Talk:MV-algebra

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This article may be too technical for most readers to understand, and needs attention from an expert on its subject. Please expand it to make it accessible to non-experts, without removing the technical details.

In my opinion, the above banner is over the top. I am wholly self taught in logic and algebra, and do NOT find this entry unusually challenging. True, a residuated lattice is not an easy concept, but there is no need to grasp it in order to understand what an MV algebra is. An MV algebra is a flavour of commutative monoid, near-baby talk, mathematically speaking

Unrelated question. Is an MV algebra theorem? If yes, MV algebras are well connected either to groups or to Boolean algebras.202.36.179.65 22:37, 17 August 2006 (UTC)

[edit] Wrong Wrong Wrong Wrong Wrong!

There are many errors in this article.

MV algebras were not invented by Jan Łukasiewicz, but by Chang.

Chang's Completeness Theorem is that Łukasiewicz logic is complete for MV algebras.

Change's Representation Theorem is that all MV algebras are isomorphic to the subdirect product of two linear MV algebras.

Identities which hold in a linear MV algebra will hold in (sub)direct product of that linear algebra.

Chang also proved (1959) that any identity in the MV algebra R[1] (an MV algebra for the interval [0,1]) is an identity for any linear MV algebra.

Hence identities in R[1] hold for all MV algebras.

This entry is badly written, and also contains several mistakes. There is no mention of the fact that MV algebras are term-wise equivalent to Wajsberg algebras, which provide the equivalent algebraic semantics for Lukasiewicz logic (in the Sense of Blok and Pigozzi). Chang's Completeness Theorem states that the variety of MV-algebras is generated by the MV-algebra over the real unit interval in which the monoidal operation is interpreted as the bounded sum, and the negation corresponds to the standard involutive negation. There is no mention of the categorical equivalence with lattice-ordered groups with strong unit. Not a word about the celebrated MacNaughton Theorem that states that the free MV-algebra over n generators corresponds to the set of piecewise linear polynomial functions with integer coefficients over the nth-cube. No mention about results on semisimple MV-algebras.

I reworded the lead to clarify the Chang-Lukasiewicz confusion (a mistake which I didn't make in my editing, BTW ; ), although others could be mine). Beyond that, feel free to add the material you discuss yourself, it would be greatly appreciated. Best, Smmurphy^(Talk)

[edit] Too technical

Perhaps some examples would help to remove this criticism... —The preceding unsigned comment was added by 141.30.225.67 (talk) 14:20, August 22, 2007 (UTC)