In mathematics, a **module** is a generalization of the concept of a vector space. While vector spaces are defined over a field, modules allow for the scalars to be elements of a more general algebraic structure called a ring.
A **monogenic semigroup** is a particular type of algebraic structure in the field of abstract algebra. Specifically, a semigroup is a set equipped with an associative binary operation. In the case of a monogenic semigroup, there is a specific defining feature: the semigroup is generated by a single element.
As of my last update in October 2023, "Monus" could refer to a few different things depending on the context. It may refer to: 1. **Monus (Currency)**: In some contexts, "Monus" might refer to a digital currency or token. It's essential to check specific cryptocurrency platforms or forums for the most recent developments in digital currencies.
A Moufang polygon is a type of combinatorial structure that generalizes certain properties of projective planes and certain geometric configurations. More specifically, Moufang polygons can be viewed as a particular kind of building in the theory of buildings in geometric group theory, related closely to groups of Lie type and algebraic structures. A Moufang polygon can be defined as a finite, strongly regular combinatorial structure defined with respect to a set of vertices and certain incidence relations among them.
A **multiplicative group** is a mathematical structure consisting of a set equipped with a binary operation that satisfies certain properties. Specifically, a multiplicative group is a set \( G \) along with a binary operation (commonly denoted as multiplication) that has the following characteristics: 1. **Closure**: For any two elements \( a, b \in G \), the result of the operation \( a \cdot b \) is also in \( G \).
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.
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.
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 \( N \).
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 *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 **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 \( \mathbb{N}_0 = \{0, 1, 2, \ldots\} \).
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.).
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.
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 \( S \)).
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.
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.
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 **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.