The canonical FEP construction

Abstract

A class K of algebras has the finite embeddability property (FEP) if every finite partial subalgebra of some member of K can be embedded into some finite member of K⁠. We prove the FEP for varieties of decreasing residuated lattice-ordered algebras using a construction based on the canonical extension. This construction produces a (generally) different finite member of the class from alternative FEP constructions for similar classes of algebras. Additionally, the constructed algebra is internally compact, in contrast to other FEP constructions. We give a description of the σ- and π-extensions of operations that do not rely on the notions of closed and open elements and we use this to obtain a syntactic description of a class of inequalities s≤t that are preserved by the construction.