The Fibonacci sequence

$\left\{{F}_{n}\right\}$ is defined as follows:

${F}_{0}=0$,

${F}_{1}=1$,

${F}_{k}={F}_{k-1}+{F}_{k-2}$ for

$k\ge 2$. A well-known theorem, due to Zeckendorf, states that every natural number has a unique representation as a sum of distinct Fibonacci numbers, if we stipulate that

${F}_{0}$ and

${F}_{1}$ are not used in the representation and that if

${F}_{a}$ and

${F}_{b}$ are used then

$|a-b|>1\xb7$ If the Zeckendorf representations of m and n are

$m={F}_{jq}+\xb7\xb7\xb7+{F}_{j1}$ and

$n={F}_{kr}+\xb7\xb7\xb7+{F}_{k1}$, then the “circle product” of m and n is defined as follows:

$m\circ n={\sum}_{b=1}^{q}{\sum}_{c=1}^{r}{F}_{jb+kc}\xb7$ In particular,

${F}_{j}\circ {F}_{k}={F}_{j+k}$ if

$j\ge 2$ and

$k\ge 2$. It is proved in this paper that circle multiplication is an associative operation.