Cancellation in semigroups in which \(x^ 2=x^ 3\). (English) Zbl 0695.20033
The author presents a discussion of the semigroup B(k,m,n), which is the free semigroup generated by k elements and satisfying \(x^ m=x^ n\), where m is a nonnegative integer and n is an integer which is greater than m. Results relating to cancellation in the semigroup which is the union of B(k,2,3) for \(k=1,2,..\). are discussed in detail.
Reviewer: J.A.Hildebrant
MSC:
20M05 | Free semigroups, generators and relations, word problems |
20M07 | Varieties and pseudovarieties of semigroups |
Keywords:
locally finite; word problem; variety of semigroups; generators; finite semigroup; alphabet; word length; golden mean; free semigroup; cancellationReferences:
[1] | Brown, T. C.,A semigroup union of disjoint locally finite subsemigroups which is not locally finite, Pacific J. of Math. 22 (1967), 11–14. · Zbl 0189.02101 |
[2] | Brzozowski, J. A., K. Culik II, and A. Gabrielian,Classification of noncounting events, J. Comput. Syst. Sci. 5 (1971), 243–271. · Zbl 0241.94050 · doi:10.1016/S0022-0000(71)80006-5 |
[3] | Dean, R.,A sequence without repeats on x, x, y, y, Am. Math. Mon. 72 (1965), 383–385. · Zbl 0135.01301 · doi:10.2307/2313498 |
[4] | Dejean, F.,Sur un théorème de Thue, J. Comb. Theory A13 (1972), 90–99. · Zbl 0245.20052 · doi:10.1016/0097-3165(72)90011-8 |
[5] | Gerhard, J. A.,The word problem for semigroups satisfying x3=x, Math. Proc. Camb. Phil. Soc. 84 (1978), 11–19. · Zbl 0384.20040 · doi:10.1017/S0305004100054827 |
[6] | Green, J. A., and D. Rees,On semigroups in which xr=x, Proc. Camb. Philos. Soc. 48 (1952), 35–40. · Zbl 0046.01903 · doi:10.1017/S0305004100027341 |
[7] | Lallement, G.,Semigroups and Combinatorial Applications, Wiley-Interscience, New York, 1979. · Zbl 0421.20025 |
[8] | Leech, L.,A problem on strings of beads, Math. Gaz. 41 (1957), 277–278. · Zbl 0079.01101 · doi:10.2307/3610126 |
[9] | Morse, M., and G. A. Hedlund,Unending chess, symbolic dynamics and a problem in semigroups, Duke Math. J. 11 (1944), 1–7. · Zbl 0063.04115 · doi:10.1215/S0012-7094-44-01101-4 |
[10] | Shevrin, L. N., personal communication, 1967. |
[11] | Thue, A.,Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Skr. Vid. Kristiania, I Mat. Naturv. Klasse 8 (1912), 1–67. · JFM 44.0462.01 |
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.