MTL-algebras that define the dual monoidal operation

As is well-known standard MTL-algebras in general do not define the t-conorm +* associated with their t-norm *. As +* is defined by x+* y = 1−((1−x) * (1−y)), we address the generalised problem of characterising those MTL-algebras with monoidal operation * that define the dual monoidal operation x+* y = ∼(∼x*∼y) for some order-reversing involution ∼. The barest requirement on such structures is clearly that they are subdirect products of order-anti-automorphic chains (o.a.a., for short). We deal with the case of BL-algebras, stating two properties of involutions and fully characterising those BL-algebras defining the dual monoidal operation when the involution satisfies both properties. We also exhibit a BL-chain defining the dual monoidal operation determined by an involution failing both properties. We further prove that all o.a.a. algebras in the variety generated by EMTL-algebras and IMTL-algebras define the dual monoidal operation uniformly with the same term. By contrast, we present a variety whose o.a.a. chains define the dual monoidal operation, but with distinct terms for distinct algebras, generally. If we require definability of the dual residual operation, too, we are left with IMTL-algebras as the only known examples.