Agreeable semigroups. (English) Zbl 1041.20043
This paper continues the investigation of \(RC\)-semigroups introduced by the authors [in Semigroup Forum 62, No. 2, 279-310 (2001; Zbl 0982.20051)]. “Spiritually”, this paper is close to previous investigations of B. Schweizer and A. Sklar [e.g., Bull. Am. Math. Soc. 73, 510-515 (1967; Zbl 0217.01703)].
“Agreeable semigroups” are structures with an additional binary operation \(*\) such that, if \(a\) and \(b\) are elements of the semigroup, then \(a*b\) is the identity partial transformation on the domain of \(a\cap b\). Various properties of such systems are found.
The authors do not mention a substantial corpus of research papers (mainly in the 1960ies and 1970ies) on the same or very similar subjects that provide a complete axiomatization of the natural analog of the equivalence relation \(\overline{\mathcal L}\) on arbitrary semigroups of partial transformations and binary relations, the “\(C\)-order”, etc. Semigroups of partial transformations closed under an additional binary operation \(\wedge\) of the set-theoretical intersection were characterized too in the seventies. A full list of references would be too long to give here but an interested reader may want to look up a survey article by the reviewer [Semigroup Forum 1, 1-62 (1970; Zbl 0197.29404)], also [Izv. Vyssh. Uchebn. Zaved., Mat. 1970, No. 4(95), 91-102 (1970; Zbl 0199.33501)] and various papers on the so-called “restrictive” semigroups and bisemigroups by V. V. Wagner and especially his students.
“Agreeable semigroups” are structures with an additional binary operation \(*\) such that, if \(a\) and \(b\) are elements of the semigroup, then \(a*b\) is the identity partial transformation on the domain of \(a\cap b\). Various properties of such systems are found.
The authors do not mention a substantial corpus of research papers (mainly in the 1960ies and 1970ies) on the same or very similar subjects that provide a complete axiomatization of the natural analog of the equivalence relation \(\overline{\mathcal L}\) on arbitrary semigroups of partial transformations and binary relations, the “\(C\)-order”, etc. Semigroups of partial transformations closed under an additional binary operation \(\wedge\) of the set-theoretical intersection were characterized too in the seventies. A full list of references would be too long to give here but an interested reader may want to look up a survey article by the reviewer [Semigroup Forum 1, 1-62 (1970; Zbl 0197.29404)], also [Izv. Vyssh. Uchebn. Zaved., Mat. 1970, No. 4(95), 91-102 (1970; Zbl 0199.33501)] and various papers on the so-called “restrictive” semigroups and bisemigroups by V. V. Wagner and especially his students.
Reviewer: Boris M. Schein (Fayetteville)
MSC:
| 20M20 | Semigroups of transformations, relations, partitions, etc. |
| 20M07 | Varieties and pseudovarieties of semigroups |
References:
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