Non-commutative fuzzy structures and pairs of weak negations

Weak pseudo BL-algebras (WPBL-algebras) are non-commutative fuzzy structures which arise from pseudo-t-norms (i.e. the non-commutative versions of triangular norms). In this paper, we study the pairs of weak negations on WPBL-algebras, extending the case of weak negations on Esteva–Godo MTL-algebras. A geometrical characterization of the pairs of weak negations in bounded chains is provided. Our main result characterizes the pairs of weak negations compatible with the multiplication of totally ordered WPBL-algebras giving a way to obtain new examples of WPBL-algebras. We use this to identify all the compatible pairs of weak negations on finite MV-chains.