On the linear Lindenbaum algebra of Basic Propositional Logic

Abstract

We study the linear Lindenbaum algebra of Basic Propositional Calculus, called linear basic algebra. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

n‐linear weakly Heyting algebras

“…The following result generalizes similar results given in [2] for linear Basic algebras, and results given in [10] and [12] for linear Heyting algebras. www.mlq-journal.org P r o o f. 1.…”

“…On the other hand, in [2] it was proved that every linear Basic algebra satisfies the identities (L1) and (L2). Now we shall prove that all these identities are equivalent in every WH-algebra, and consequently in every Basic algebra.…”

“…On the other hand, in [2] was introduced the linear Basic logic BPC and it was proved that it is complete with respect to the linear Basic algebras, i. e., Basic algebras satisfying the linear identity (x → y) ∨ (y → x) = 1. The logic BPC and the linear Basic algebras are a generalization of the Dummett logic [6] and the linear Heyting algebras [11], respectively.…”

“…First we introduce a more extensive class of frames for the logics wK σ and sK σ . The representation results given in [3] for WH-algebras and the results given in [2] for linear basic algebras suggest to consider ordered WH-frames. These are relational structures F = X, ≤, R , where X, ≤ is a poset and R is a binary relation defined on X satisfying the condition that for any x, y, z ∈ X, if x ≤ y and (y, z) ∈ R, then (x, z) ∈ R. Considering the results of [4] and [3] it is easy to see that the logic wK σ is complete with respect to the class of all ordered WH-frames.…”

“…In the last section we study the variety WH 2 . This variety contains as subvarieties the variety of the linear Heyting algebras (see [10] and [12]) and the variety of linear Basic algebras studied in [2].…”