Recent developments in the categorical foundations of universal algebra have given an impetus to an understanding of the lambda calculus coming from categorical logic: an interpretation is a semi-closed algebraic theory. Scott’s representation theorem is then completely natural and leads to a precise Fundamental Theorem showing the essential equivalence between the categorical and more familiar...

Birkhoff’s variety theorem from universal algebra characterises equational subcategories of varieties. We give an analogue of Birkhoff’s theorem in the setting of enrichment in categories. For a suitable notion of an equational subcategory we characterise these subcategories by their closure properties in the ambient algebraic category.

The ChangLoś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and...

The recursion theorem in the weak form {e}(z) = x(e, z) (universal function not needed) and in Rogers form φφf(n)(z) = φn(z), and Rice theorem are proved a first time using programs in C, and a second time with scripts in Bash.

The main result is a fixed point theorem for compositions of chain faithful mttltifunctions (Corollary 2.3). The theorem is then applied to get sufficient conditions for the fixed point property of the product of two partially ordered sets. For the Special Issue on the NIH Conference on Universal Algebra & Lattice Theory

In this paper we study diagonal quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin the Conway–Schneeberger 15 theorem.

In classical information theory, entropy rate and algorithmic complexity per symbol are related by a theorem of Brudno. In this paper, we prove a quantum version of this theorem, connecting the von Neumann entropy rate and two notions of quantum Kolmogorov complexity, both based on the shortest qubit descriptions of qubit strings that, run by a universal quantum Turing machine, reproduce them a...

We generalize a theorem of Mazur concerning the universal norms of an abelian variety over a Zp-extension of a complete local field. Then we apply it to the proof of a control theorem for abelian varieties over global function fields.

In this article, a brief history and recent developments on universal forms are introduced. Recent developments include Conway-Schneeberger’s fifteen theorem, Bhargava’s finiteness theorem on representability of an infinite set of positive integers by positive definite integral quadratric forms, and their generalizations to higher rank representability. Some interesting corollaries and applicat...