Discussiones Mathematicae General Algebra and Applications 24(2) (2004) 153-176
doi: 10.7151/dmgaa.1082

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COMMUTATION OF OPERATIONS AND ITS RELATIONSHIP WITH MENGER AND MANN SUPERPOSITIONS

Fedir M. Sokhatsky

Tymirazieva str., 27, apt. 6
Vinnytsia 21001, Ukraine

e-mail: fedir@vinnitsa.com

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.

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.

2000 Mathematics Subject Classification: 08A40, 08A55, 20N05, 20N15, 20M20, 20M30.

References

[1] V.D. Belousov, Conjugate operations (Russian), p 37-52 in: "Studies in General Algebra" (Russian), Akad. Nauk Moldav. SSR Kishinev (Chishinau) 1965.
[2] V.D. Belousov, Balanced identities in quasigroups, (Russian) Mat. Sb. (N.S.) 70 (112) (1966), 55-97.
[3] V.D. Belousov, Systems of orthogonal operations (Russian), Mat. Sb. (N.S.) 77 (119) (1968), 38-58.
[4] K. Denecke and P. Jampachon, N-solid varietes of free Menger algebras of rank n, Eastwest J. Math. 5 (2003), 81-88.
[5] W.A. Dudek and V.S. Trokhimenko, Functional Menger P-algebras, Comm. Algebra 30 (2003), 5921-5931.
[6] K. Głazek, Morphisms of general algebras without fixed fundamental operations, p. 89-112 in: ``General Algebra and Applications", Heldermann-Verlag, Berlin 1993.
[7] K. Głazek, Algebras of Algebraic Operations and Morphisms of Algebraic System (Polish), Wydawnictwo Uniwersytetu Wroc awskiego, Wrocaw 1994 (146 pp.).
[8]A. Knoebel, Cayley-like representations are for all algebras, not morely groups, Algebra Universalis 46 (2001), 487-497.
[9] H. Mann, On orthogonal latin squares, Bull. Amer. Math. Soc. 50 (1944), 249-257.
[10] K. Menger, The algebra of functions: past, present and future, Rend. Mat. Appl. 20 (1961), 409-430.
[11] M.B. Schein and V.S. Trohimenko, Algebras of multiplace functions, Smigroup Forum 17 (1979), 1-64.
[12] F.N. Sokhatsky, An abstract characterization (2,n)-semigroups of n-ary operations (Russian), Mat. Issled. no. 65 (1982), 132-139 .
[13] V.S. Trokhimenko, On algebras of binary operations (Russian), Mat. Issled. no. 24 (1972), 253-261.
[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] K.A. Zaretski, An abstract characterization of the bisemigroup of binaryoperations (Russian), Mat. Zametki 1 (1965), 525-530.

Received 14 July 2001
Revised 27 December 2001
Revised 3 August 2004
Revised 27 November 2004