Talk:Newman's lemma

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What exactly is meant by connected, directed graph? — Preceding unsigned comment added by 136.159.108.219 (talk) 22:46, 8 May 2012 (UTC)[reply]

I checked the sources (Huet, and Cohen’s “Further algebra and applications” en lieu of “Universal algebra” which I don’t have access to) to figure out what is meant by the “graphs of a preorder”, and as I suspected, it turned out that the current formulation in the article using preorders is thoroughly nonsensical. The sources, as well as our own article Abstract rewriting system, all state the lemma using variations of “well-founded locally confluent relations are confluent”. The relations are not assumed to be preorders, and I have no idea how one would derive this result from some property of preorders. The only preorder that comes to mind when given a general reduction relation is its transitive reflexive closure, but the “diamond property” for this preorder amounts to confluence of the original ARS, not local confluence. It seems that the clumsy wording with “graphs of a preorder” is trying to say that connected components are in 1–1 correspondence with minimal elements (which expresses a form of the Church–Rosser property), but then the lemma is trivially true for transitive relations (as local confluence and confluence coincide by definition for them), and moreover, it is pointless to talk about preorders as the noetherian condition then makes it a proper partial order (which is obvious anyway, since minimal elements cannot possibly be in bijection with components if distinct minimal elements can be equivalent). I’ll try to fix the article.—Emil J. 18:08, 14 June 2013 (UTC)[reply]

Also, diamond property is defined differently in the literature, and it is much stronger that the condition in the assumption of Newman’s lemma.—Emil J. 18:35, 14 June 2013 (UTC)[reply]

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