Monoids over which products of indecomposable acts are indecomposable

Year 2017, Volume: 46 Issue: 2, 229 - 237, 01.04.2017

Mojtaba Sedaghatjoo Ahmad Khaksari

https://izlik.org/JA45PL27UJ

Abstract

In this paper we prove that for a monoid $S$, products of indecomposable right $S$-acts are indecomposable if and only if $S$ contains a right
zero. Besides, we prove that subacts of indecomposable right $S$-acts are indecomposable if and only if $S$ is left reversible. Ultimately, we prove that the one element right $S$-act $\Theta_S$ is product flat if and only if $S$ contains a left zero.

Keywords

Indecomposable act , left reversible monoid , Baer criterion , product at , super flat

References

• Adamek, J., Herrlich, H. and Strecker, G. Abstract and Concrete Categories The Joy of Cats (John Wiley and Sons, New York, 1990).
• Bulman-Fleming, S. Products of projective S-systems, Comm. Algebra 19 (3), 951964, 1991.
• Bulman-Fleming, S. and Laan, V. Tensor products and preservation of limits, for acts over monoids, Semigroup Forum 63, 161179, 2001.
• Bulman-Fleming, S and McDowell, K. Coherent monoids, in: Latices, Semigroups and Universal Algebra, ed. J. Almeida et al. (Plenum Press, New York, 1990).
• Gould, V. Coherent monoids, J. Austral. Math. Soc. 53(Series A), 166182, 1992.
• Kilp, M., Knauer, U. and Mikhalev, A. Monoids, Acts and Categories, (W. de gruyter, Berlin, 2000).
• Nico, W.R. A classication of indecomposable S-sets, J. Algebra 54(1), 260272, 1978.
• Renshaw, J. Monoids for which condition (P)acts are projective, Semigroup Forum 61(1), 4656, 1998.
• Sedaghatjoo, M., Khosravi, R. and Ershad, M. Principally weakly and weakly coherent monoids, Comm. Algebra 37(12), 42814295, 2009.