A **regular semigroup** is a specific type of algebraic structure in the field of abstract algebra, particularly in the study of semigroups. A semigroup is defined as a set equipped with an associative binary operation.

A **rational monoid** is a type of algebraic structure that arises in the context of formal language theory and automata. It can be defined as a monoid that can be represented by a finite automaton or described by a regular expression. ### Definitions: 1. **Monoid**: A monoid is an algebraic structure consisting of a set equipped with an associative binary operation and an identity element.

A quantum groupoid is a mathematical structure that generalizes both groups and groupoids within the framework of quantum algebra. It combines aspects of noncommutative geometry and the theory of quantum groups. To unpack this concept, let's first define some relevant terms: 1. **Groupoid**: A groupoid is a category where every morphism (arrow) is invertible.

Quantum differential calculus is a mathematical framework that extends traditional differential calculus into the realm of quantum mechanics and quantum systems. It provides tools and techniques to study functions and mappings that behave according to the principles of quantum theory, particularly in contexts such as quantum mechanics, quantum field theory, and quantum geometry.

A pseudogroup is a concept that appears in various contexts, primarily in the realm of mathematics, particularly in group theory and geometry. However, the exact meaning can differ based on the field of study. 1. **In Group Theory**: A pseudogroup is often defined as a set that behaves like a group but does not satisfy all the group axioms.

A pseudo-ring is a mathematical structure that generalizes some properties of rings but does not satisfy all the axioms that typically define a ring. More formally, a pseudo-ring is a set equipped with two binary operations, usually denoted as addition and multiplication, such that it satisfies certain ring-like properties but may lack others.

A **primitive ring** is a type of ring in which the process of "building up" the ring can be viewed as being generated by a single element, specifically, it is a ring that has a faithful module that is simple. Here is a more formal definition and some details: 1. **Definition**: A ring \( R \) is called primitive if it has no nontrivial two-sided ideals and it is simple as a module over itself.

In category theory, a **pointed set** is a type of set that has a distinguished element, often referred to as the "base point." Formally, a pointed set can be defined as a pair \((X, x_0)\) where: - \(X\) is a set. - \(x_0 \in X\) is a distinguished element of \(X\) called the base point.

A **planar ternary ring** (PTR) is a specific type of algebraic structure that generalizes some of the properties of linear algebra to more complex relationships involving three elements. Here are the key aspects of planar ternary rings: 1. **Ternary Operation**: A PTR involves a ternary operation, which means it takes three inputs from the set and combines them according to specific rules or axioms.

A partial groupoid is a generalization of a groupoid in the context of category theory and algebra. To understand what a partial groupoid is, we first need to recall the definition of a groupoid. A **groupoid** is a category in which every morphism (arrow) is invertible. Formally, a groupoid consists of a set of objects and a set of morphisms between these objects that allow for composition and inverses.

Partial algebra, often referred to as partial algebraic structures, is a mathematical framework that deals with algebraic systems where the operations are not necessarily defined for all possible pairs of elements in the set. In contrast to traditional algebraic structures (like groups, rings, or fields), where operations (e.g., addition, multiplication) are defined for every pair of elements, partial algebra allows for operations that are only partially defined.

In mathematics, particularly in the field of ring theory, an **overring** is a type of ring that contains another ring as a subring. More formally, given a ring \( R \), an overring \( S \) is defined such that: 1. \( R \) is a subring of \( S \) (i.e., every element of \( R \) is also an element of \( S \)).

Algebraic structures are fundamental concepts in abstract algebra, a branch of mathematics that studies algebraic systems in a broad manner. Here’s an outline of key algebraic structures: ### 1. **Introduction to Algebraic Structures** - Definition and significance of algebraic structures in mathematics. - Examples of basic algebraic systems. ### 2. **Groups** - Definition of a group: A set equipped with a binary operation satisfying closure, associativity, identity, and invertibility.

An **ordered exponential field** is a mathematical structure that extends the concepts of both fields and order theory. In particular, it refers to an ordered field equipped with a particular function that behaves like the exponential function. ### Key Components: 1. **Field**: A set equipped with two operations, typically addition and multiplication, which satisfy certain properties (like associativity, commutativity, the existence of additive and multiplicative identities, etc.).

A **numerical semigroup** is a special type of subset of the non-negative integers. Specifically, it is a subgroup of the non-negative integers under addition that is closed under addition and contains the identity element 0. More formally, a numerical semigroup is defined as follows: 1. It is a subset \( S \) of the non-negative integers \( \mathbb{N}_0 = \{0, 1, 2, \ldots\} \).

A *nowhere commutative semigroup* is a type of algebraic structure characterized by its non-commutative nature. In algebra, a semigroup is defined as a set equipped with an associative binary operation. Specifically, a semigroup \( S \) is a set with a binary operation \( \cdot \) such that: 1. **Closure**: For all \( a, b \in S \), the product \( a \cdot b \in S \).

A near-semiring is an algebraic structure similar to a semiring but with a relaxed definition of some of its properties. Specifically, a near-semiring is defined as a set equipped with two binary operations, typically called addition and multiplication, that satisfy certain axioms. Here are the main characteristics of a near-semiring: 1. **Set**: A near-semiring consists of a non-empty set \( S \).

A **near-ring** is a mathematical structure similar to a ring, but it relaxes some of the conditions that define a ring. Specifically, a near-ring is equipped with two binary operations, typically called addition and multiplication, but it does not require that all the properties of a ring hold. Here are the main features of a near-ring: 1. **Set**: A near-ring consists of a non-empty set \( N \).

In mathematics, particularly in the context of algebra and number theory, a "near-field" may refer to a structure similar to a field, but with weaker properties. A near-field typically satisfies most properties of a field except for certain requirements, such as the existence of multiplicative inverses for all non-zero elements. However, the concept of "near-field" is not as widely recognized or standardized as fields, rings, or groups.

An N-ary group is a generalization of the concept of a group in abstract algebra. In group theory, a group is defined as a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility.