In the context of abelian groups, the term "height" can refer to a couple of different concepts depending on the specific area of mathematics being considered, such as group theory or algebraic geometry. 1. **In Group Theory**: The height of an abelian group can refer to a measure of the complexity of the group, particularly when it comes to finitely generated abelian groups.

A Barsotti–Tate group is an important concept in the area of algebraic geometry and representation theory, particularly in the study of p-adic representations and finite field extensions. Named after mathematicians Francesco Barsotti and John Tate, these groups are essentially a kind of p-divisible group that has additional structure, allowing them to be classified and understood in terms of their representation theory.

In the context of algebraic groups, the **radical** refers to a specific type of subgroup that is closely related to the structure of the group itself. More formally, the radical of an algebraic group \( G \) is defined as the largest normal solvable subgroup of \( G \). ### Key Concepts: 1. **Algebraic Group**: An algebraic group is a group that is also an algebraic variety, meaning it can be defined by polynomial equations.

In the context of algebraic groups and Lie algebras, a **root datum** is a structured way of encoding certain aspects of the symmetries and properties of these mathematical objects. Specifically, a root datum consists of the following components: 1. **A finite set of roots**: These are usually vectors in a Euclidean space, which can be thought of as directions that reflect the symmetries of the system.

A **subgroup series** in group theory is a sequence of subgroups of a given group \( G \) that is organized such that each subgroup is a normal subgroup of the next one in the series.

The affine group is a mathematical concept that arises in the context of geometry and linear algebra. It is essentially a group that consists of affine transformations, which are a generalization of linear transformations that include translations.

In the context of group theory, particularly in the study of algebraic groups and Lie groups, a diagonal subgroup is typically a subgroup that is constructed from the diagonal elements of a product of groups. For example, consider the direct product of two groups \( G_1 \) and \( G_2 \).

The Five Lemma is a result in the field of homological algebra, particularly in the context of derived categories and spectral sequences. It provides a criterion for when a five-term exact sequence of chain complexes splits. This lemma is commonly used in the study of abelian categories and the derived functor theory.

The Nine Lemma is a result in algebraic topology, specifically in the study of homotopy theory. It deals with examining the relationships between spaces that can be constructed from homotopy types and the existence of certain maps between them. The lemma provides a way to relate complications in a certain diagram of spaces and maps to giving conditions under which some homotopy properties hold true.

The Short Five Lemma is a tool in algebraic topology, specifically in the context of homological algebra and the theory of derived functors. It deals with the properties of cohomology in the setting of a commutative diagram of chain complexes and can be used to derive relationships between cohomology groups of various objects.

A triangulated category is a particular type of category that arises in the context of homological algebra and derived categories. It provides a framework to study homological properties by relating them to geometric intuition through triangles, similar to how one uses exact sequences in abelian categories.

A semimodular lattice is a special type of lattice in the field of order theory and abstract algebra. A lattice \( L \) is a partially ordered set (poset) in which any two elements have a unique supremum (join) and an infimum (meet). The term "semimodular" specifically refers to a certain condition that relates to the structure of the lattice.

In the context of databases and relational algebra, a relation of degree zero is a special case of a relation where there are no attributes (or columns). In relational database terminology, the degree of a relation (or table) refers to the number of attributes it contains. When a relation has a degree of zero, it means that it is essentially an empty set without any data or structure.

Auslander–Reiten theory is a branch of representation theory in mathematics, particularly within the field of algebra and category theory. It is named after the mathematicians Maurice Auslander and Idun Reiten, who made significant contributions to the understanding of module theory and the representation theory of algebras. At its core, Auslander–Reiten theory deals with the study of certain special kinds of categories called abelian categories, particularly the category of modules over a fixed ring.

A signature block is a section at the end of an email, document, or letter that contains information about the author or sender. It typically includes the sender's name, title or position, company or organization name, contact information (such as phone number or email address), and sometimes additional details like company logo, social media links, or website URL. In emails, signature blocks help recipients identify the sender, and they can also serve as a professional finishing touch.

"Geek girl" is a term that generally refers to a woman who is passionate about subjects typically associated with geek culture, such as technology, gaming, science fiction, fantasy, comic books, and various other niche interests. The term can encompass a wide range of activities and interests, from programming and gaming to participating in fandoms or attending conventions.

A "purity test" can refer to different concepts depending on the context. Here are a few interpretations: 1. **Social/Cultural Context**: In many college and social environments, a purity test is a questionnaire or a set of questions designed to gauge a person's sexual experiences or attitudes about intimacy. These tests often humorously assess someone's perceived "purity" based on their answers, and they can sometimes serve as conversation starters within a group.

Mass amateurization refers to the phenomenon where everyday individuals gain access to tools and platforms that enable them to produce, create, and share content, products, or services that were previously the domain of professionals or specialists. This trend has been facilitated by advancements in technology, particularly the internet, social media, and affordable software and hardware.

Netprov, short for "networked improvisation," is a form of digital performance art and storytelling that takes place in online environments. It combines elements of improvisational theater with online communication, utilizing platforms such as social media, chat rooms, and interactive websites to create spontaneous narratives and collaborative storytelling experiences. In netprov, participants often take on roles and contribute to a story in real-time, responding to each other’s contributions in an unscripted manner.

Nettime is an online community and mailing list that focuses on discussions of digital culture, politics, and technology. Founded in 1995, it serves as a platform for activists, artists, academics, and other thinkers to share ideas, critique contemporary issues related to the internet and technology, and explore the implications of digital media on society. Members engage in discussions ranging from internet art, cyberpolitics, to the effects of technology on social relations.