Constrained abstract representation problems in semigroups and partial groupoids

In this paper different constrained abstract representation theorems for partial groupoids and semigroups are proved. Methods for improving the retract properties of the structures are also developed. These have strong class-theoretical implications for many types of generalized periodic semigroups and related partial semigroups. The results are significant in a model-theoretical setting

Tarski's Q-relation algebras and Thompson's groups

The connections between Tarski's Q-relation algebras and Thompson's groups F, T, V, and monoid M are reviewed here, along with Jonsson-Tarski algebras, fork algebras, true pairing algebras, and tabular relation algebras. All of these are related to the finitization problem and Tarski's formalization of set theory without variables. Most of the technical details occur in the variety of J-algebras, which is obtained from relation algebras by omitting union and complementation and adopting a set of axioms created by Jonsson. Every relation algebra or J-algebra that contains a pair of conjugated quasiprojections satisfying the Domain and Unicity conditions, such as those that arise from J\'onsson-Tarski algebras or fork algebras, will also contain homomorphic images of F, T, V, and M. The representability of tabular relation algebras is extended here to J-algebras, using a notion of tabularity equivalent among relation algebras to the original definition.Comment: 64 pages, 4 figures, 1 tabl

Issues in commonsense set theory

The success of set theory as a foundation for mathematics inspires its use in artificial intelligence, particularly in commonsense reasoning. In this survey, we briefly review classical set theory from an AI perspective, and then consider alternative set theories. Desirable properties of a possible commonsense set theory are investigated, treating different aspects like cumulative hierarchy, self-reference, cardinality, etc. Assorted examples from the ground-breaking research on the subject are also given. © 1995 Kluwer Academic Publishers

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

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