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Orientations of chain groups

Abstract

We prove generalizations to chain groups, of Minty's Arc Colouring Lemma and its extension, the well-known Farkas Lemma. In these the orientation of the edges is replaced by an arbitrary chain.

A function ϕ on a chain groupN isrepresentable if there exists a chainR such that ϕ(X)=R·X for allXN. Anorientation is a chain with values ±1. We prove that for a regular chain group a linear function that is representable by an orientation for each chainXN locally, is representable by an orientation globally.

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Communicated by I. Rival

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Pretzel, O. Orientations of chain groups. Order 12, 135–147 (1995). https://doi.org/10.1007/BF01108623

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Mathematics Subject Classifications (1991)

  • 05B35
  • 90C27
  • 05C38

Key words

  • Chain group
  • orientation
  • circuit
  • inversion
  • invariant