The paper investigates associativity properties of operations derived from algorithms concerning interval filling sequences whose definition is due to Z. Daróczy, A. Járai, and I. Kátai [Acta Sci. Math. (Szeged) 50, 337-350 (1986; Zbl 0619.10007)].
A sequence \(\lambda=(\lambda_n)\) is called interval filling if \(L(\lambda):=\sum_{n=1}^\infty\lambda_n<\infty\) and, for any \(x\in I:=[0,L(\lambda)]\), there exists a sequence \(\delta_n\in\{0,1\}\) such that \(x=\sum_{n=1}^\infty \delta_n\lambda_n\). A sequence of functions \(\alpha_n:I\to\{0,1\}\) is called an algorithm with respect to \(\lambda\) if \(x=\sum_{n=1}^\infty \alpha_n(x)\lambda_n\) for all \(x\in I\).
Having an algorithm \(\alpha=(\alpha_n)\), an operation \(\circ:I\times I\to I\) can be defined in the following way: \[ x\circ y=\sum_{n=1}^\infty \alpha_n(x)\alpha_n(y)\lambda_n \qquad (x,y\in I). \] The first main result of the paper shows that \(\alpha\) is associative if and only if \(\alpha_n(x\circ y)=\alpha_n(x)\alpha_n(y)\) for all \(n\geq 1\), \(x,y\in I\). The second main result states that the regular algorithm \(\varepsilon=(\varepsilon_n)\) defined by the recursion \[ \varepsilon_n(x) =\begin{cases} 0, &\qquad\text{if } x<\sum_{i=1}^{n-1}\varepsilon_i(x)\lambda_i+\lambda_n, \\ 1, &\qquad\text{if } x\geq\sum_{i=1}^{n-1}\varepsilon_i(x)\lambda_i+\lambda_n, \end{cases} \qquad (n\in\mathbb{N}, x\in I) \] is associative for all interval filling sequences.