-ary groupoid; (extented) Menger multisemigroup (of operations); commutation of an operation; unar (of commutations); Mann superposition; abstract characterization of Menger algebras; function algebra
UR - http://eudml.org/doc/287629
ER -

References

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  1. [1] V.D. Belousov, Conjugate operations (Russian), 'Studies in General Algebra' (Russian), Akad. Nauk Moldav. SSR Kishinev (Chishinau) 1965, 37-52. 
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  5. [5] W.A. Dudek and V.S. Trokhimenko, Functional Menger P-algebras, Comm. Algebra 30 (2003), 5921-5931. Zbl1018.20057
  6. [6] K. Głazek, Morphisms of general algebras without fixed fundamental operations, 'General Algebra and Applications', Heldermann-Verlag, Berlin 1993, 89-112. Zbl0793.08006
  7. [7] K. Głazek, Algebras of Algebraic Operations and Morphisms of Algebraic System (Polish), Wydawnictwo Uniwersytetu Wroc awskiego, Wrocaw 1994 (146 pp.). 
  8. [8] A. Knoebel, Cayley-like representations are for all algebras, not morely groups, Algebra Universalis 46 (2001), 487-497. Zbl1058.08002
  9. [9] H. Mann, On orthogonal latin squares, Bull. Amer. Math. Soc. 50 (1944), 249-257. Zbl0060.32307
  10. [10] K. Menger, The algebra of functions: past, present and future, Rend. Mat. Appl. 20 (1961), 409-430. Zbl0113.03904
  11. [11] M.B. Schein and V.S. Trohimenko, Algebras of multiplace functions, Smigroup Forum 17 (1979), 1-64. Zbl0397.08001
  12. [12] F.N. Sokhatsky, An abstract characterization (2,n)-semigroups of n-ary operations (Russian), Mat. Issled. no. 65 (1982), 132-139. 
  13. [13] V.S. Trokhimenko, On algebras of binary operations (Russian), Mat. Issled. no. 24 (1972), 253-261. 
  14. [14] T. Yakubov, About (2,n)-semigroups of n-ary operations (Russian), Izvest. Akad. Nauk Moldav. SSR (Bul. Akad. Stiince RSS Moldaven) 1974, no. 1, 29-46. 
  15. [15] K.A. Zaretski, An abstract characterization of the bisemigroup of binaryoperations (Russian), Mat. Zametki 1 (1965), 525-530. 

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