Modal operators on bounded residuated $\rm l$-monoids

summary:Bounded residuated lattice ordered monoids (${\rm R\ell}$-monoids) form a class of algebras which contains the class of Heyting algebras, i.e.\ algebras of the propositional intuitionistic logic, as well as the classes of algebras of important propositional fuzzy logics such as pseudo $\mathop{\rm MV}$-algebras (or, equivalently, $\mathop{\rm GMV}$-algebras) and pseudo $\mathop{\rm BL}$-algebras (and so, particularly, $\mathop{\rm MV}$-algebras and $\mathop{\rm BL}$-algebras). Modal operators on Heyting algebras were studied by Macnab (1981), on $\mathop{\rm MV}$-algebras were studied by Harlenderová and Rachůnek (2006) and on bounded commutative ${\rm R\ell}$-monoids in our paper which will apear in Math. Slovaca. Now we generalize modal operators to bounded ${\rm R\ell}$-monoids which need not be commutative and investigate their properties also for further derived algebras