Monadic Bounded Commutative Residuated l-monoids

An algebra M = (M ; ,∨,∧,→, 0, 1) of type 〈2, 2, 2, 2, 0, 0〉 is called a 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 Lukasiewicz infinite valued logic), and Heyting algebras, i.e. algebras of the propositional 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−)−. Then, among others, (M ;⊕) is a semigroup and x⊕0 = x−− for each x ∈ M . Further, we say that an R`-monoid M is normal iff M satisfies identity (x y)−− = x−− y−−. (Both BL-algebras and Heyting algebras are normal R`-monoids.)