Existence of a non-lattice λ-lattice satisfying (C1) and (C2) whose Sasaki operations form an adjoint pair

Determine whether there exists a λ-lattice (A,⊔,⊓,′) that is not a lattice, satisfies the identities (C1) y ⊔ (x ⊔ y′) ⊓ y ≈ x ⊔ y and (C2) x ⊔ (x ⊓ y) ⊓ x ≈ x ⊓ y for all x,y ∈ A, and whose Sasaki operations defined by (S2) x ⊙ y := (x ⊔ y′) ⊓ y and x → y := x ⊔ (x ⊓ y) form an adjoint pair. Ascertain existence by constructing such an example or prove that no such λ-lattice can exist.

Background

The paper studies when Sasaki operations form an adjoint pair across several algebraic structures beyond orthomodular lattices, including lattices with a unary operation, λ-lattices, and ordered semirings. For λ-lattices (A,⊔,⊓,′), the authors define Sasaki operations by (S2): x ⊙ y := (x ⊔ y′) ⊓ y and x → y := x ⊔ (x ⊓ y). They also introduce strengthened identities (C1) and (C2) as λ-lattice analogues of conditions used in the lattice case.

While examples show that adjointness can hold in non-lattice λ-lattices without (C1) and (C2), and that (C2) is not necessary for (A2), the authors do not know any non-lattice λ-lattice satisfying both (C1) and (C2) whose Sasaki operations form an adjoint pair. Moreover, Theorem 3.7 shows that if the unary operation is surjective and (C1) and (C2) hold together with adjointness, then the structure must be a lattice, suggesting strong constraints on possible counterexamples.