Abstract
Given an integral commutative residuated lattices L=(L,∨,∧), its full twist-product (L2,⊔,⊓) can be endowed with two binary operations ⊙ and ⇒ introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a∈L we define a certain subset Pa(L) of L2. We characterize when Pa(L) is a sublattice of the full twist-product (L2,⊔,⊓). In this case Pa(L) together with some natural antitone involution ′ becomes a pseudo-Kleene lattice. If L is distributive then (Pa(L),⊔,⊓,′) becomes a Kleene lattice. We present sufficient conditions for Pa(L) being a subalgebra of (L2,⊔,⊓,⊙,⇒) and thus for ⊙ and ⇒ being a pair of adjoint operations on Pa(L). Finally, we introduce another pair ⊙ and ⇒ of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law, and we investigate when Pa(L) is closed under these new operations.
Highlights
Kalman ([1]) as a special kind of De Morgan lattices that serves as an algebraic axiomatization of a certain propositional logic satisfying the double negation law but not necessarily the excluded middle law
If the underlying lattice is not distributive, such lattices are called pseudoKleene. It is a question if certain binary operations can be introduced in a Kleene or pseudo-Kleene lattice such that they form an adjoint pair
We apply an approach using the full twist-product construction and another construction extending a distributive lattice to a Kleene one
Summary
If the underlying lattice is not distributive, such lattices are called pseudoKleene (see e.g., [2]) It is a question if certain binary operations can be introduced in a Kleene or pseudo-Kleene lattice such that they form an adjoint pair. To solve this problem, we apply an approach using the full twist-product construction and another construction extending a distributive lattice to a Kleene one. M. Wille ([4]) introduced binary operations and ⇒ on the full twist-product ( L2 , t, u) to be converted into a residuated lattice ( L2 , t, u, , ⇒, (1, 1)).
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