A general approach to define binders using matching logic

We propose a novel shallow embedding of binders using matching logic, where the binding behavior of object-level binders is obtained for free from the behavior of the built-in existential binder of matching logic. We show that binders in various logical systems such as lambda-calculus, System F, pi-calculus, pure type systems, etc., can be defined in matching logic. We show the correctness of our definitions by proving conservative extension theorems, which state that a sequent/judgment is provable in the original system if and only if it is provable in matching logic. An appealing aspect of our embedding of binders in matching logic is that it yields models to all binders, also for free. We show that models yielded by matching logic are deductively complete to the formal reasoning in the original systems. For lambda-calculus, we further show that the yielded models are representationally complete---a desired property that is not enjoyed by many existing lambda-calculus semantics.Ope

A Set-Theoretical Definition of Application

This paper is in two parts. Part 1 is the previously unpublished 1972 memorandum [41], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary set-theoretical model of the λβ-calculus. Functions are modelled in a similar way to that normally employed in set theory, by their graphs; difficulties are caused in this enterprise by the axiom of foundation. Next, based on that model, a model of the λβη-calculus is constructed by means of a natural deduction method. Finally, a theorem is proved giving some general properties of those non-trivial models of the λβη-calculus which are continuous complete lattices. In the second part we begin with a brief discussion of models of the λ-calculus in set theories with anti-foundation axioms. Next we review the model of the λβ-calculus of Part 1 and also the closely related- but different!- models of Scott [51, 52] and of Engeler [19, 20]. Then we discuss general frameworks in which elementary constructions of models can be given. Following Longo [36], one can employ certain Scott-Engeler algebras. Following Coppo, Dezani-Ciancaglini, Honsell and Longo [13], one can obtain filter models from their Extended Applicative Type Structures. We give an extended discussion of various ways of constructing models of the λβη-calculus, and the connections between them. Finally we give extensions of the theorem to complete partial orders. Throughout we concentrate on means of constructing models. We hardly consider any analysis of their properties; we do not at all consider their application

A Survey on the Conjecture of Lehmer and the Conjecture of Schinzel-Zassenhaus

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Quaternion Algebras

This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout