Cui, Ranran; Luo, Yanfeng Gelfand-Kirillov dimension of some primitive abundant semigroups. (English) Zbl 1310.20047 Indian J. Pure Appl. Math. 44, No. 6, 809-822 (2013). Summary: The growth and Gelfand-Kirillov dimension of some primitive abundant semigroups are investigated. It is shown that for certain primitive abundant (regular) semigroup \(S\), \(S\) as well as the semigroup algebra \(K[S]\) has polynomial growth if and only if all of its cancellative submonoids (subgroups) \(T\) as well as \(K[T]\) have polynomial growth. As applications, it is shown that if \(S\) is a finitely generated primitive inverse monoid having the permutational property, then \(\mathrm{clK}\dim K[S]=\mathrm{GK}\dim K[S]=\mathrm{rk}(S)\). Cited in 2 Documents MSC: 20M05 Free semigroups, generators and relations, word problems 16P90 Growth rate, Gelfand-Kirillov dimension 20M25 Semigroup rings, multiplicative semigroups of rings 16S36 Ordinary and skew polynomial rings and semigroup rings 20M17 Regular semigroups Keywords:Gelfand-Kirillov dimension; growth of abundant semigroups; primitive regular semigroups; completely 0-simple semigroups; inverse monoids; semigroup algebras PDFBibTeX XMLCite \textit{R. Cui} and \textit{Y. Luo}, Indian J. Pure Appl. Math. 44, No. 6, 809--822 (2013; Zbl 1310.20047) Full Text: DOI References: [1] S. A. Amitusur and L. W. Small, GK-dimension of comers and ideals, Israel J. Math., 69 (1990), 152-160. · Zbl 0697.16025 · doi:10.1007/BF02937301 [2] F. Cedo, E. Jespers and J. Okniński, The Gelfand-Kirillov dimension of quadratic algebras satisfying the cyclic condition, Proceedings of the American Mathematical Society, 134 (2006), 653-663. · Zbl 1092.16014 · doi:10.1090/S0002-9939-05-08003-2 [3] A.H. Clifford and G. B. Preston, The Algebraic theory of semigroups, Vol.1 of Mathematical Surveys, no.7, American Mathematical Society, Providence, RI, USA (1961). · Zbl 0111.03403 [4] A. El-Qallali and J. B. Fountain, Idempotent-connected abundant semigroups, Proceedings of the Royal Society of Edinburgh, 91A (1981), 79-90. · Zbl 0501.20043 · doi:10.1017/S0308210500012646 [5] J. B. Fountain, Abundant semigroups, Proc. London Math. Soc, 44 (1982), 103-129. · Zbl 0481.20036 · doi:10.1112/plms/s3-44.1.103 [6] R. I. Grigorchuk, Cancellative semigroups of power growth, Mat. Zametki, 43 (1988), 305-319. · Zbl 0643.20036 [7] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. JHES, 53(1) (1985), 53-73. · Zbl 0474.20018 · doi:10.1007/BF02698687 [8] X. J. Guo, F. S. Huang and K. P. Shum, Type-A semigroups whose full subsemi-groups form a chain under set inclusion, Asian-European J. Math., 1 (2008), 359-367. · Zbl 1166.20053 · doi:10.1142/S179355710800031X [9] J. M. Howie, An introduction to semigroup theory, Academic Press, London (1976). · Zbl 0355.20056 [10] A. V. Kelarev and J. Okniński, A combinatorial property and growth for semigroups of matrices, Communications in Algebra, 26(9) (1998), 2789-2805. · Zbl 0942.20043 · doi:10.1080/00927879808826310 [11] G. R. Krause and T. H. Lenagan, Growth of Algebras and Gelfand-Kirillov Dimension, Pitman, London (2000). · Zbl 0957.16001 [12] J. Okniński, A note on the Pi-property of semigroup algebras. In: Perspectives in ring theory, Proc. NATO Adv. Res. Workshop, Antwerp/Belg. 1987, NATO ASI Ser., Ser. C, 233 (1988), 275-278. [13] J. Okniński, Gelfand-Kirillov dimension of noetherian semigroup algebras, Journal of algebra, 162 (1993), 302-316. · Zbl 0801.16029 · doi:10.1006/jabr.1993.1255 [14] J. Okniński, Semigroup algebras, Marcel Dekker, New York (1991). · Zbl 1178.16025 [15] J. Okniński, Semigroups of matrices, Series in Algebra 6, World Sci. Publ. (1998). · Zbl 0911.20042 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.