New Structures Based on Completions Bertrand Gilles Bertrand - PowerPoint PPT Presentation

New Structures Based on Completions Gilles New Structures Based on Completions Bertrand Gilles Bertrand Universit´ e Paris-Est Laboratoire d’Informatique Gaspard-Monge D´ epartement Informatique et T´ el´ ecommunications ESIEE Paris March 22, 2013

New Structures Based on We investigate an axiomatic approach related to Completions combinatorial topology and simple homotopy. Gilles Bertrand We use completions as a ”language” for describing collections of objects. We consider objects which are simplicial complexes. �� �� �� �� � �� �� �� �� �� �� � � � � �� ��

Plan of the presentation New Structures Based on Completions Gilles Bertrand Simplicial complexes and completions Dendrites Dyads Confluence Relative dendrites Conclusion

New Structures Based on Completions Gilles Bertrand Simplicial complexes and completions

Simplicial complexes New Structures Based on Let X be a finite family composed of finite sets, X is a Completions simplicial complex if x ∈ X whenever x ⊆ y and y ∈ X . Gilles Bertrand We write S for the collection of all simplicial complexes. Let X ∈ S . An element of X is a face of X . A complex A ∈ S is a cell if A = ∅ or if A has precisely one non-empty maximal face x . We write C for the collection of all cells. ��� ��� ��� ��� �� �� �� �� ��� ��� ��� ��� �� �� ��� ��� ��� ��� ����� ����� ��� ��� ��� ��� �� �� � � �� �� � ��� ��� ��� ��� ����� ����� ��� ��� �� �� � � ��� ��� ��� ��� ����� ����� ��� ��� �� �� � � ��� ��� ��� ��� ����� ����� ��� ��� �� �� �� �� �� �� �� �� � � �� �� � � � � � � ��� ��� ��� ��� ����� ����� ��� ��� �� ���� � �� � �� �� �� �� �� �� �� �� � � � � � � ��� ��� ���� ���� ��� ��� ���� ��� ���

Completions New Structures Based on Completions Gilles Bertrand A completion may be seen as a rewriting rule which permits to derive collections of objects. Completions allows to formulate, in an easy way, inductive definitions.

Completions New Let K be an arbitrary sub-collection of S , K is a dedicated Structures Based on symbol (a kind of variable). Completions Gilles We say that a property � K � is a completion (on S ) if � K � Bertrand may be expressed as the following property: − > If F ⊆ K , then G ⊆ K whenever Cond ( F , G ). � K � where Cond ( F , G ) is a condition on a finite collection F and an arbitrary collection G . Theorem : Let � K � be a completion on S and let X ⊆ S . There exists, under the subset ordering, a unique minimal collection which contains X and which satisfies � K � . We write � X ; K � for this unique minimal collection. If � K � and � Q � are two completions, � K � ∧ � Q � is a completion, the symbol ∧ standing for the logical “and”. We write � X ; K , Q � for � X ; K ∧ Q � .

Completions New Let K be an arbitrary sub-collection of S , K is a dedicated Structures Based on symbol (a kind of variable). Completions Gilles We say that a property � K � is a completion (on S ) if � K � Bertrand may be expressed as the following property: − > If F ⊆ K , then G ⊆ K whenever Cond ( F , G ). � K � where Cond ( F , G ) is a condition on a finite collection F and an arbitrary collection G . Theorem : Let � K � be a completion on S and let X ⊆ S . There exists, under the subset ordering, a unique minimal collection which contains X and which satisfies � K � . We write � X ; K � for this unique minimal collection. If � K � and � Q � are two completions, � K � ∧ � Q � is a completion, the symbol ∧ standing for the logical “and”. We write � X ; K , Q � for � X ; K ∧ Q � .

Completions New Let K be an arbitrary sub-collection of S , K is a dedicated Structures Based on symbol (a kind of variable). Completions Gilles We say that a property � K � is a completion (on S ) if � K � Bertrand may be expressed as the following property: − > If F ⊆ K , then G ⊆ K whenever Cond ( F , G ). � K � where Cond ( F , G ) is a condition on a finite collection F and an arbitrary collection G . Theorem : Let � K � be a completion on S and let X ⊆ S . There exists, under the subset ordering, a unique minimal collection which contains X and which satisfies � K � . We write � X ; K � for this unique minimal collection. If � K � and � Q � are two completions, � K � ∧ � Q � is a completion, the symbol ∧ standing for the logical “and”. We write � X ; K , Q � for � X ; K ∧ Q � .

Example of a Completion: Connectedness We observe that: New Structures Based on A cell is connected; and Completions If S and T are connected, then S ∪ T is connected Gilles Bertrand whenever S ∩ T is non-empty; and All connected complexes may be obtained by iteratively applying the preceding rule. We define the completion � Υ � as follows: − > If S , T ∈ K , then S ∪ T ∈ K whenever S ∩ T � = {∅} . � Υ � We set Π = � C ; Υ � , Π is precisely the collection of all simplicial complexes which are (path) connected. We see that this completion is an alternative to the classical definition of connectedness. Furthermore it provides a constructive way for generating all connected complexes.

Example of a Completion: Connectedness We observe that: New Structures Based on A cell is connected; and Completions If S and T are connected, then S ∪ T is connected Gilles Bertrand whenever S ∩ T is non-empty; and All connected complexes may be obtained by iteratively applying the preceding rule. We define the completion � Υ � as follows: − > If S , T ∈ K , then S ∪ T ∈ K whenever S ∩ T � = {∅} . � Υ � We set Π = � C ; Υ � , Π is precisely the collection of all simplicial complexes which are (path) connected. We see that this completion is an alternative to the classical definition of connectedness. Furthermore it provides a constructive way for generating all connected complexes.

Example of a Completion: Connectedness We observe that: New Structures Based on A cell is connected; and Completions If S and T are connected, then S ∪ T is connected Gilles Bertrand whenever S ∩ T is non-empty; and All connected complexes may be obtained by iteratively applying the preceding rule. We define the completion � Υ � as follows: − > If S , T ∈ K , then S ∪ T ∈ K whenever S ∩ T � = {∅} . � Υ � We set Π = � C ; Υ � , Π is precisely the collection of all simplicial complexes which are (path) connected. We see that this completion is an alternative to the classical definition of connectedness. Furthermore it provides a constructive way for generating all connected complexes.

New Structures Based on Completions Gilles Bertrand Dendrites Motivation: To describe a remarkable collection of acyclic complexes.

Dendrites: the basic idea New Structures Let X and Y be two trees, and let Z = X ∩ Y . Based on Completions Gilles Bertrand � �� �� � �� �� � � �� �� � � �� �� �� �� � � �� �� �� �� � � �� �� �� �� � � �� �� � � � � �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� � � � � � � � �� �� �� �� X Y X ∩ Y ☛ ✟ X ∪ Y is a tree whenever X ∩ Y is a tree X ∩ Y is a tree whenever X ∪ Y is a tree ✡ ✠

Dendrites: the axioms New Structures Based on Completions Gilles We define the completions � D 1 � and � D 2 � as follows: Bertrand For any S , T ∈ S , − > If S , T ∈ K , then S ∪ T ∈ K whenever S ∩ T ∈ K . � D 1 � − > If S , T ∈ K , then S ∩ T ∈ K whenever S ∪ T ∈ K . � D 2 � We set D = � C ; D 1 , D 2 � . Each element of D is a dendrite. It may be shown that a complex is a dendrite if and only if it is acyclic in the sense of integral homology.

Dendrites: the axioms New Structures Based on Completions Gilles We define the completions � D 1 � and � D 2 � as follows: Bertrand For any S , T ∈ S , − > If S , T ∈ K , then S ∪ T ∈ K whenever S ∩ T ∈ K . � D 1 � − > If S , T ∈ K , then S ∩ T ∈ K whenever S ∪ T ∈ K . � D 2 � We set D = � C ; D 1 , D 2 � . Each element of D is a dendrite. It may be shown that a complex is a dendrite if and only if it is acyclic in the sense of integral homology.