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Betweenness and order in semigroups

Published online by Cambridge University Press:  24 October 2008

J. Gilder
J. Gilder
Affiliation:
College of Science and Technology, Manchester
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Extract

A betweenness semigroup is a semigroup possessing a ternary relation, ‘b lies between a and c’, which is invariant under the semigroup operation. We take as our axioms of betweenness those suggested by Shepperd in (2) and given below in 1·01–1·04. It was shown in (2) that these axioms lead to just two types of betweenness; namely a linear betweenness corresponding to the order of points on an undirected straight line or a cyclic betweenness of just four points on a circle. In (3), Shepperd characterized betweenness groups and we here generalize his results to betweenness semigroups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 1 , January 1965 , pp. 13 - 28
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

REFERENCES

(1)Clifford, A. H.Ordered commutative semigroups of the second kind. Proc. Amer. Math. Soc. 9 (1958), 682–687.CrossRefGoogle Scholar
(2)Shepperd, J. A. H.Transitivities of betweenness and separation and the definition of betweenness and separation groups. J. London Math. Soc. 31 (1956), 240–248.CrossRefGoogle Scholar
(3)Shepperd, J. A. H.Betweenness groups. J. London Math. Soc. 32 (1957), 277–285.CrossRefGoogle Scholar