Abstract
We prove that for every archimedean positively ordered semigroup \(S\) with a maximal element, if \(S\) satisfies one minor condition, then \(S\) is isomorphic to a mixture of \(P_1\) and \(P_1^*\) in the notation of [2]. We also discuss a variation of metrizable spaces where we use a positively ordered semigroup to describe distances.
Similar content being viewed by others
References
Alimov, N.G.: On ordered semigroups. Izvestiya Akad. Nauk SSSR. Ser. Mat. 14, 569–576 (1950)
Clifford, A.H.: Naturally totally ordered commutative semigroups. Amer. J. Math. 76, 631–646 (1954). https://doi.org/10.2307/2372706
Fuchs, L.: Note on fully ordered semigroups. Acta Math. Acad. Sci. Hungar. 12, 255–259 (1961). https://doi.org/10.1007/BF02066688
Hofmann, K.H., Lawson, J.D.: Linearly ordered semigroups: historical origins and A. H. Clifford’s influence. In: Semigroup Theory and Its Applications (New Orleans, LA, 1994). London Math. Soc. Lecture Note Ser., vol. 231, pp. 15–39. Cambridge Univ. Press, Cambridge (1996)
Kowalski, O.: On archimedean positively fully ordered semigroups. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 8, 97–99 (1965)
Nyikos, P., Reichel, H.C.: On uniform spaces with linearly ordered bases. II. (\(\omega _{\mu }\)-metric spaces). Fund. Math. 93(1), 1–10 (1976). https://doi.org/10.4064/fm-93-1-1-10
Satyanarayana, M.: Positively Ordered Semigroups. Lecture Notes in Pure and Applied Mathematics, vol. 42. Marcel Dekker, New York (1979)
Sikorski, R.: Remarks on some topological spaces of high power. Fund. Math. 37, 125–136 (1950). https://doi.org/10.4064/fm-37-1-125-136
Stevenson, F.W., Thron, W.J.: Results on \(\omega _{\mu }\)-metric spaces. Fund. Math. 65, 317–324 (1969). https://doi.org/10.4064/fm-65-3-317-324
Acknowledgements
The author thanks Miami University for support via the faculty research grant program.
He is also grateful to the anonymous referee for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jimmie D. Lawson.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ishiu, T. Archimedean positively ordered semigroups with maximal elements. Semigroup Forum 105, 244–264 (2022). https://doi.org/10.1007/s00233-022-10297-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-022-10297-z