# Article

 Title: On finiteness conditions for Rees matrix semigroups (English) Author: Ayik, Hayrullah Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 55 Issue: 2 Year: 2005 Pages: 455-463 Summary lang: English . Category: math . Summary: Let $T=\mathcal {M}[S;I,J;P]$ be a Rees matrix semigroup where $S$ is a semigroup, $I$ and $J$ are index sets, and $P$ is a $J\times I$ matrix with entries from $S$, and let $U$ be the ideal generated by all the entries of $P$. If $U$ has finite index in $S$, then we prove that $T$ is periodic (locally finite) if and only if $S$ is periodic (locally finite). Moreover, residual finiteness and having solvable word problem are investigated. (English) Keyword: Rees matrix semigroup Keyword: periodicity Keyword: local finiteness Keyword: residual finiteness Keyword: word problem MSC: 20M05 MSC: 20M10 idZBL: Zbl 1081.20065 idMR: MR2137151 . Date available: 2009-09-24T11:24:24Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/127991 . Reference: [1] H. Ayık and N.  Ruškuc: Generators and relations of Rees matrix semigroups.Proc. Edinburgh Math. Soc. 42 (1999), 481–495. MR 1721767 Reference: [2] É. A. Golubov: Finitely separable and finitely approximatable full 0-simple semigroups.Math. Notes 12 (1972), 660–665. 10.1007/BF01093670 Reference: [3] J. M.  Howie: Fundamentals of Semigroup Theory.Oxford University Press, Oxford, 1995. Zbl 0835.20077, MR 1455373 Reference: [4] M. V.  Lawson: Rees matrix semigroups.Proc. Edinburgh Math. Soc. 33 (1990), 23–37. Zbl 0668.20049, MR 1038762 Reference: [5] J. Meakin: Fundamental regular semigroups and the Rees construction.Quart. J.  Math. Oxford 33 (1985), 91–103. Zbl 0604.20060, MR 0780353 Reference: [6] D. Rees: On semi-groups.Proc. Cambridge Philos. Soc. 36 (1940), 387–400. Zbl 0028.00401, MR 0002893 Reference: [7] N. Ruškuc: On large subsemigroups and finiteness conditions of semigroups.Proc. London Math. Soc. 76 (1998), 383–405. MR 1490242 Reference: [8] E. F.  Robertson, N. Ruškuc and J.  Wiegold: Generators and relations of direct products of semigroups.Trans. Amer. Math. Soc. 350 (1998), 2665–2685. MR 1451614, 10.1090/S0002-9947-98-02074-1 .

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