Every 𝑚-permutable variety satisfies the congruence identity 𝛼𝛽_{ℎ}=𝛼𝛾_{ℎ}

Lipparini, Paolo

doi:10.1090/S0002-9939-07-09337-9

Every $m$-permutable variety satisfies the congruence identity $\alpha \beta _h= \alpha \gamma _h$
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Proc. Amer. Math. Soc. 136 (2008), 1137-1144 Request permission

Abstract:

It is known that congruence lattices of algebras in $m$-permutable varieties satisfy non-trivial identities; however, the identities discovered so far are rather artificial and seem to have little intrinsic interest. We show here that every $m$-permutable variety satisfies the well-known and well-studied congruence identity $\alpha \beta _h= \alpha \gamma _h$. We also get a new condition equivalent to $m$-permutability.

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