# zbMATH — the first resource for mathematics

Power graphs and semigroups of matrices. (English) Zbl 1043.20042
By the power graph of a semigroup $$S$$ the authors mean the directed graph with the set of vertices $$S$$ and with edges $$(u,v)$$ where $$u,v\in S$$ and $$v$$ is a power of $$u$$ but $$v\neq u$$. Infinite groups whose power graphs satisfy a certain technical finiteness condition were characterized by the first two authors [in Contributions to general algebra 12. Klagenfurt, Verlag Johannes Heyn, 229-235 (2000; Zbl 0966.05040)]. In this note the result is extended to the case where $$S$$ is an infinite semigroup of $$n\times n$$ matrices over a division ring or a semigroup of $$n\times n$$ monomial matrices over a group.

##### MSC:
 20M20 Semigroups of transformations, relations, partitions, etc. 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
##### Keywords:
semigroups of matrices; power graphs
Zbl 0966.05040
Full Text:
##### References:
 [1] Howie, Fundamentals of semigroup theory (1995) · Zbl 0835.20077 [2] de Luca, Finiteness and regularity in semigroups and formal languages (1998) [3] de Luca, Handbook of Formal Languages 1 pp 747– (1997) [4] Chartland, Graphs and digraphs (1996) [5] Robinson, A Course in the theory of groups (1982) · Zbl 0483.20001 [6] Okniński, Semigroups of matrices (1998) [7] Jespers, Proc. Roy. Soc. Edinburgh Sect.A 129 pp 1185– (1999) · Zbl 0943.20068 [8] Lothair, Combinatorics on words (1982) [9] Kelarev, Contrib. General Algebra 12 pp 229– (2000) [10] Lidl, Applied abstract algebra (1998) [11] DOI: 10.1090/S0002-9939-96-03036-5 · Zbl 0845.16041 [12] Kelarev, Combinatorics, Complexity and Logic pp 289– (1997) [13] Justin, Southeast Asian Bull. Math. 18 pp 91– (1994) [14] Neumann, J. Austral. Math. Soc. 21 pp 467– (1976)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.