Concept

Coherent space

Résumé
In proof theory, a coherent space (also coherence space) is a concept introduced in the semantic study of linear logic. Let a set C be given. Two subsets S,T ⊆ C are said to be orthogonal, written S ⊥ T, if S ∩ T is ∅ or a singleton. The dual of a family F ⊆ ℘(C) is the family F ⊥ of all subsets S ⊆ C orthogonal to every member of F, i.e., such that S ⊥ T for all T ∈ F. A coherent space F over C is a family of C-subsets for which F = (F ⊥) ⊥. In Proofs and Types coherent spaces are called coherence spaces. A footnote explains that although in the French original they were espaces cohérents, the coherence space translation was used because spectral spaces are sometimes called coherent spaces. Definitions As defined by Jean-Yves Girard, a coherence space \mathcal{A} is a collection of sets satisfying down-closure and binary completeness in the following sense:
  • Down-closure: all subsets of a set in \mathcal A remain in \mathcal A:
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