Ordered inverse semigroups
HTML articles powered by AMS MathViewer
- by Tôru Saitô PDF
- Trans. Amer. Math. Soc. 153 (1971), 99-138 Request permission
Abstract:
In this paper, we consider two questions: one is to characterize the structure of ordered inverse semigroups and the other is to give a condition in order that an inverse semigroup is orderable. The solution of the first question is carried out in terms of three types of mappings. Two of these consist of mappings of an $\mathcal {R}$-class onto an $\mathcal {R}$-class, while one of these consists of mappings of a principal ideal of the semilattice $E$ constituted by idempotents onto a principal ideal of $E$. As for the second question, we give a theorem which extends a well-known result about groups that a group $G$ with the identity $e$ is orderable if and only if there exists a subsemigroup $P$ of $G$ such that $P \cup {P^{ - 1}} = G,P \cap {P^{ - 1}} = \{ e\}$ and $xP{x^{ - 1}} \subseteqq P$ for every $x \in G$.References
- A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
- Paul Conrad, Right-ordered groups, Michigan Math. J. 6 (1959), 267–275. MR 106954
- Tôru Saitô, Ordered idempotent semigroups, J. Math. Soc. Japan 14 (1962), 150–169. MR 144993, DOI 10.2969/jmsj/01420150
- Tôru Saitô, Regular elements in an ordered semigroup, Pacific J. Math. 13 (1963), 263–295. MR 152598, DOI 10.2140/pjm.1963.13.263
- Tôru Saitô, Proper ordered inverse semigroups, Pacific J. Math. 15 (1965), 649–666. MR 191977, DOI 10.2140/pjm.1965.15.649
- D. M. Smirnov, One-sided orders in groups with ascending central series, Algebra i Logika Sem. 6 (1967), no. 2, 77–88 (Russian, with English summary). MR 0214522
- James Wiegold, Semigroup coverings of groups. I, II, Mat.-Fyz. C̆asopis Sloven. Akad. Vied 11 (1961), 3-13; ibid. 12 (1961), 217–223 (English, with Russian summary). MR 0158946
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 153 (1971), 99-138
- MSC: Primary 06.70
- DOI: https://doi.org/10.1090/S0002-9947-1971-0270990-5
- MathSciNet review: 0270990