Summary
Certain t* elements of an abstract algebra are called independent if every equation satisfied by these elements is identically true in the algebra [2]. For finite algebras we have: Given an integert*>3, everyt* elements are independent if every operation oft* variables is trivial, i. e. if it is identically equal to one of the variables (Th. 1). Ift*≤3, then there exists an algebra in which everyt* elements are independent but not every operation oft* variables is trivial; moreover ift*=3, then n is the number of elements of such an algebra if n=2 or 4 (mod 6 and n>3 (Th. 2).
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References
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Swierczkowski, S. On two numerical costants associated with finite algebras. Annali di Matematica 61, 241–245 (1963). https://doi.org/10.1007/BF02410653
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DOI: https://doi.org/10.1007/BF02410653