Factorization of operations of medial and abelian algebras.
DOI:
https://doi.org/10.31558/1817-2237.2017.1-2.7Keywords:
mediality, medial law, medial algebras, algebra of endomorphisms, Abelian universal algebraAbstract
Let A be an m x n matrix of variables. An n-ary operation f and m-ary operation g are said to satisfy the medial law if two results are the same: 1) an application of f to the rows of A then an application of g to the obtained column and 2) an application of g to the columns of A then an application of f to the obtained row. A universal algebra (A; Ω) is called: medial if every two operations from Ω satisfy the medial law; abelian if it is medial and has a one-element subalgebra. Criteria for being medial and for being Abelian are found for universal algebras (A; Ω) which have 0 Q and f Ω such that the term f(x0,…, xn) defines a quasigroup operation if all variables are 0 except xi and xp and it defines a permutation of Q if all variables are f(0,…,0) except xi or except xp for some different i, p.
References
J. Aczel, J. Dhombres Functional equations in several variables. Cambridge University Press.- 1989 - 462p.
J. Aczel, V. D. Belousov, M. Hosszu, Generalized associativity and bisymmetry on quasigroups,
Acta Math. Acad. Sci. Hungar. 11 (1960), 127-136.
Belousov V.D. n-ary quasigroups. Kishinev: Stiintsa, 1972, 225 p. (in russian)
Ehsani A, Movsisyan Yu. M. Linear representation of medial-like algebras. Comm Algebra, 2013, 41, 3429–44.
Ehsani A., Characterization of regular medial algebras. ScienceAsia 40 (2014): 175–181.
Kurosh A.G. General algebra. Lectures of 1969-1970 academic years. Moscou: Nauka, 1974, 159 p. (in russian)
F.N. Sokhatskii (F.M. Sokhatsky) About of group isotopes II., Ukrainian Math. J., 47(12), 1995, pp. 1935-1948.
F.M. Sokhatsky About mediality of universal algebras and cross isotopes of groups, Reports of the National Academy of Sciences of Ukraine, 2006. No 11. P. 29-35. (in Ukrainian)
