Digraphs and the semigroup of all functions on a finite set.

*(English)*Zbl 0634.20034Let \(T_ n\) denote the full transformation semigroup on the finite set \(\bar n=\{1,2,...,n\}\); that is the set of all selfmaps on \(\bar n\) under composition. In a previous paper [Can. Math. Bull. 29, 344-351 (1986; Zbl 0552.20045)] the author introduced an algorithm that constructed all square roots of a given member of \(T_ n\) as an alternative to the necessary and sufficient conditions by M. Snowden and J. Howie for a map to be a square [Glasg. Math. J. 23, 137-149 (1982; Zbl 0486.20040)]. In this paper the method, which uses the representation of members of \(T_ n\) as digraphs with functional components, is extended to furnish to algorithms that find all solutions to the equations \(ax^ mb=c\) and \(ax=xb(a,b,c\) in \(T_ n)\). Examples are used to motivate and illustrate the successive steps in the algorithm. As an application the members of \(T_ n\) which commute with no members of \(T_ n\) other than the powers of a are determined.

Reviewer: P.M.Higgins

##### MSC:

20M20 | Semigroups of transformations, relations, partitions, etc. |

20M05 | Free semigroups, generators and relations, word problems |

05C20 | Directed graphs (digraphs), tournaments |

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\textit{P. M. Higgins}, Glasg. Math. J. 30, No. 1, 41--57 (1988; Zbl 0634.20034)

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##### References:

[1] | Howie, Proc. Roy. Soc. Edinburgh, Sect. A 81 pp 317– (1978) · Zbl 0403.20038 · doi:10.1017/S0308210500010647 |

[2] | DOI: 10.1112/jlms/s1-41.1.707 · Zbl 0146.02903 · doi:10.1112/jlms/s1-41.1.707 |

[3] | Snowden, Glasgow Math. J. 23 pp 137– (1982) |

[4] | Harary, Graph theory (1969) |

[5] | Clifford, The algebraic theory of semigroups I (1961) |

[6] | Higgins, Bull. Canad. Math. Soc. 29 pp 344– (1986) · Zbl 0552.20045 · doi:10.4153/CMB-1986-053-2 |

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