The Finite Embeddability Property for Some Noncommutative Knotted Varieties of RL and DRL

Residuated lattices, although originally considered in the realm of algebra providing a general setting for studying ideals in ring theory, were later shown to form algebraic models for substructural logics. The latter are non-classical logics that include intuitionistic, relevance, many-valued, and linear logic, among others. Most of the important examples of substructural logics are obtained by adding structural rules to the basic logical calculus FL. We denote by � the varieties of knotted residuated lattices. Examples of these knotted rules include integrality and contraction. The extension of �� by the rules corresponding to these two equations is equivalent to Gentzen’s original system �� for intuitionism. Apart from applications to logic and to abstract ring theory, residuated lattices are connected to mathematical linguistics, computer science, and quantum mechanics, among other areas. Even thought the connections to other disciplines are abundant, the current document is of purely algebraic nature. Results in [17] establish the finite model property (FMP) for the implicational fragment of ��� extended by some knotted rules. The finite embeddability property (FEP) is known to hold for commutative ��^�_� (�� = ��); the strong finite model property follows for the corresponding logics. Recent results by Horčík show that the word problem is undecidable for the varieties ��^�_� when 1 ≤ � \u3c � or 2 ≤ � \u3c �. Therefore these varieties do not have the FEP. We refer the reader to [16] for details on how this is connected to the Burnside problems in group theory and to regularity of languages in automata theory. In the present document, using purely algebraic methods, we prove the FEP for subvarieties of ��^�_� and ���^�_� that satisfy properties weaker than commutativity. The proof uses the theory of residuated frames introduced by Galatos and Jipsen . In Chapter 1, we present the basic definitions and constructions that will be used throughout the full document. We point the reader towards Section 1.4, where we list a relevant list of varieties for which the FEP holds or not. Chapter 2 presents a proof of the FEP for subvarieties of ��^�_� that satisfy the identity ��� = �^2�. The proof of this case relies on finding the free object over the class of pomonoids that satisfy the previous equality and �� ≤ ��. Chapter 3 focuses on the study of the noncommutative equation that we use to define the varieties studied in the following two chapters. This equation arises as a natural generalization of the basic equation ��� = �^2�. Chapter 4 presents the FEP for ��^�_�. In the general case, the free object in the class is fairly complicated, so we identify instead an object outside the class, which is both free and structured enough to allow us to prove the result. In the last section, we extend our result to cover some other subvarieties of knotted residuated lattices. These subvarieties include the cyclic, cyclic-involutive, and representable ones. Chapter 5 details a proof for the fully distributive case. Here we enrich the free object discovered in Chapter 4 by creating the meet semilattice generated by it. We remark that the FEP for a variety � is equivalent to the condition that all finitely presented algebras in � are residually finite. Varieties of semigroups with this property have been fully characterized in [20]. In particular, the variety of monoids axiomatized by ��� = �^2� has been studied and has the FEP. These results do not imply the FEP for the corresponding variety of residuated lattices, which also serves as the simplest case of our analysis. 1. Science, the magazine itself is not showing up at the top, but at the botto

Distributive Residuated Frames and Generalized Bunched Implication Algebras

We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions, we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames