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Structure of Chinese algebras. (English) Zbl 1246.16022

The Chinese monoid ${M}_{n}$ of rank $n$ is generated by ${a}_{1},{a}_{2},\cdots ,{a}_{n}$ subject to the relations

${a}_{j}{a}_{i}{a}_{k}={a}_{j}{a}_{k}{a}_{i}={a}_{k}{a}_{j}{a}_{i},\phantom{\rule{1.em}{0ex}}i\le k\le j·$

The monoid ${M}_{n}$ is infinite and has polynomial growth.

In the paper under review the authors study the structure of the monoid algebra $K\left[{M}_{n}\right]$ of ${M}_{n}$ over a field $K$. The authors show that $K\left[{M}_{n}\right]$ has only finitely many prime ideals and completely describe them using certain homogeneous congruences on ${M}_{n}$. Further, it is shown that the prime radical of $K\left[{M}_{n}\right]$ coincides with the Jacobson radical. As a consequence the authors derive a new representation of ${M}_{n}$ as a submonoid of the product ${B}^{k}×{ℤ}^{l}$ for some $k,l\in ℕ$, where $B$ denotes the bicyclic monoid, and show that ${M}_{n}$ satisfies a nontrivial identity.

##### MSC:
 16S36 Ordinary and skew polynomial rings and semigroup rings 16S15 Finite generation, finite presentability, normal forms 20M25 Semigroup rings, multiplicative semigroups of rings 16N60 Prime and semiprime associative rings 20M05 Free semigroups, generators and relations, word problems 16D25 2-sided ideals (associative rings and algebras)
##### References:
 [1] Adjan, S. I.: Defining relations and algorithmic problems for groups and semigroups, Tr. mat. Inst. steklova 85, 1-124 (1966) · Zbl 0204.01702 [2] Cassaigne, J.; Espie, M.; Krob, D.; Novelli, J. -C.; Hivert, F.: The chinese monoid, Internat. J. Algebra comput. 11, No. 3, 301-334 (2001) · Zbl 1024.20046 · doi:10.1142/S0218196701000425 [3] Cedó, F.; Okniński, J.: Plactic algebras, J. algebra 274, No. 1, 97-117 (2004) · Zbl 1054.16022 · doi:10.1016/j.jalgebra.2003.12.004 [4] Clifford, A. H.; Preston, G. B.: The algebraic theory of semigroups, vol. 1, (1961) · Zbl 0111.03403 [5] Duchamp, G.; Krob, D.: Plactic-growth like monoids, , 124-142 (1994) · Zbl 0875.68720 [6] Fulton, W.: Young tableaux, (1997) [7] Gateva-Ivanova, T.: A combinatorial approach to set-theoretic solutions of the Yang-Baxter equation, J. math. Phys. 45, No. 10, 3828-3858 (2004) · Zbl 1065.16037 · doi:10.1063/1.1788848 [8] Jaszuńska, J.; Okniński, J.: Chinese algebras of rank 3, Comm. algebra 34, No. 8, 2745-2754 (2006) · Zbl 1116.16031 · doi:10.1080/00927870600651760 [9] Jespers, E.; Krempa, J.; Puczylowski, E.: On radicals of graded rings, Comm. algebra 10, No. 17, 1849-1854 (1982) · Zbl 0493.16003 · doi:10.1080/00927878208822807 [10] Jespers, E.; Okniński, J.: Binomial semigroups, J. algebra 202, No. 1, 250-275 (1998) · Zbl 0910.20038 · doi:10.1006/jabr.1997.7292 [11] Jespers, E.; Okniński, J.: Noetherian semigroup algebras, (2007) [12] Karpilovsky, G.: The Jacobson radical of classical rings, (1991) [13] Krause, G. R.; Lenagan, T. H.: Growth of algebras and Gelfand-kirillov dimension, Grad stud. Math. 22 (2000) · Zbl 0957.16001 [14] Krempa, J.: On semisimplicity of tensor products, Lect. notes pure appl. Math. 51, 105-122 (1979) · Zbl 0431.16003 [15] Lam, T. Y.: A first course in noncommutative rings, Grad texts in math. 131 (2001) [16] Lascoux, A.; Leclerc, B.; Thibon, J. Y.: The plactic monoid, Encyclopedia math. Appl. 90 (2002) [17] Lascoux, A.; Schützenberger, M. P.: Le monoïde plaxique, , 129-156 (1978) · Zbl 0517.20036 [18] Okniński, J.: Semigroup algebras, (1991) [19] Sloane, N. J. A.: The on-line encyclopedia of integer sequences, (2007)