Embedding theorems and finiteness properties for residuated structures and substructural logics

Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2008.Paper 1. This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, which either treated fewer subsignatures or focussed on the conservation of theorems only. Paper 2. It is proved that the variety of relevant disjunction lattices has the finite embeddability property (FEP). It follows that Avron’s relevance logic RMImin has a strong form of the finite model property, so it has a solvable deducibility problem. This strengthens Avron’s result that RMImin is decidable. Paper 3. An idempotent residuated po-monoid is semiconic if it is a subdirect product of algebras in which the monoid identity t is comparable with all other elements. It is proved that the quasivariety SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite. The lattice-ordered members of this class form a variety SCIL, which is not locally finite, but it is proved that SCIL has the FEP. More generally, for every relative subvariety K of SCIP, the lattice-ordered members of K have the FEP. This gives a unified explanation of the strong finite model property for a range of logical systems. It is also proved that SCIL has continuously many semisimple subvarieties, and that the involutive algebras in SCIL are subdirect products of chains. Paper 4. Anderson and Belnap’s implicational system RMO can be extended conservatively by the usual axioms for fusion and for the Ackermann truth constant t. The resulting system RMO is algebraized by the quasivariety IP of all idempotent commutative residuated po-monoids. Thus, the axiomatic extensions of RMO are in one-to-one correspondence with the relative subvarieties of IP. It is proved here that a relative subvariety of IP consists of semiconic algebras if and only if it satisfies x (x t) x. Since the semiconic algebras in IP are locally finite, it follows that when an axiomatic extension of RMO has ((p t) p) p among its theorems, then it is locally tabular. In particular, such an extension is strongly decidable, provided that it is finitely axiomatized

An algebraic study of residuated ordered monoids and logics without exchange and contraction.

Thesis (Ph.D.)-University of Natal, Durban, 1998.Please refer to the thesis for the abstract

Hyperidentities and Related Concepts, II

This survey article illustrates many important current trends and perspectives for the field including classification of hyperidentities, characterizations of algebras with hyperidentities, functional representations of free algebras, structure results for bilattices, categorical questions and applications. However, the paper contains new results and open problems, too

Admissible rules and the Leibniz hierarchy

This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and signi cance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the non-algebraizable fragments of relevance logic are considered.This work is based on research supported in part by the National Research Foundation of South Africa (UID 85407).https://www.dukeupress.edu/notre-dame-journal-of-formal-logichb2016Mathematics and Applied Mathematic

Frame definability in finitely-valued modal logics

In this paper we study frame definability in finitely valued modal logics and establish two main results via suitable translations: (1) in finitely valued modal logics one cannot define more classes of frames than are already definable in classical modal logic (cf. [27, Thm. 8]), and (2) a large family of finitely valued modal logics define exactly the same classes of frames as classical modal logic (including modal logics based on finite Heyting and MV-algebras, or even BL-algebras). In this way one may observe, for example, that the celebrated Goldblatt–Thomason theorem applies immediately to these logics. In particular, we obtain the central result from [26] with a much simpler proof and answer one of the open questions left in that paper. Moreover, the proposed translations allow us to determine the computational complexity of a big class of finitely valued modal logics

Decidability for Residuated Lattices and Substructural Logics

We present a number of results related to the decidability and undecidability of various varieties of residuated lattices and their corresponding substructural logics. The context of this analysis is the extension of residuated lattices by various simple equations, dually, the extension of substructural logics by simple structural rules, with the aim of classifying simple equations by the decidability properties shared by their extensions. We also prove a number of relationships among simple extensions by showing the equational theory of their idempotent semiring reducts coincides with simple extensions of idempotent semirings. On the decidability front, we develop both semantical and syntactical methods for establishing decidability as well as tractability of decision procedures. On the undecidability front, we develop a notion of algebraic machines for which the theory of residuated frames will allow us to encode decision problems within the theories of residuated lattices and their substructural analogues. We prove the undecidability of the word problem for a broad class of simple extensions for both commutative and non-commutative residuated lattices. Furthermore, through a deduction theorem we establish the undecidability of the equational theory for a broad class of simple extensions. Translated in terms of substructural logics, we prove that the undecidability of both provability and deducibility for a multitude of extensions of FLe by simple rules

Preservation theorems for algebraic and relational models of logic

A thesis submitted to the School of Computer Science, Faculty of Science, University of the Witwatersrand, Johannesburg in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 15 May 2013In this thesis a number of different constructions on ordered algebraic structures are studied. In particular, two types of constructions are considered: completions and finite embeddability property constructions. A main theme of this thesis is to determine, for each construction under consideration, whether or not a class of ordered algebraic structures is closed under the construction. Another main focus of this thesis is, for a particular construction, to give a syntactical description of properties preserved by the construction. A property is said to be preserved by a construction if, whenever an ordered algebraic structure satisfies it, then the structure obtained through the construction also satisfies the property. The first four constructions investigated in this thesis are types of completions. A completion of an ordered algebraic structure consists of a completely lattice ordered algebraic structure and an embedding that embeds the former into the latter. Firstly, different types of filters (dually, ideals) of partially ordered sets are investigated. These are then used to form the filter (dually, ideal) completions of partially ordered sets. The other completions of ordered algebraic structures studied here include the MacNeille completion, the canonical extension (also called the completion with respect to a polarization) and finally a prime filter completion. A class of algebras has the finite embeddability property if every finite partial subalgebra of some algebra in the class can be embedded into some finite algebra in the class. Firstly, two constructions that establish the finite embeddability property for residuated ordered structures are investigated. Both of these constructions are based on completion constructions: the first on the Mac- Neille completion and the second on the canonical extension. Finally, algebraic filtrations on modal algebras are considered and a duality between algebraic and relational versions of filtrations is established