Semisimplicity, Amalgamation Property and Finite Embeddability Property of Residuated Lattices

In this thesis, we study semisimplicity, amalgamation property and finite embeddability property of residuated lattices. We prove semisimplicity and amalgamation property of residuated lattices which are of purely algebraic character, by using proof-theoretic methods and results of substructural logics. On the other hand, we show the finite model property (FMP) for various substructural logics, including fuzzy logics as a consequence of the finite embeddability property (FEP) of corresponding classes of residuated lattices. Thus all of our studies are attempts at bridging gaps between algebras and logics. The first topics of our thesis is finite embeddability property (FEP) of various classes of integral residuated lattices. A class of algebras has the FEP if every finite partial subalgebra of a member of the class can be embedded into a finite member of the same class. W. Blok and C. J. van Alten showed that the class of all partially ordered biresiduated integral groupoids has the FEP. This implies that the variety of all integral residuated lattices (IRL) has the FEP. The FEP of a given variety of IRL implies the finite model property (FMP) for the corresponding logic. We prove the FEP for various classes of the variety IRL. From this the FMP follows for various substructural logics including fuzzy logics. Next, we study the semisimplicity of free FLw-algebras. An algebra is semisimple if it has a subdirect representation with simple factors. V. N. Grǐsin proved that every free CFLew-algebra is semisimple. To show this Grǐsin introduced a new sequent system which is equivalent to CFLew and showed that the cut elimination theorem holds for the sequent system. Later, T. Kowalski and H. Ono proved that every free FLew-algebras is also semisimple using Grǐsin’s idea. By using this, they proved that the variety of all FLew-algebras is generated by it finite simple members. By using the similar technique, we show that every free FLw-algebras is semisimple. We will introduce a new sequent system FL+w which is equivalent to FLw and for which cut elimination theorem holds. Using proof-theoretic properties of FL+w , we show the semisimplicity of free FLw-algebras. Lastly, we discuss the amalgamation property (AP) of commutative residuated lattices. Kowalski showed the AP for the variety FLew of all FLe-algebras. The result is obtained by the fact that (1) the logical system FLew has the Craig’s interpolation property (CIP), and (2) the variety of FLew has the equational interpolation property (EIP). A. Wroński proved that the EIP of a variety implies the AP. Therefore the variety FLew has the AP. We show that Kowalski’s proof of the AP works well also for the variety CRL of all commutative residuated lattices. To show this result, we introduce a sequent for commutative residuated lattices and show the CIP, and using them we prove that the variety CRL has the EIP. By considering filters on residuated lattices, we can show that many important subclasses of CRL has the AP. Moreover, we can show that if L is a logic which is an extension of FLe with the CIP and K is the variety which is corresponding to L, then K has the EIP, from which the AP of K follows.