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http://dx.doi.org/10.4134/JKMS.j180256

LEFT QUASI-ABUNDANT SEMIGROUPS  

Ji, Zhulin (Department of Mathematics Xi'an University of Architecture and Technology)
Ren, Xueming (Department of Mathematics Xi'an University of Architecture and Technology)
Wang, Yanhui (College of Mathematics and Systems Science Shandong University of Science and Technology)
Publication Information
Journal of the Korean Mathematical Society / v.56, no.5, 2019 , pp. 1159-1172 More about this Journal
Abstract
A semigroup S is called a weakly abundant semigroup if its every $\tilde{\mathcal{L}}$-class and every $\tilde{\mathcal{R}}$-class contains an idempotent. Our purpose is to study an analogue of orthodox semigroups in the class of weakly abundant semigroups. Such an analogue is called a left quasi-abundant semigroup, which is a weakly abundant semigroup with a left quasi-normal band of idempotents and having the congruence condition (C). To build our main structure theorem for left quasi-abundant semigroups, we first give a sufficient and necessary condition of the idempotent set E(S) of a weakly abundant semigroup S being a left quasi-normal band. And then we construct a left quasi-abundant semigroup in terms of weak spined products. Such a result is a generalisation of that of Guo and Shum for left semi-perfect abundant semigroups. In addition, we consider a type Q semigroup which is a left quasi-abundant semigroup having the PC condition.
Keywords
weakly abundant semigroups; weak spined products; left quasiabundant semigroups; type Q semigroups;
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1 M. J. J. Branco, G. M. S. Gomes, V. Gould, and Y. H. Wang, Ehresmann monoids: adequacy and expansions, J. Algebra 513 (2018), 344-367. https://doi.org/10.1016/j.jalgebra.2018.06.036   DOI
2 A. El-Qallali and J. B. Fountain, Idempotent-connected abundant semigroups, Proc. Roy. Soc. Edinburgh Sect. A 91 (1981/82), no. 1-2, 79-90. https://doi.org/10.1017/S0308210500012646   DOI
3 J. Fountain, Abundant semigroups, Proc. London Math. Soc. (3) 44 (1982), no. 1, 103-129. https://doi.org/10.1112/plms/s3-44.1.103   DOI
4 J. Fountain, G. Gomes, and V. Gould, Membership of a A ${\vee}$ G for classes of finite weakly abundant semigroups, Period. Math. Hungar. 59 (2009), no. 1, 9-36. https://doi.org/10.1007/s10998-009-9009-1   DOI
5 X. Guo and K. P. Shum, Left semi-perfect abundant semi-groups of type W, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), no. 3, 603-613. https://doi.org/10.1017/S0308210500004029   DOI
6 J. M. Howie, An Introduction to Semigroup Theory, Academic Press, London, 1976.
7 M. V. Lawson, Semigroups and ordered categories. I. The reduced case, J. Algebra 141 (1991), no. 2, 422-462. https://doi.org/10.1016/0021-8693(91)90242-Z   DOI
8 S. Ma, X. Ren, and Y. Yuan, On U-ample ${\omega}$-semigroups, Front. Math. China 8 (2013), no. 6, 1391-1405. https://doi.org/10.1007/s11464-013-0337-3   DOI
9 M. Petrich, Lectures in Semigroups, Akademie-Verlag, Berlin, 1977.
10 M. Petrich, Malcev products of unipotent monoids and varieties of bands, Semigroup Forum 83 (2011), no. 2, 161-189. https://doi.org/10.1007/s00233-011-9318-6   DOI
11 M. Petrich, On weakly ample semigroups, J. Aust. Math. Soc. 97 (2014), no. 3, 404-417. https://doi.org/10.1017/S1446788714000202   DOI
12 M. Petrich and N. R. Reilly, Completely Regular Semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts, 23, John Wiley & Sons, Inc., New York, 1999.
13 X. Ren, Y. Wang, and K. P. Shum, On U-orthodox semigroups, Sci. China Ser. A 52 (2009), no. 2, 329-350. https://doi.org/10.1007/s11425-009-0025-7   DOI
14 X. Ren, Y. Wang, and K. P. Shum, U-concordant semigroups, Algebra Colloq. 25 (2018), no. 2, 295-318. https://doi.org/10.1142/S1005386718000214   DOI
15 X. Ren, Q. Yin, and K. P. Shum, On $U^{\sigma}$-abundant semigroups, Algebra Colloq. 19 (2012), no. 1, 41-52. https://doi.org/10.1007/s00233-012-9414-2   DOI
16 M. V. Lawson, Rees matrix semigroups, Proc. Edinburgh Math. Soc. (2) 33 (1990), no. 1, 23-37. https://doi.org/10.1017/S0013091500028856   DOI
17 Y. Wang, Weakly B-orthodox semigroups, Period. Math. Hungar. 68 (2014), no. 1, 13-38. https://doi.org/10.1007/s10998-014-0023-6   DOI
18 Y. Wang, Hall-type representations for generalised orthogroups, Semigroup Forum 89 (2014), no. 3, 518-545. https://doi.org/10.1007/s00233-014-9583-2   DOI
19 Y. Wang, Beyond regular semigroups, Semigroup Forum 92 (2016), no. 2, 414-448. https://doi.org/10.1007/s00233-015-9714-4   DOI
20 Y. Wang and D. Abdulkadir, Restriction ${\omega}$-semigroups, Semigroup Forum 97 (2018), no. 2, 307-324. https://doi.org/10.1007/s00233-018-9961-2   DOI