Means compatible with semigroup laws
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2020-03-31 13:54
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PADMANABHAN, Ranganathan, SHUKLA, Alok. Means compatible with semigroup laws. In: Quasigroups and Related Systems, 2019, vol. 27, nr. 2(42), pp. 317-324. ISSN 1561-2848.
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Quasigroups and Related Systems
Volumul 27, Numărul 2(42) / 2019 / ISSN 1561-2848

Means compatible with semigroup laws

CZU: 515.142.22+514.172.45

Pag. 317-324

Padmanabhan Ranganathan, Shukla Alok
 
University of Manitoba
 
 
Disponibil în IBN: 25 martie 2020


Rezumat

A binary mean operation m(x; y) is said to be compatible with a semigroup law , if  satises the Gauss' functional equation m(x; y) m(x; y) = x  y for all x; y. Thus the arithmetic mean is compatible with the group addition in the set of real numbers, while the geometric mean is compatible with the group multiplication in the set of all positive real numbers. Using one of the Jacobi theta functions, Tanimoto [6], [7] has constructed a novel binary operation  compatible with the arithmetico-geometric mean agm(x; y) of Gauss. Tanimoto shows that it is only a loop operation, but not associative. A natural question is to ask if there exists a group law  compatible with arithmetic-geometric mean. In this paper we prove that there is no semigroup law compatible with agm and hence, in particular, no group law either. Among other things, this explains why Tanimoto's operation  using theta functions must be non-associative.