Structural characterization of (finite) commutative idempotent involutive residuated lattices

Olim Tuyt

(Joint work with Peter Jipsen and Diego Valota)

Let \(\mathsf{CIdInRL}\) be the variety of commutative idempotent involutive residuated lattices. Interesting subvarieties of \(\mathsf{CIdInRL}\) include Sugihara monoids, the algebraic semantics of relevance logic \(\textsf{RM}^t\) [1]. In this talk we present some results on the structure of commutative idempotent involutive residuated lattices, resulting in a structural characterization of all finite members of \(\mathsf{CIdInRL}\).

The algebras \(\mathbf{A} \in \mathsf{CIdInRL}\) are studied by considering their monoidal reduct. For \(a, b \in A\), consider the monoidal order \(\sqsubseteq\), where \(a \sqsubseteq b\) if and only if \(a \cdot b = a\), which forms a meet-semilattice ([2]). For each \(a \in A\), define \(0_a = a \cdot - a\) and \(1_a = a \vee - a\). Then \(\mathbf{A}\) can be partitioned into Boolean algebras \(B_a = \{ b \in A \mid 0_a \sqsubseteq b \sqsubseteq 1_a \}\). Moreover, \(\{ 0_a \mid a \in A \}\) forms a distributive sublattice of \(\mathbf{A}\). These properties are exploited to define a construction combining two members of \(\mathsf{CIdInRL}\) into a new commutative idempotent involutive residuated lattice. It is then shown that this construction characterizes all finite members of the variety.

  1. A.R. Anderson and N.D. Belnap, Entailment. Volume I: The logic of relevance and necessity , Princeton University Press 1975.

  2. J. Gil-Férez and P. Jipsen and G. Metcalfe Structure Theorems for Idempotent Residuated Lattices, preprint.

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