aequationes mathematicae

, Volume 36, Issue 2–3, pp 251–267 | Cite as

Standard threads and distributivity

  • King-Tim Mak
  • Kermit Sigmon
Research Papers


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.

AMS (1980) subject classification

Primary 39B40 Secondary 22A30, 13J99 


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Copyright information

© Birkhäuser Verlag 1988

Authors and Affiliations

  • King-Tim Mak
    • 1
    • 2
  • Kermit Sigmon
    • 1
    • 2
  1. 1.Department of Information & Decision SciencesUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of FloridaGainesvilleUSA

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