Papers › Structural aspects of semigroups based on digraphs
Structural aspects of semigroups based on digraphs
James East, Maximilien Gadouleau, James D. Mitchell
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Given any digraph D without loops or multiple arcs, there is a natural construction of a semigroup ⟨D⟩ of transformations. To every arc (a,b) of D is associated the idempotent transformation (a→b) mapping a to b and fixing all vertices other than a. The semigroup ⟨D⟩ is generated by the idempotent transformations (a→b) for all arcs (a,b) of D. In this paper, we consider the question of when there is a transformation in ⟨D⟩ containing a large cycle, and, for fixed k∈ℕ, we give a linear time algorithm to verify if ⟨D⟩ contains a transformation with a cycle of length k. We also classify those digraphs D such that ⟨D⟩ has one of the following properties: inverse, completely regular, commutative, simple, 0-simple, a semilattice, a rectangular band, congruence-free, is 𝒦-trivial or 𝒦-universal where 𝒦 is any of Green's ℋ-, ℒ-, ℛ-, or 𝒥-relation, and when ⟨D⟩ has a left, right, or two-sided zero.
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