Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let g: D × D → R be a symmetric function on a finite set D satisfying g(x, x) = 0 for all x ∈ D. A switch gσ of g w.r.t. a local valuation σ: D → R is defined by gσ(x, y) = σ(x) + g(x, y) + σ(y) for x ≠ y and gσ(x, x) = 0 for all x.  show that every symmetric function g has a unique minimal semimetric switch, and, moreover, there is a switch of g that is isometric to a finite Manhattan metric. Also, for each metric on D,  associate an extension metric on the set of all nonempty subsets of D, and  show that this extended metric inherits the switching classes on D.
['Andrzej Ehrenfeucht', 'Tero Harju', 'Grzegorz Rozenberg']
I am writing a research paper on a idea Finite metrics in switching classes
[{"from": "human", "value": "I am writing a research paper on a idea Finite metrics in switching classes"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let g: D \u00d7 D \u2192 R be a symmetric function on a finite set D satisfying g(x, x) = 0 for all x \u2208 D. A switch g\u03c3 of g w.r.t. a local valuation \u03c3: D \u2192 R is defined by g\u03c3(x, y) = \u03c3(x) + g(x, y) + \u03c3(y) for x \u2260 y and g\u03c3(x, x) = 0 for all x.  show that every symmetric function g has a unique minimal semimetric switch, and, moreover, there is a switch of g that is isometric to a finite Manhattan metric. Also, for each metric on D,  associate an extension metric on the set of all nonempty subsets of D, and  show that this extended metric inherits the switching classes on D."}]