  
  [1X13 [33X[0;0YGraph inverse semigroups[133X[101X
  
  [33X[0;0YIn  this  chapter  we  describe  a class of semigroups arising from directed
  graphs.[133X
  
  
  [1X13.1 [33X[0;0YCreating graph inverse semigroups[133X[101X
  
  [1X13.1-1 GraphInverseSemigroup[101X
  
  [29X[2XGraphInverseSemigroup[102X( [3XE[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA graph inverse semigroup.[133X
  
  [33X[0;0YIf  [3XE[103X  is a digraph (i.e. it satisfies [2XIsDigraph[102X ([14XDigraphs: IsDigraph[114X)) then
  [10XGraphInverseSemigroup[110X  returns  the  graph  inverse  semigroup  [10XG(E)[110X  where,
  roughly speaking, elements correspond to paths in the graph [10XE[110X.[133X
  
  [33X[0;0YGiven a digraph [10X[3XE[103X[10X = (E ^ 0, E ^ 1, r, s)[110X the [13Xgraph inverse semigroup [10XG(E)[110X of
  [10XE[110X[113X is the semigroup with zero generated by the sets [10XE ^ 0[110X and [10XE ^ 1[110X, together
  with  a  set  of  variables  [22X{e  ^  -1 ∣ e∈ E ^ 1}[122X, satisfying the following
  relations for all [22Xv, w∈ E ^ 0[122X and [22Xe, f∈ E ^ 1[122X:[133X
  
  [8X(V)[108X
        [33X[0;6Y[22Xvw = δ_v,w⋅ v[122X,[133X
  
  [8X(E1)[108X
        [33X[0;6Y[22Xs(e)⋅ e=e⋅ r(e)=e[122X,[133X
  
  [8X(E2)[108X
        [33X[0;6Y[22Xs(e)⋅ e = e⋅ r(e) =e[122X,[133X
  
  [8X(CK1)[108X
        [33X[0;6Y[22Xe^-1f=δ_e,f⋅ r(e)[122X.[133X
  
  [33X[0;0Y(Here  [22Xδ[122X is the Kronecker delta.) We define [22Xv^-1=v[122X for each [22Xv ∈ E^0[122X, and for
  any path [22Xy=e_1dots e_n[122X ([22Xe_1dots e_n ∈ E^1[122X) we let [22Xy^-1 = e_n^-1 dots e_1^-1[122X.
  With this notation, every nonzero element of [22XG(E)[122X can be written uniquely as
  [22Xxy^-1[122X for some paths [22Xx, y[122X in [22XE[122X, by the CK1 relation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10], [1], [127X[104X
    [4X[25X>[125X [27X                  [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [127X[104X
    [4X[25X>[125X [27X                  [1, 4], [1, 5, 9], [1, 2, 7, 8], [3, 5]]);[127X[104X
    [4X[28X<digraph with 10 vertices, 37 edges>[128X[104X
    [4X[25Xgap>[125X [27XS := GraphInverseSemigroup(gr);[127X[104X
    [4X[28X<infinite graph inverse semigroup with 10 vertices, 37 edges>[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfInverseSemigroup(S);[127X[104X
    [4X[28X[ e_1, e_2, e_3, e_4, e_5, e_6, e_7, e_8, e_9, e_10, e_11, e_12, [128X[104X
    [4X[28X  e_13, e_14, e_15, e_16, e_17, e_18, e_19, e_20, e_21, e_22, e_23, [128X[104X
    [4X[28X  e_24, e_25, e_26, e_27, e_28, e_29, e_30, e_31, e_32, e_33, e_34, [128X[104X
    [4X[28X  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 [128X[104X
    [4X[28X ][128X[104X
    [4X[25Xgap>[125X [27XAssignGeneratorVariables(S);[127X[104X
    [4X[25Xgap>[125X [27Xe_1 * e_1 ^ -1;[127X[104X
    [4X[28Xe_1e_1^-1[128X[104X
    [4X[25Xgap>[125X [27Xe_1 ^ -1 * e_1 ^ -1;[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27Xe_1 ^ -1 * e_1;[127X[104X
    [4X[28Xv_2[128X[104X
  [4X[32X[104X
  
  [1X13.1-2 Range[101X
  
  [29X[2XRange[102X( [3Xx[103X ) [32X attribute
  [29X[2XSource[102X( [3Xx[103X ) [32X attribute
  
  [4X[32X  Example  [32X[104X
  [4X[32X[104X
  
  [1X13.1-3 IsVertex[101X
  
  [29X[2XIsVertex[102X( [3Xx[103X ) [32X attribute
  
  [4X[32X  Example  [32X[104X
  [4X[32X[104X
  
  [1X13.1-4 IsGraphInverseSemigroup[101X
  
  [29X[2XIsGraphInverseSemigroup[102X( [3Xx[103X ) [32X filter
  [29X[2XIsGraphInverseSemigroupElement[102X( [3Xx[103X ) [32X filter
  
  [4X[32X  Example  [32X[104X
  [4X[32X[104X
  
  [1X13.1-5 GraphOfGraphInverseSemigroup[101X
  
  [29X[2XGraphOfGraphInverseSemigroup[102X( [3Xx[103X ) [32X filter
  
  [4X[32X  Example  [32X[104X
  [4X[32X[104X
  
