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We study substitutive systems generated by nonprimitive substitutions and show that transitive subsystems of substitutive systems are substitutive. As an application we obtain a complete characterisation of the sets of words that can appear…
The notion of a \emph{$G$-completely reducible} subgroup is important in the study of algebraic groups and their subgroup structure. It generalizes the usual idea of complete reducibility from representation theory: a subgroup $H$ of a…
An abstract machine is a theoretical model designed to perform a rigorous study of computation. Such a model usually consists of configurations, instructions, programs, inputs and outputs for the machine. In this paper we formalize these…
The standard assumptions that underlie many conceptual and quantitative frameworks do not hold for many complex physical, biological, and social systems. Complex systems science clarifies when and why such assumptions fail and provides…
In this paper, based on ideals, we investigate residuated lattices from fuzzy set theory and lattice theory point of view. Ideals are important concepts in the theory of algebraic structures used for formal fuzzy logic and first, we…
In this article we explain the theory of rigid residue complexes in commutative algebra and algebraic geometry, summarizing the background, recent results and anticipated future results. Unlike all previous approaches to Grothendiec…
We deal with equations over free semilattice of infinite rank and prove that any infinite consistent system of equations is equivalent to its finite subsystem. Moreover, we describe irreducible algebraic sets and solve some algorithmic…
To better understand the algebra $\mathcal{M}_n$ of all $n\times n$ complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to its role in complementing those results…
This article will be a continuation of our research into self-justifying systems. It will introduce several new theorems and their applications. (One of these results will transform our previous infinite-sized self-verifying formalisms into…
We present module theory and linear maps as a powerful generalised and computationally efficient framework for the relational data model, which underpins today's relational database systems. Based on universal constructions of modules we…
In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new…
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…
We prove the complete intersection theorem and complete nontrivial-intersection theorem for systems of set partitions
This paper investigates the class of finitely presented monoids defined by homogeneous (length-preserving) relations from a computational perspective. The properties of admitting a finite complete rewriting system, having finite derivation…
The present paper refers to the theory and the practice of the systems regarding non-linear systems and their applications. We aimed the integration of these systems to elaborate their response as well as to highlight some outstanding…
Classical Ramsey theory has successfully extended to relational structures, yielding a wealth of results that have profoundly influenced other areas of mathematics. Interestingly, the same development has not occurred in the case of dual…
Nominal sets provide a foundation for reasoning about names. They are used primarily in syntax with binders, but also, e.g., to model automata over infinite alphabets. In this paper, nominal sets are related to nominal renaming sets, which…
Grey system theory is an important mathematical tool for describing uncertain information in the real world. It has been used to solve the uncertainty problems specially caused by lack of information. As a novel theory, the theory can deal…
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…
A complex system is a system composed of many interacting parts, often called agents, which displays collective behavior that does not follow trivially from the behaviors of the individual parts. Examples include condensed matter systems,…