Standard threads and distributivity

Summary

On a given standard thread (J, ∘), all operations * over which ∘ distributes are determined and among such operations those which are continuous are identified. A standard thread is a topological semigroup on a closed real number interval whose largest element is an identity and smallest element is a zero for the semigroup. A quotient operation can be defined forx ⩾ y on a standard thread by

$$\frac{y}{x}: = \min \{ w \in J|y = x \circ w\} .$$

The operations * in question are shown to be generated by pairs of functionsp, q:J → J such that

$$x * y = \left\{ {\begin{array}{*{20}c} {x \circ q\left( {\frac{y}{x}} \right) if x \geqslant y} \\ {p\left( {\frac{x}{y}} \right) \circ y if x \leqslant y.} \\ \end{array} } \right.$$

Those functionsp andq which generate operations * over which ∘ distributes are completely identified.