Amalgamation in Semilinear Residuated Lattices

TL;DR

This paper surveys the current understanding of amalgamation properties in semilinear residuated lattices, focusing on idempotent and cancellative varieties, and highlights open questions in the cancellative case.

Contribution

It provides a comprehensive overview of amalgamation in semilinear residuated lattices, applying general tools to specific cases and identifying remaining open problems.

Findings

The variety of commutative semilinear residuated lattices lacks the amalgamation property.

Amalgamation is well understood in most semilinear varieties.

Open questions mainly remain in the cancellative varieties.

Abstract

We survey the state of the art on amalgamation in varieties of semilinear residuated lattices. Our discussion emphasizes two prominent cases from which much insight into the general picture may be gleaned: idempotent varieties and their generalizations ($n$-potent varieties, knotted varieties), and cancellative varieties and their relatives (MV-algebras, BL-algebras). Along the way, we illustrate how general-purpose tools developed to study amalgamation can be brought to bear in these contexts and solve some of the remaining open questions concerning amalgamation in semilinear varieties. Among other things, we show that the variety of commutative semilinear residuated lattices does not have the amalgamation property. Taken as a whole, we see that amalgamation is well understood in most interesting varieties of semilinear residuated lattices, with the last few outstanding open questions remaining principally in the cancellative setting.