In this chapter we describe a class of semigroups arising from directed graphs.
‣ GraphInverseSemigroup ( E ) | ( operation ) |
Returns: A graph inverse semigroup.
If E is a digraph (i.e. it satisfies IsDigraph
(Digraphs: IsDigraph)) then GraphInverseSemigroup
returns the graph inverse semigroup G(E)
where, roughly speaking, elements correspond to paths in the graph E
.
Given a digraph E = (E ^ 0, E ^ 1, r, s)
the graph inverse semigroup G(E)
of E
is the semigroup with zero generated by the sets E ^ 0
and E ^ 1
, together with a set of variables \(\{e ^ {-1} \mid e\in E ^ 1\}\), satisfying the following relations for all \(v, w\in E ^ 0\) and \(e, f\in E ^ 1\):
\(vw = \delta_{v,w}\cdot v\),
\(s(e)\cdot e=e\cdot r(e)=e\),
\(s(e)\cdot e = e\cdot r(e) =e\),
\(e^{-1}f=\delta_{e,f}\cdot r(e)\).
(Here \(\delta\) is the Kronecker delta.) We define \(v^{-1}=v\) for each \(v \in E^0\), and for any path \(y=e_1\dots e_n\) (\(e_1\dots e_n \in E^1\)) we let \(y^{-1} = e_n^{-1} \dots e_1^{-1}\). With this notation, every nonzero element of \(G(E)\) can be written uniquely as \(xy^{-1}\) for some paths \(x, y\) in \(E\), by the CK1 relation.
gap> gr := Digraph([[2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10], [1], > [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], > [1, 4], [1, 5, 9], [1, 2, 7, 8], [3, 5]]); <digraph with 10 vertices, 37 edges> gap> S := GraphInverseSemigroup(gr); <infinite graph inverse semigroup with 10 vertices, 37 edges> gap> GeneratorsOfInverseSemigroup(S); [ e_1, e_2, e_3, e_4, e_5, e_6, e_7, e_8, e_9, e_10, e_11, e_12, e_13, e_14, e_15, e_16, e_17, e_18, e_19, e_20, e_21, e_22, e_23, e_24, e_25, e_26, e_27, e_28, e_29, e_30, e_31, e_32, e_33, e_34, e_35, e_36, e_37, v_1, v_2, v_3, v_4, v_5, v_6, v_7, v_8, v_9, v_10 ] gap> AssignGeneratorVariables(S); gap> e_1 * e_1 ^ -1; e_1e_1^-1 gap> e_1 ^ -1 * e_1 ^ -1; 0 gap> e_1 ^ -1 * e_1; v_2
‣ Range ( x ) | ( attribute ) |
‣ Source ( x ) | ( attribute ) |
‣ IsVertex ( x ) | ( attribute ) |
‣ IsGraphInverseSemigroup ( x ) | ( filter ) |
‣ IsGraphInverseSemigroupElement ( x ) | ( filter ) |
‣ GraphOfGraphInverseSemigroup ( x ) | ( filter ) |
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