Relational Representation Theorems for General Lattices with Negations

The Beth property and interpolation in lattice-based algebras and logics

“…[1]). Representation theorems for algebras in the above classes can be found in [1,2]. We also consider subclasses of these classes based on distributive lattices, i.e., those satisfying the following identities (D):…”

“…We consider some classes of (not necessarily distributive) lattices with additional operations as well as logics based on these classes (see [1,2]). A bounded lattice is a lattice A = (A, ∨, ∧, 0, 1), where 0 is the least element and 1 is the greatest element of A.…”

“…In [1], general lattices with negation were considered, where the negation meets the following condition (N 2):…”

“…In the present paper, we will look at the logics based on lattices with additional operations introduced in [1,2]. Interrelations of different versions of interpolation, the Beth property, and amalgamation were investigated in [3,4] in relation to modal logics and varieties of modal algebras, superintuitionistic logics and varieties of Heyting algebras, positive logics and varieties of relatively pseudocomplemented lattices.…”

Unification in Intermediate Logics

“…Whether there is an intermediate logic with infinitary unification type is not known. Dzik located logics with unitary unification in the lattice of intermediate logics by showing that all extensions of De Morgan logic have nullary or unitary unification, and that for the latter the converse holds too [6]. There exist modal logics for which the unification problem is undecidable [33], but whether there exists an intermediate logic with this property is unknown.…”