Related papers: A Unified Approach to Algebraic Set Theory
Categorical sequence clustering plays a crucial role in various fields, but the lack of interpretability in cluster assignments poses significant challenges. Sequences inherently lack explicit features, and existing sequence clustering…
Combinatorial design theory studies set systems with certain balance and symmetry properties and has applications to computer science and elsewhere. This paper presents a modular approach to formalising designs for the first time using…
The theory of associative $n$-categories has recently been proposed as a strictly associative and unital approach to higher category theory. As a foundation for a proof assistant, this is potentially attractive, since it has the potential…
We construct a topology on a given algebraically closed field with a distinguished subfield which is also algebraically closed. This topology is finer than Zariski topology and it captures the sets definable in the pair of algebraically…
Bishop's informal set theory is briefly discussed and compared to Lawvere's Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive…
Classification and invariants, with respect to basis changes, of finite dimensional algebras are considered. An invariant open, dense (in the Zariscki topology) subset of the space of structural constants is defined. The algebras with…
In the present paper, we propose Abstract Algebraic Logic (AAL) as a general logical framework for Judgment Aggregation. Our main contribution is a generalization of Herzberg's algebraic approach to characterization results in on judgment…
In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along…
We study a new model theory for formal mathematical systems that we developed in a previous paper. We introduce isomorphic and homomorphic structures for formal languages, present some results and examples and conclude our paper with a…
Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…
Classical algebraic structures require exact satisfaction of their defining axioms. We propose similarity algebra, a framework extending algebraic and Lie structures to settings where operations satisfy quantitative bounds up to a tolerance…
This is an introduction to cluster algebras and their common triangular bases. These bases are Kazhdan-Lusztig-type and serve as the canonical bases of cluster algebras from the representation-theoretic point of view. We review seeds…
Using ideas from synthetic topology, a new approach to descriptive set theory is suggested. Synthetic descriptive set theory promises elegant explanations for various phenomena in both classic and effective descriptive set theory.…
Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to…
This chapter lays out a framework for discussing (\ast)-structures on module-algebras over a Hopf (\ast)-algebra (H). We define a complex conjugation functor (V \mapsto \bar{V}), which is an involution on the module category (\hmod), and…
When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theorems from Maskin's 2004 survey article on Auction Theory using the Isabelle/HOL…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
Algebraic logic studies algebraic theories related to proposition and first-order logic. A new algebraic approach to first-order logic is sketched in this paper. We introduce the notion of a quantifier theory, which is a functor from the…
The accurate representation of epistemic uncertainty is a challenging yet essential task in machine learning. A widely used representation corresponds to convex sets of probabilistic predictors, also known as credal sets. One popular way of…
The kernel of analysis, to me anyway, is the following idea: A point is arbitrarily close to a set if every neighborhood of the point intersects the set. Defining ``arbitrarily close'' in this way provides a foundation for classical results…