Monadic bounded commutative residuated `-monoids

bounded commutative R`-monoid iff (i) (M;¯, 1) is a commutative monoid, (ii) (M;∨,∧, 0, 1) is a bounded lattice, and (iii) x ¯ y ≤ z ⇐ ⇒ x ≤ y → z, (iv) x ¯ (x → y) = x ∧ y, for each x, y, z ∈M. In the sequel, by an R`-monoid we will mean a bounded commutative R`-monoid. (Note that bounded commutative R`-monoids are just bounded integral generalized BL-algebras in the sense of [1] or [2].) Recall that the class of all R`-monoids is a variety of algebras of type 〈2, 2, 2, 2, 0, 0〉. Especially, BL-algebras, i.e. algebraic counterpart of the propositional basic fuzzy logic (and, consequently, MV-algebras, i.e. algebras of the ÃLukasiewicz infinite valued logic), and Heyting algebras, i.e. algebras of the proposi-tional intuitionistic logic, are particular cases of R`-monoids. Therefore, R`-monoids could be taken as algebras of a more general propositional logic than the basic and intuitionistic logic. On any R`-monoid M we define the unary operation − (negation) by x −: = x → 0, and the binary operation ⊕ (addition) by x ⊕ y: = (x− ¯ y−)−