The finite embeddability property for some noncommutative knotted extensions of FL

Abstract

We consider the knotted structural rule x<sup>m</sup>≤x<sup>n</sup> for n different than m and m greater or equal than 1. Previously van Alten proved that commutative residuated lattices that satisfy the knotted rule have the finite embeddability property (FEP). Namely, every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. In our work we replace the commutativity property by some slightly weaker conditions. Particularly, we prove the FEP for the variety of residuated lattices that satisfy the equation xyx=x<sup>2</sup>y and the knotted rule. Furthermore, we investigate some generalizations of this noncommutative property by working with equations that allow us to move variables. We also note that the FEP implies the finite model property. Hence the logics modeled by these residuated lattices are decidable.