Finiteness Properties for Idempotent Residuated Structures

A class K of similar algebras is said to have the finite embeddability property (briefly, the FEP) if every finite subset of an algebra in K can be extended to a finite algebra in K, with preservation of all partial operations. If a finitely axiomatized variety or quasivariety of finite type has the FEP, then its universal first order theory is decidable, hence its equational and quasi-equational theories are decidable as well. Where the algebras are residuated ordered groupoids, these theories are often interchangeable with logical systems of independent interest. Partly for this reason, there has been much recent investigation of finiteness properties such as the FEP in varieties of residuated structures. A residuated partially ordered monoid is said to be idempotent if its monoid operation is idempotent. In this case, the partial order is equationally definable, so the structures can be treated as pure algebras. Such an algebra is said to be conic if each of its elements lies above or below the monoid identity t ; it is semiconic if it is a subdirect product of conic algebras. We prove that