Tense operators on non-commutative residuated lattices

In this paper, the algebraic properties of tense operators on a non-commutative residuated lattice are investigated. First, some examples and basic properties are given. Next, it is proved that the Dedekind–MacNeille completion of a tense non-commutative residuated lattice is again a tense non-commutative residuated lattice, together with suitable operations. In the sequel, the concept of tense filter is introduced and some structural theorems are given. Several characterizations of normal tense filters and maximal tense filters are obtained, as well. Also, some characterizations of tense non-commutative residuated lattices using normal tense filters are given.