Group Representation for Even and Odd Involutive Commutative Residuated Chains

Abstract

For odd and for even involutive, commutative residuated chains a representation theorem is presented in this paper by means of direct systems of abelian o-groups equipped with further structure. This generalizes the corresponding result of J. M. Dunnabout finite Sugihara monoids.

Amalgamation in classes of involutive commutative residuated lattices

“…In order to provide an insight on why algebras in I c 0 ∪ I c 1 can be represented by bunches of layer groups, in addition to bunches of layer groups, also the notion of bunches of layer algebras has been introduced in [8]. There, algebras in I c 0 ∪ I c 1 are first decomposed into their layer algebras and then the layer algebras are deformed to become layer groups.…”

“…This intermediate clarifying step will not be needed in the present paper. Therefore, the two steps in the main representation theorem of [8] are reformulated here in Theorem 2.5 in a more condensed one-step form (i.e., without referring to layer algebras) to make the proof of Lemma 3.2 much shorter. (A) Given an odd or an even involutive FL e -chain X = (X,…”

“…Denote by ½ the trivial (one-element) group. Since the representation of totally ordered even or odd Sugihara monoids is of the form ½ u , ½ u , ς u→v ∅,∅,κ,≤κ or ½ u , ½ u , ς u→v {t},∅,κ\{t},≤κ , respectively [8], in every layer the groups in the induced V-formation are trivial, and the construction of Theorem 3.5 produces the trivial group as their amalgamation in every layer. Hence the amalgamation, as done in this paper, of a V-formation of totally ordered even or odd Sugihara monoids is also a totally ordered even or odd Sugihara monoid, respectively.…”

“…On that basis, we consider two subclasses which contain only algebras which are 'idempotent symmetric', and show that those classes (and hence also the generated semilinear variety) have the amalgamation property. The proofs make use of a representation theorem, in which odd or even totally ordered involutive commutative pointed residuated lattices are represented as direct systems of abelian o-groups equipped with some further structure [8] (see Theorem 2.5 below). After some preliminaries on residuated lattices, amalgamation, and direct systems in Section 2, the main construction and results are presented in Section 3.…”

Logics of Variable Inclusion

“…[8]). Recently, the tools offered by Płonka sums have fruitfully been extended to the structural analysis of residuated structures, establishing a natural connection with substructural logics (see [21], [18]). In line with this trend, we will further extend the application of the method.…”