GAP (semigroups) - Chapter 13: Graph inverse semigroups

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13 Graph inverse semigroups
13.1 Creating graph inverse semigroups
13.1-1 GraphInverseSemigroup
13.1-2 Range
13.1-3 IsVertex
13.1-4 IsGraphInverseSemigroup
13.1-5 GraphOfGraphInverseSemigroup

### 13 Graph inverse semigroups

In this chapter we describe a class of semigroups arising from directed graphs.

#### 13.1 Creating graph inverse semigroups

##### 13.1-1 GraphInverseSemigroup

`‣ 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 \(e, f\in E ^ 1\):

(V)

  \(vw = \delta_{v,w}\cdot v\),

(E1)

  \(s(e)\cdot e=e\cdot r(e)=e\),

(E2)

  \(s(e)\cdot e = e\cdot r(e) =e\),

(CK1)

  \(e^{-1}f=\delta_{e,f}\cdot r(e)\).

(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

##### 13.1-2 Range

`‣ Range`( x ) | ( attribute )

`‣ Source`( x ) | ( attribute )

##### 13.1-3 IsVertex

`‣ IsVertex`( x ) | ( attribute )

##### 13.1-4 IsGraphInverseSemigroup

`‣ IsGraphInverseSemigroup`( x ) | ( filter )

`‣ IsGraphInverseSemigroupElement`( x ) | ( filter )

##### 13.1-5 GraphOfGraphInverseSemigroup

`‣ GraphOfGraphInverseSemigroup`( x ) | ( filter )

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