Algebraic and relational structures

代数和关系结构

• 批准号: RGPIN-2015-05661
• 负责人: Valeriote, Matthew
• 金额: $ 1.82万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=680246
• 关键词: Algebraic; relational; structures

Algebraic and relational structures arise in many different contexts, not only within mathematics, but in most of the sciences, and even in day to day life. For example, algebras known as groups are used to represent the symmetries of crystals and other physical systems and Boolean algebras are used to codify rules of logic. Many other common systems, in particular those arising from questions in computer science can be viewed in an algebraic way. Thus research on algebraic and relational structures can have an impact on several branches of the sciences and in particular on computer science.***  *Several important properties of algebraic systems can be expressed via equations. For example, the fact that addition is commutative is expressed by the equation x + y = y + x. It has proved useful to organize and classify algebraic systems according to the equations that they satisfy. These equationally defined classes of algebras, called varieties, are objects that I study in my research. I am interested in understanding the connection between various types of complexities that can arise in varieties and the structure of the algebras that belong to them.***  *A more recent direction that my research has taken relates to a problem from computational complexity. An important class of***problems, known as constraint satisfaction problems (CSPs), has a natural expression in terms of finite algebras and relational***structures. CSPs are ubiquitous in many areas of artificial intelligence, computer science, and discrete mathematics, such as***database theory, scheduling, and networking. A system of linear equations can be viewed as a special type of CSP. I am investigating a conjectured correlation between manageable complexity of subclasses of CSPs and the structure of the corresponding algebraic and relational systems.

代数和关系结构出现在许多不同的环境中，不仅在数学中，而且在大多数科学中，甚至在日常生活中。例如，称为群的代数被用来表示晶体和其他物理系统的对称性，而布尔代数被用来编码逻辑规则。许多其他常见的系统，特别是那些由计算机科学中的问题引起的系统，可以用代数的方式来看待。因此，对代数和关系结构的研究可以对科学的几个分支产生影响，特别是对计算机科学。代数系统的几个重要性质可以用方程来表示。例如，加法是可交换的这一事实被表示为方程x+y=y+x。事实证明，根据代数系统所满足的方程来组织和分类代数系统是有用的。这些由方程定义的代数类，称为簇，是我在研究中研究的对象。我感兴趣的是了解变种中可能出现的各种类型的复杂性与属于它们的代数的结构之间的联系。我的研究采取的一个较新的方向与计算复杂性的问题有关。一类重要的*问题称为约束满足问题(CSPs)，它有一个用有限代数和关系*结构表示的自然表达式。CSP在人工智能、计算机科学和离散数学的许多领域中无处不在，例如*数据库理论、调度和网络。线性方程组可以看作是CSP的一种特殊类型。我正在研究CSP子类的可管理复杂性与相应的代数系统和关系系统的结构之间的猜想相关性。