Tang, Xilin Identities for a class of regular unary semigroups. (English) Zbl 1157.20036 Commun. Algebra 36, No. 7, 2487-2502 (2008). A semigroup \(S\) with an additional unary operation \(x\mapsto x^\circ\) is called a regular unary semigroup if \(S\) satisfies the identities (RU1) \(xx^\circ x=x\) and (RU2) \(x^\circ xx^\circ=x^\circ\). The author studies the following unary semigroup identities: (T) \((x_1^\circ\cdots x_n^\circ)^\circ=x_n^{\circ\circ}\cdots x_1^{\circ\circ}\), (LRB) \(xx^\circ yy^\circ xx^\circ=xx^\circ yy^\circ\), (RRB) \(x^\circ xy^\circ yx^\circ x=y^\circ yx^\circ x\). He shows that a regular unary semigroup \(S\) satisfies (T), (LRB) and (RRB) iff the set \(S^\circ=\{s^\circ\mid s\in S\}\) is an inverse subsemigroup containing a unique inverse of each element of \(S\) and gives various characterizations and properties of regular unary semigroups satisfying (LRB) and (RRB) or (LRB) and (T). Reviewer’s remark. The author writes (p. 2501) that it is still open whether or not (RU1) is a corollary of (RU2), (T), (LRB) and (RRB). However, it is clear that (RU2), (T), (LRB) and (RRB) hold in the 2-element semilattice \(\{0,1\}\) with the unary operation \(x\mapsto x^\circ\) defined by \(x^\circ=0\) for all \(x\), while (RU1) fails. Reviewer: Mikhail Volkov (Ekaterinburg) Cited in 2 ReviewsCited in 6 Documents MSC: 20M17 Regular semigroups 20M05 Free semigroups, generators and relations, word problems 20M10 General structure theory for semigroups 20M07 Varieties and pseudovarieties of semigroups Keywords:regular unary semigroups; unary semigroup identities; inverse transversals PDFBibTeX XMLCite \textit{X. Tang}, Commun. Algebra 36, No. 7, 2487--2502 (2008; Zbl 1157.20036) Full Text: DOI References: [1] Blyth T. S., Proc. Roy. Soc. Edinburgh 92 pp 253– (1982) · Zbl 0507.20026 · doi:10.1017/S0308210500032522 [2] Blyth T. S., Comm. Algebra 29 (2) pp 611– (2001) · Zbl 0988.20049 · doi:10.1081/AGB-100001527 [3] Blyth T. S., Comm. Algebra 29 (2) pp 799– (2001) · Zbl 0988.20050 · doi:10.1081/AGB-100001544 [4] Hall T. E., Bull. Austral. Math. Soc. 40 pp 59– (1989) · Zbl 0666.20028 · doi:10.1017/S000497270000349X [5] McAlister D. B., J. Algebra 51 pp 491– (1978) · Zbl 0391.20043 · doi:10.1016/0021-8693(78)90118-7 [6] McAlister D. B., Q. J. Math. Oxford 34 (2) pp 459– (1983) · Zbl 0537.20033 · doi:10.1093/qmath/34.4.459 [7] McFadden R., Proc. Roy. Soc. Edinburgh 91 pp 107– (1981) · Zbl 0503.20025 · doi:10.1017/S0308210500012671 [8] Pastijn F., Simon Stevin 56 pp 3– (1982) [9] Saito T., Semigroup Forum 33 pp 149– (1986) · Zbl 0581.20057 · doi:10.1007/BF02573188 [10] Saito T., Proc. Edinburgh Math. Soc. 32 pp 41– (1989) · Zbl 0647.20063 · doi:10.1017/S0013091500006891 [11] Tang X. L., Semigroup Forum 55 pp 24– (1997) · Zbl 0897.20040 · doi:10.1007/PL00005909 [12] Tang X. L., Comm. Algebra 23 (11) pp 4157– (1995) · Zbl 0834.20061 · doi:10.1080/00927879508825455 [13] Tang X. L., SEA. Bull. Math. 25 pp 321– (2001) [14] Wang L. M., Algebra Colloquium 7 (4) pp 441– (2000) [15] Wang L. M., Comm. Algebra 26 (4) pp 1243– (1998) · Zbl 0935.20054 · doi:10.1080/00927879808826196 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.