Modal MTL-algebras

A modal MTL-algebra is an algebra in the variety generated by the modal MTL-chains—linearly ordered commutative, bounded, integral, residuated lattices equipped with a unary order-preserving operation. Reverse modal MTL-algebras can be defined similarly by equipping a unary order-reversing operation instead. We axiomatize the variety of (reverse) modal MTL-algebras. Two constructions are considered on (reverse) modal MTL-chains: the MacNeille completion of the underlying order and a finite embeddability construction. In both cases we define a suitable extension of the unary order-preserving (-reversing) operation. Properties preserved via these constructions are investigated using approximations. In particular, a large class of identities preserved by each of the constructions is described syntactically.