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 allX∈N. 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 chainX∈N 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
Mathematics Subject Classifications (1991)
- Chain group