The subvariety of commutative residuated lattices represented by twist-products

Abstract

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Existentially closed De Morgan algebras

“…where D L is the underlying lattice of D. Note that this construction is sometimes referred to as twist-product (see, for example, [BC14]) and is attributed to Kalman [Kal58].…”

Adjoint Operations in Twist-Products of Lattices

“…• When can P a (L) be equipped with these operations forming an adjoint pair? • Can we define the operations and ⇒ in a way different from that of [3] or [4] to obtain an integral residuated lattice on the full twist-product (L 2 , , )?…”

“…Assume now that (L, ∨, ∧, •, →, 1) is an integral commutative residuated lattice. In Theorem 3.1 in [3] which is a particular case of Corollary 3.6 in [4], Busaniche and Cignoli introduced two additional binary operations and ⇒ on its full twist-product (L 2 , , ) as follows:…”

“…They showed that (L 2 , , , , ⇒, (1, 1)) is again a commutative residuated lattice, i.e., and ⇒ form an adjoint pair. For the convenience of the reader we provide a proof since it is not explicitly contained in [3,4].…”

Constructions of Kleene lattices

“…Consider a distributive lattice L = (L, ∨, ∧) and an arbitrary element a of L. We can construct the so-called full twist-product of L. By the full twist-product of L (see e.g. [2] and [7]) is meant the lattice (L 2 , ⊔, ⊓) where ⊔ and ⊓ are defined as follows:…”