Adjoint Operations in Twist-Products of Lattices

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 R. Cignoli as well C. Tsinakis A. Wille such that it becomes a lattice. For every a∈L we define certain subset Pa(L) of L2. We characterize when is sublattice the (L2,⊔,⊓). In this case together some natural antitone involution ′ pseudo-Kleene If L distributive then (Pa(L),⊔,⊓,′) Kleene present sufficient conditions for being subalgebra (L2,⊔,⊓,⊙,⇒) thus pair adjoint on Pa(L). Finally, introduce another bounded lattice resulting algebra satisfying double negation law, investigate closed under these new operations.