Truth values on generalizations of some commutative fuzzy structures (Q869115)

Basic logic was introduced by Hájek as the logic of continuous t-norms and their adjoint implications. BL-algebras are the algebras of basic logic. To capture the notion of ``very true'', Hájek also introduced \(\text{BL}_{\text{vt}}\)-algebras, adding a suitable unary operation. A generalization of \(\text{BL}_{\text{vt}}\)-algebras is introduced in this paper by equipping every bounded commutative residuated lattice-ordered monoid with a unary subdiagonal and monotone map.