Cycloidal algebras

Rok, strany: 2013, 33 - 40

In this paper we introduce for an arbitrary algebra (groupoid, binary system) $(X;*)$ a sequence of algebras $(X;*)_n=(X;\circ)$, where $x\circ y=[x*y]_n=x*[x*y]_n-1$, $[x*y]₀=y$. For several classes of examples we study the cycloidal index $(m,n)$ of $(X;*)$, where $(X;*)_m=(X;*)_n$ for $m>n$ and $m$ is minimal with this property. We show that $(X;*)$ satisfies the left cancellation law, then if $(X;*)_m=(X;*)_n$, then also $(X;*)_m-n=(X;*)₀$, the right zero semigroup. Finite algebras are shown to have cycloidal indices (as expected). $B$-algebras are considered in greater detail. For commutative rings $R$ with identity, $x*y=ax+by+c$, $a,b,c\in\mathbb{R}$ defines a linear product and for such linear products the commutativity condition $[x*y]_n=[y*x]_n$ is observed to be related to the golden section, the classical one obtained for $\mathbb{R}$, the real numbers, $n=2$ and $a=1$ as the coefficient $b$.

ISO 690:

Han, J., Kim, H., Neggers, J. 2013. Cycloidal algebras. In Mathematica Slovaca, vol. 63, no.1, pp. 33-40. 0139-9918. DOI: https://doi.org/10.2478/s12175-012-0079-9

APA:

Han, J., Kim, H., Neggers, J. (2013). Cycloidal algebras. Mathematica Slovaca, 63(1), 33-40. 0139-9918. DOI: https://doi.org/10.2478/s12175-012-0079-9