2016
DOI: 10.1007/s10801-016-0703-9
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Lengths of words in transformation semigroups generated by digraphs

Abstract: Given a simple digraph D on n vertices (with n ≥ 2), there is a natural construction of a semigroup of transformations D . For any edge (a, b) of D, let a → b be the idempotent of rank n − 1 mapping a to b and fixing all vertices other than a; then, define D to be the semigroup generated bybe the minimal length of a word in E(D) expressing α. It is well known that the semigroup Sing n of all transformations of rank at most n − 1 is generated by its idempotents of rank n − 1. When D = K n is the complete undire… Show more

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Cited by 5 publications
(3 citation statements)
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“…For condition (3), recall that by Proposition 3.1, the D-classes correspond to the isomorphism classes of the induced subgraphs. If is a graph on at least two vertices, then any induced subgraph of with two vertices either has no edge or has a single edge between the two vertices.…”
Section: When Does An Inverse Monoid Of Partial Permutations Coincide...mentioning
confidence: 99%
See 1 more Smart Citation
“…For condition (3), recall that by Proposition 3.1, the D-classes correspond to the isomorphism classes of the induced subgraphs. If is a graph on at least two vertices, then any induced subgraph of with two vertices either has no edge or has a single edge between the two vertices.…”
Section: When Does An Inverse Monoid Of Partial Permutations Coincide...mentioning
confidence: 99%
“…Finally, let us mention papers devoted to finite monoids and semigroups and their relation to graphs that take a different approach from ours. Graphs play a fundamental role in studying certain classes of finite monoids and semigroups (see, e.g., [3,23,24]), while monoids attached to graphs have also been considered (see, e.g., [2]). The monoids and semigroups appearing in this type of research consist almost exclusively of transformations (e.g., transition monoids of automata, transformation semigroups generated by idempotent transformations coming from graphs, endomorphism monoids of graphs) rather than of partial permutations considered in our paper.…”
Section: Introductionmentioning
confidence: 99%
“…Several authors have classified those digraphs D such that D has a specific semigroup property. For instance, in [YY06] those digraphs D such that D is regular are classified; and in [CCGM16] those D where D is a band are classified. In [YY06,YY09], necessary and sufficient conditions on digraphs D and D ′ are given so that D = D ′ or D ∼ = D ′ , respectively.…”
Section: Introductionmentioning
confidence: 99%