The variety of integral residuated lattices fails the amalgamation property

We answer negatively the open question whether integral residuated lattices have the amalgamation property, recorded by Metcalfe, Paoli, and Tsinakis in their 2023 monograph and explicitly posed by Jipsen and Santschi in Residuated lattices do not have the amalgamation property. A span of finite integral residuated lattices, with a three-element common algebra and extensions of sizes five and six, has no amalgam even among all residuated lattices. Thus the three-element Gödel chain, in the signature without a bottom constant, is not an amalgamation base. The same example excludes amalgamation in every variety containing the five-element chain, including the varieties of n-potent integral residuated lattices for all n ≥ 2. We also isolate a local obstruction without global integrality assumptions and characterize amalgamation of a common-bottom quotient-product span by extension of the corresponding congruence.

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