相关论文: Cospans and spans of graphs: a categorical algebra…
We suggest that the canonical parallel operation of processes is composition in a well-supported compact closed category of spans of reflexive graphs. We present the parallel operations of classical process algebras as derived operations…
We show that Span(Graph)*, an algebra for open transition systems introduced by Katis, Sabadini and Walters, satisfies a universal property. By itself, this is a justification of the canonicity of this model of concurrency. However, the…
A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and…
We discuss the notion of a span of cospans and define, for them, horizonal and vertical composition. These compositions satisfy the interchange law if working in a topos $\mathbf{C}$ and if the span legs are monic. A bicategory is then…
Quadratic algebras associated to graphs have been introduced by I. Gelfand, S. Gelfand, and Retakh in connection with decompositions of noncommutative polynomials. Here we show that, for each graph with rare triangular subgraphs, the…
One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce "structured cospans" as a way to study networks with inputs and outputs. Given a functor $L \colon…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
We introduce and investigate generalizations of interval and proper interval graphs to simplicial complexes, including strong interval, unit interval, and under closed variants. Through equivalent combinatorial and algebraic…
In this paper we present the set of intervals as a normed vector space. We define also a four-dimensional associative algebra whose product gives the product of intervals in any cases. This approach allows to give a notion of divisibility…
In this note we define a process algebra TCP (Truly Concurrent Processes) which corresponds closely with the automata model of concurrency based on Span(RGraph), the category of spans of reflexive graphs. In TCP, each process has a fixed…
We give parallel algorithms for string diagrams represented as structured cospans of ACSets. Specifically, we give linear (sequential) and logarithmic (parallel) time algorithms for composition, tensor product, construction of diagrams from…
This paper is devoted to a new approach of the arithmetic of intervals. We present the set of intervals as a normed vector space. We define also a four-dimensional associative algebra whose product gives the product of intervals in any…
New notions are introduced in algebra in order to better study the congruences in number theory. For example, the <special semigroups> makes an important such contribution.
We introduce rigid algebras, a generalization of rigid categories to arbitrary symmetric monoidal $(\infty,2)$-categories. We develop their general theory, showing in particular that the a priori $(\infty,2)$-category of rigid algebras is…
Following our previous work, we suggest here a large class of algebras of scalars in which simultaneous and correlated computations can be performed owing to the existence of surjective algebra homomorphisms. This may replace the currently…
What is Sequence Algebra? This is a question that any teacher or student of mathematics or computer science can engage with. Sequences are in Calculus, Combinatorics, Statistics and Computation. They are foundational, a step up from number…
In this paper we deal with a subclass of chordal graphs, which are simultaneously strictly chordal and interval, the strictly interval graphs. We present a new characterization of the class that leads to a simple linear recognition…
In this book super interval matrices using the special type of intervals of the form [0, a] are introduced. Several algebraic structures like semigroups, groups, semirings, rings, semivector spaces and vector spaces are introduced. Special…
This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The…
Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…