# Standard threads and distributivity

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## Summary

On a given standard thread ( The operations * in question are shown to be generated by pairs of functions Those functions

*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 for*x ⩾ y*on a standard thread by$$\frac{y}{x}: = \min \{ w \in J|y = x \circ w\} .$$

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

*p*and*q*which generate operations * over which ∘ distributes are completely identified.## AMS (1980) subject classification

Primary 39B40 Secondary 22A30, 13J99## Preview

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## References

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## Copyright information

© Birkhäuser Verlag 1988