Unification and Projectivity in De Morgan and Kleene Algebras

Abstract

We provide a complete classification of solvable instances of the equational unification problem over De Morgan and Kleene algebras with respect to unification type. The key tool is a combinatorial characterization of finitely generated projective De Morgan and Kleene algebras

Distributive bilattices from the perspective of natural duality theory

“…nullary if there exists u, a unifier of A, such that u v for each mg-unifier of A (in symbols, type A (A) = 0); unitary if there exists a unifier u of A such that v u for each unifier v of A (type A (A) = 1); finitary if there exists a finite set U of mg-unifiers of A such that for each unifier v of A there exists u ∈ U with v u, and for each v of A there exists w unifier of A with w v (type A (A) = ω); and infinitary otherwise (type A (A) = ∞). In [4], an algorithm to classify finitely presented bounded distributive lattices by their unification type was presented. Since the unification type of an algebra is a categorical invariant (see [21]), the results in [4] can be combined with the equivalence between DB and D to investigate the unification types of finite distributive bilattices.…”

Admissibility via natural dualities

“…We remark that the algebra E DMA (Y, ≤, f , P(Y )) defined in this lemma not only embeds into F DMA , it is also a retract of F DMA . A characterization of finite retracts of free De Morgan algebras, the finite projective algebras of this variety, is given in [7]. 5.2.…”

“…So η ∧ µ is constantly 0; that is, c ∧ d = ⊥. But (η ∨ µ)(x) = η(x) ∨ µ(x) = 1, and therefore (η ∨ µ)(x) = 1 ¬(η ∨ µ)(x) = 0; that is, c ∨ d ¬(c ∨ d) which contradicts(7). (ii) ⇒ (i).…”

“…(b) For every x ∈ X, there exists y ∈ X such that y ≤ x, f (x). (iii) A satisfies (6) and(7).Proof. (ii) ⇔ (iii).…”

Unification on Subvarieties of Pseudocomplemented Distributive Lattices

“…Unification in different fragments of intuitionistic logic that include the implication were investigated in [5,11,15,18]. The variety of bounded distributive lattices was proved to have nullary type (that is, there exists a unification problem that does not admit a minimal complete set of unifiers) in [7], and the type of each unification problem was calculated in [4].…”