Some properties of (strong) semi-divisible residuated lattices and modal operators

In this paper, we introduce a new class of residuated lattices called strong semi-divisible residuated lattices and investigate some of their properties. We show that contrarily to modal operators defined on divisible residuated lattices, not all modal operators defined on (strong) semi-divisible residuated lattices are closure operators. Moreover, we give an equivalent condition for a modal operator to be a closure operator on a semi-divisible residuated lattice. We then define some operators on a normal semi-divisible residuated lattice and show that they are strong modal operators. Finally, after giving some connections between the set of idempotent and the set of semi-idempotent elements of a normal residuated lattice, we construct a (strong) semi-divisible residuated lattice which is not an Rℓ-monoid and define a monotone modal operator on it.