# Article

Full entry | PDF   (0.3 MB)
Keywords:
Rees matrix semigroup; periodicity; local finiteness; residual finiteness; word problem
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.
References:
[1] H. Ayık and N.  Ruškuc: Generators and relations of Rees matrix semigroups. Proc. Edinburgh Math. Soc. 42 (1999), 481–495. MR 1721767
[2] É. A. Golubov: Finitely separable and finitely approximatable full 0-simple semigroups. Math. Notes 12 (1972), 660–665. DOI 10.1007/BF01093670
[3] J. M.  Howie: Fundamentals of Semigroup Theory. Oxford University Press, Oxford, 1995. MR 1455373 | Zbl 0835.20077
[4] M. V.  Lawson: Rees matrix semigroups. Proc. Edinburgh Math. Soc. 33 (1990), 23–37. MR 1038762 | Zbl 0668.20049
[5] J. Meakin: Fundamental regular semigroups and the Rees construction. Quart. J.  Math. Oxford 33 (1985), 91–103. MR 0780353 | Zbl 0604.20060
[6] D. Rees: On semi-groups. Proc. Cambridge Philos. Soc. 36 (1940), 387–400. MR 0002893 | Zbl 0028.00401
[7] N. Ruškuc: On large subsemigroups and finiteness conditions of semigroups. Proc. London Math. Soc. 76 (1998), 383–405. MR 1490242
[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. DOI 10.1090/S0002-9947-98-02074-1 | MR 1451614

Partner of