相关论文: Identity and Categorification

Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…

In his 1879 paper on the Begriffsschrift, Gottlob Frege introduced a notation to formalize mathematical arguments. In this note we explain Frege's notation by using the nowadays common notions from elementary propositional logic. We compare…

We introduce a new way of formalizing the intensional identity type based on the fact that a entity known as computational paths can be interpreted as terms of the identity type. Our approach enjoys the fact that our elimination rule is…

The basic concepts of category theory are developed and examples of them are presented to illustrate them using measurement theory and probability theory tools. Motivated by Perrone's workarXiv:1912.10642 where notes on category theory are…

The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…

In these lecture notes, we give a brief introduction to some elements of category theory. The choice of topics is guided by applications to functional programming. Firstly, we study initial algebras, which provide a mathematical…

Families of operator identities appeared as a consequence of an existence of finite-dimensional representation of (super) Lie algebras of first-order differential operators and $q$-deformed (quantum) algebras of first-order…

Menon's identity is a classical identity involving gcd sums and the Euler totient function $\phi$. In a recent paper, Zhao and Cao derived the Menon-type identity $\sum\limits_{\substack{k=1}}^{n}(k-1,n)\chi(k) = \phi(n)\tau(\frac{n}{d})$,…

Using the Hilbert-Bernays account as a spring-board, we first define four ways in which two objects can be discerned from one another, using the non-logical vocabulary of the language concerned. (These definitions are based on definitions…

The proof identity problem asks: When are two proofs the same? The question naturally occurs when one reflects on mathematical practice. The problem understandably can be seen as a challenge for mathematical logic, and indeed various…

Frolicher and Nijenhuis recognized well in the middle of the previous century that the Lie bracket and its Jacobi identity could and should exist beyond Lie algebras. Nevertheless the conceptual meaning of their discovery has been obscured…

When the discriminants $\Delta$ and $\Delta p^2$ are idoneal, Patane proved a theorem which connects the theta series associated to binary quadratic forms of each discriminant. This paper generalizes the main theorem of Patane by no longer…

In this paper we describe the the category of Lie algebras of group algebras and the category of Plesken Lie algebras and explore the categorical relations between them. Further we provide the examples of the Lie algebra of the group…

In this paper I consider some logical and mathematical aspects of the discussion of the identity and individuality of quantum entities. I shall point out that for some aspects of the discussion, the logical basis cannot be put aside; on the…

We solve the word problem of the identity $x(yz) = (xy)(yz)$ by investigating a certain group describing the geometry of that identity. We also construct a concrete realization of the free system of rank~1 relative to the above identity

Here are considered some categorical aspects of "Differential calculus" archetype of local approximation of arbitrary morphisms by "linear" ones.

We present a broader framework for the Cauchy identity derived from the determinant expansion of collocation matrices. This approach yields an infinite family of identities, where the original Cauchy identity stands as a particular case. To…

In this paper I study the connection between logic and metaphysics in Plato's participation theory, from the structural properties of the latter. Although Plato was the first ever to formulate the contradiction principle explicitly (in the…

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.

Arithmetic groups are groups of matrices with integral entries. We shall first discuss their origin in number theory (Gauss, Minkowski) and their role in the "reduction theory of quadratic forms". Then we shall describe these groups by…