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Skew Products on Directed Graphs

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The document discusses skew products of directed graphs. Specifically, it defines skew products as taking a directed graph E and group G, along with a function f from edges of E to G. The skew product graph E ×f G has vertices that are pairs of vertices in E and elements of G, and edges that are pairs of edges in E and elements of G where the range and source of the edge are modified by the group element associated to the edge by f. Several examples of skew products are given to illustrate this construction. The document also discusses inverse semigroups associated with directed graphs.

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SKEW PRODUCTS OF DIRECTED GRAPHSCaridad Arroyo QuijanoKelvin Díaz Cruz
For what are the directed graph useful? From a practical point of view, directed graphs let us study the unity relations that interact with each other. For instance, a computer net can be represented and studied using directed graphs, where the vertices would be the terminals and the path the connections which can be cables or wireless connections. After all, a wide range of problems can be solve with this application since directed graphs have vectorial spaces characteristics.
Directed GraphA directed graph E = (V, P, r , s) consists of a countable set of vertices V, a countable set of edges, P, and maps r, s : PV identifying the range and source of each edge.
The Path SpaceA finite path is a sequence α of edges e1···ek where s(ei + 1) = r (ei) for i=1,, k – 1. We write α = e1···ek. The length l(α) of α is just k. Each vertex v is regarded as a finite path of length zero. We denote the set of all finite paths in E by E*, and it is called the path space of E.
A Partial Operation on E * For paths α and β in E * the expression αβis defined only when r(α) = s(β). In that case the product αβ, is just juxtaposition of α and β.
Examplee1e3α = e1e2e3 , β = e4e1 , r(α) = v4 , s(β) = v4 ,αβ = e1e2e3 e4e1 ,the expression, βα is not defined.v1v2e4e2v4 v3
Skew ProductsLet E = (V, P, r , s) be a directed graph , let (G, •) be a group and let f : P -> G be a function. The skew product graph E ×f G is defined as follows: The set of vertices is V × G, and the set of edges is E1 × G where the structure maps are: s’ : P × G -> V × G and r’ : P × G -> V × G, are defined by the equations,s’ (e, g) = (s (e), g) and r’ (e, g) = (r (e), g • f(e)).
Examples of Skew ProductsG=(Z2,+), E = e v , f(e)=[1] E ×f Z2 =(V × Z2,P× Z2, s’, r’)V× Z2 ={(v, [0]), (v, [1])}P× Z2 ={(e, [0]), (e, [1])} s’(e,[0]) = (s(e),[0]) = (v,[0]) r’(e,[0]) = (r(e),[0]+[1]) = (v,[1]) s’(e,[1]) = (s(e),[1]) = (v,[1]) r’(e,[1]) = (r(e),[1]+[1]) = (v,[0])
• (v, [1])(e,[1])(e,[0]) • (v, [0])
ExampleG = (Z, +), E = e v , f(e)=1 E ×f Z=(V × Z,P× Z, s’, r’)V× Z={(v, n): n  Z}P× Z ={(e, n): n  Z} s’(e, n)=(s(e), n)=(v, n) r’(e, n)=(r(e), n + f(e))=(v, n+1)
(v, 0) (e, -1) (e, 0) • (e, 1)(v, 1) • ••(v,-1)••• • •
Example G = (Z3 ˟ , •), E = e v , f(e)=[1]E ×f (Z3 ˟)=(V × (Z3 ˟),P× (Z3 ˟), s’, r’)V× (Z3 ˟) ={(v, [1]), (v, [2]) }P× (Z3 ˟) ={(e, [1]), (e, [2]) } s’(e, [1]) = (v, [1]) = r’ (e, [1]) s’(e, [2]) = (v, [2]) = r’ (e, [2])
• (v, [1])(e,[1])(e,[2]) • (v, [2])
A semigroup S is called inverse semigroup if for each s  S, there exists a unique element s*  S such that ss*s = s and s*ss* = s*.Inverse Semigroups
TheoremTh. Every group is an inverse semigroup Let G be any group. For each g∊G, there is aunique element g-1 ∊ G such that g g-1 = g-1g = e. Then, g g-1g = g(g-1g) = g e = gg-1g g-1= g-1(1g g-1) = g-1 e = g-1
The Inverse Semigroup SE of a Directed Graph E SE= { (α, β) : α, βE*, r(α) = r(β) }U {z}Involution is defined on SE by: z* = z, (α, β)* = (β,α), and product is defined by: (α, δμ) (δ, β) = (α, βμ) (α, δ) (δμ, β) = (αμ, β) .The other products on SE are defined to be z.
Example: The Inverse Semigroup of a LoopThe inverse semigroup SEassociated to the directed graph E = e v can be identified with S (N).
ExampleE = e  v
en := v if n=0 and en :=e…e if n>0
E* = {en : n ≥ 0}

More Related Content

Skew Products on Directed Graphs

  • 1. SKEW PRODUCTS OF DIRECTED GRAPHSCaridad Arroyo QuijanoKelvin Díaz Cruz
  • 2. For what are the directed graph useful? From a practical point of view, directed graphs let us study the unity relations that interact with each other. For instance, a computer net can be represented and studied using directed graphs, where the vertices would be the terminals and the path the connections which can be cables or wireless connections. After all, a wide range of problems can be solve with this application since directed graphs have vectorial spaces characteristics.
  • 3. Directed GraphA directed graph E = (V, P, r , s) consists of a countable set of vertices V, a countable set of edges, P, and maps r, s : PV identifying the range and source of each edge.
  • 4. The Path SpaceA finite path is a sequence α of edges e1···ek where s(ei + 1) = r (ei) for i=1,, k – 1. We write α = e1···ek. The length l(α) of α is just k. Each vertex v is regarded as a finite path of length zero. We denote the set of all finite paths in E by E*, and it is called the path space of E.
  • 5. A Partial Operation on E * For paths α and β in E * the expression αβis defined only when r(α) = s(β). In that case the product αβ, is just juxtaposition of α and β.
  • 6. Examplee1e3α = e1e2e3 , β = e4e1 , r(α) = v4 , s(β) = v4 ,αβ = e1e2e3 e4e1 ,the expression, βα is not defined.v1v2e4e2v4 v3
  • 7. Skew ProductsLet E = (V, P, r , s) be a directed graph , let (G, •) be a group and let f : P -> G be a function. The skew product graph E ×f G is defined as follows: The set of vertices is V × G, and the set of edges is E1 × G where the structure maps are: s’ : P × G -> V × G and r’ : P × G -> V × G, are defined by the equations,s’ (e, g) = (s (e), g) and r’ (e, g) = (r (e), g • f(e)).
  • 8. Examples of Skew ProductsG=(Z2,+), E = e v , f(e)=[1] E ×f Z2 =(V × Z2,P× Z2, s’, r’)V× Z2 ={(v, [0]), (v, [1])}P× Z2 ={(e, [0]), (e, [1])} s’(e,[0]) = (s(e),[0]) = (v,[0]) r’(e,[0]) = (r(e),[0]+[1]) = (v,[1]) s’(e,[1]) = (s(e),[1]) = (v,[1]) r’(e,[1]) = (r(e),[1]+[1]) = (v,[0])
  • 9. • (v, [1])(e,[1])(e,[0]) • (v, [0])
  • 10. ExampleG = (Z, +), E = e v , f(e)=1 E ×f Z=(V × Z,P× Z, s’, r’)V× Z={(v, n): n  Z}P× Z ={(e, n): n  Z} s’(e, n)=(s(e), n)=(v, n) r’(e, n)=(r(e), n + f(e))=(v, n+1)
  • 11. (v, 0) (e, -1) (e, 0) • (e, 1)(v, 1) • ••(v,-1)••• • •
  • 12. Example G = (Z3 ˟ , •), E = e v , f(e)=[1]E ×f (Z3 ˟)=(V × (Z3 ˟),P× (Z3 ˟), s’, r’)V× (Z3 ˟) ={(v, [1]), (v, [2]) }P× (Z3 ˟) ={(e, [1]), (e, [2]) } s’(e, [1]) = (v, [1]) = r’ (e, [1]) s’(e, [2]) = (v, [2]) = r’ (e, [2])
  • 14. A semigroup S is called inverse semigroup if for each s  S, there exists a unique element s*  S such that ss*s = s and s*ss* = s*.Inverse Semigroups
  • 15. TheoremTh. Every group is an inverse semigroup Let G be any group. For each g∊G, there is aunique element g-1 ∊ G such that g g-1 = g-1g = e. Then, g g-1g = g(g-1g) = g e = gg-1g g-1= g-1(1g g-1) = g-1 e = g-1
  • 16. The Inverse Semigroup SE of a Directed Graph E SE= { (α, β) : α, βE*, r(α) = r(β) }U {z}Involution is defined on SE by: z* = z, (α, β)* = (β,α), and product is defined by: (α, δμ) (δ, β) = (α, βμ) (α, δ) (δμ, β) = (αμ, β) .The other products on SE are defined to be z.
  • 17. Example: The Inverse Semigroup of a LoopThe inverse semigroup SEassociated to the directed graph E = e v can be identified with S (N).
  • 18. ExampleE = e  v
  • 19. en := v if n=0 and en :=e…e if n>0
  • 20. E* = {en : n ≥ 0}
  • 21. e n e m = e n + m
  • 22. SE = {(en, e m) : n, m N U {0} } (en , e m + k) (e m , e j) = (en , e j + k) , (en , e m) (e m + k, e j) = (en + k, e j) ; n, m, j, k N {0}.SE= S (N)Future ProjectIn the literature there are algebraic structures associated to directed graphs, so called inverse semigroups; which allow us to interpret directed graphs in dynamical terms. This motivates us to investigate inverse semigroups associated to skew products. Exactly looking for patterns between the original directed graph and the product and their respective inverse semigroups.
  • 23. ReferencesAllan L. T. Patterson. (2003). Graph Inverse Semigroups, Grupoids and their C* Algebras.

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