Algebraic structures

Algebraic structures are fundamental concepts in abstract algebra that provide a framework for understanding and analyzing mathematical systems in terms of their operations and properties. An algebraic structure consists of a set accompanied by one or more binary operations that satisfy specific axioms.

The term "algebras" can refer to several different concepts depending on the context, but it generally relates to a branch of mathematics that deals with symbols and the rules for manipulating those symbols. Here are some common interpretations: 1. **Algebra in Mathematics**: This is the most common use of the term. Algebra is a field of mathematics that involves studying mathematical symbols and the rules for manipulating these symbols. It includes solving equations, working with polynomials, and understanding functions.

Coalgebras are a mathematical concept primarily used in the fields of category theory and theoretical computer science. They generalize the notion of algebras, which are structures used to study systems with operations, to structures that focus on state-based systems and behaviors. ### Basic Definition: A **coalgebra** for a functor \( F \) consists of a set (or space) \( C \) equipped with a structure map \( \gamma: C \to F(C) \).

A finite group is a mathematical structure in the field of abstract algebra that consists of a set of elements equipped with a binary operation (often called group operation) that satisfies four key properties: closure, associativity, identity, and invertibility. Specifically, a group \( G \) is called finite if it contains a finite number of elements, which we denote as \( |G| \) (the order of the group).

Hypercomplex numbers are a generalization of complex numbers that extend beyond the traditional two-dimensional complex plane into higher dimensions. They can be understood as numbers that incorporate additional dimensions through the introduction of new units, much like complex numbers extend the real numbers with the imaginary unit \(i\), where \(i^2 = -1\). ### Key Types of Hypercomplex Numbers 1.

Lie groups are mathematical structures that combine concepts from algebra and geometry. They are named after the Norwegian mathematician Sophus Lie, who studied them in the context of continuous transformation groups. ### Basic Definition A **Lie group** is a group that is also a smooth manifold, meaning that the group operations (multiplication and inversion) are smooth (infinitely differentiable) functions. This combination allows for the study of algebraic structures (like groups) with the tools of calculus and differential geometry.

Non-associative algebra refers to a type of algebraic structure where the associative law does not necessarily hold. In mathematics, the associative law states that for any three elements \( a \), \( b \), and \( c \), the equation \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \) should be true for all operations \( \cdot \).

Ockham algebras are algebraic structures that arise in the study of formal logic, particularly in connection with concepts of nominalism and the philosophy of mathematics. They are named after the philosopher William of Ockham, who is known for advocating simplicity in explanations, often referred to as Ockham's Razor.

Ordered algebraic structures are mathematical structures that combine the properties of algebraic operations with a notion of order. These structures help to study and characterize the relationships between elements not just through algebraic operations, but also through the relationships denoted by comparisons (like "less than" or "greater than").

In the context of group theory in mathematics, a group is a set equipped with an operation that combines any two elements to form a third element while satisfying four fundamental properties. These properties define the structure of a group.

Action algebra is not a standard term widely recognized in conventional mathematical literature, but it could refer to several possible concepts depending on the context. In mathematics and theoretical computer science, the term could relate to the study of algebraic structures that involve actions, such as in group theory or the algebra of operations. 1. **Group Actions and Algebraic Structures**: In the context of group theory, an "action" often refers to how a group operates on a set.

An additive group is a mathematical structure that consists of a set equipped with an operation (usually referred to as addition) that satisfies four key properties: closure, associativity, the existence of an identity element, and the existence of inverses.

An **affine monoid** is an algebraic structure that arises in the context of algebraic geometry, commutative algebra, and combinatorial geometry. Specifically, an affine monoid is a certain type of commutative monoid that can be characterized by its geometric interpretation and algebraic properties.

BCK algebra is a type of algebraic structure that is derived from the theory of logic and set theory. Specifically, it is a variant of binary operations that generalizes certain properties of Boolean algebras. The term "BCK" comes from the properties of the operations defined within the structure.

A BF-algebra is a particular type of algebraic structure that arises in the study of functional analysis and operator theory, especially in the context of bounded linear operators on Banach spaces. The term "BF-algebra" is short for "bounded finite-dimensional algebra," and it can be understood in the context of specific properties of the algebras it describes.

In algebra, specifically in the study of rings and modules, a **band** refers to a particular type of algebraic structure that can be characterized as a set equipped with a binary operation that behaves in a specific way. More formally, a **band** is a type of monoid where every element is idempotent.

The Baumslag–Gersten group is an example of a type of group that can be defined by a particular presentation involving generators and relations. Specifically, it can be denoted as \(BG(m, n)\) where \(m\) and \(n\) are positive integers.

A biordered set is a mathematical structure that is a type of ordered set with two compatible order relations. More formally, a set \( S \) is called a biordered set if it is equipped with two binary relations \( \leq \) and \( \preceq \) that satisfy certain axioms.

Biracks and biquandles are algebraic structures used in the study of knots and 3-manifolds, particularly in the field of knot theory and topological quantum field theories. They provide a framework for understanding symmetries of knots and links through combinatorial methods. ### Birack A **birack** is a set \( X \) equipped with two binary operations \( \blacktriangledown \) and \( \blacktriangleleft \) that satisfy certain axioms.

Boolean algebra is a branch of mathematics that deals with variables that have two possible values: typically represented as true (1) and false (0). It was introduced by mathematician George Boole in the mid-19th century and serves as a foundational structure in fields such as computer science, electrical engineering, and logic. ### Basic Structure of Boolean Algebra: 1. **Elements**: The elements of Boolean algebra are single bits (binary variables) that can take values of true or false.

A cancellative semigroup is a specific type of algebraic structure used in the field of abstract algebra. A semigroup is defined as a set equipped with an associative binary operation. A semigroup \( S \) is called cancellative if it satisfies the cancellation property. Here's a more formal definition: Let \( S \) be a semigroup with a binary operation \( \cdot \).

In mathematics, particularly in the field known as category theory, a "category" is a fundamental structure that encapsulates abstract mathematical concepts and their relationships. Categories provide a unifying framework for various areas of mathematics by focusing on the relationships (morphisms) between objects rather than on the objects themselves. A category consists of: 1. **Objects**: These can be any mathematical entities, such as sets, groups, topological spaces, or other structures.

In group theory, a **class of groups** typically refers to a specific category or type of groups that share certain properties or characteristics. Here are a few common classes of groups: 1. **Abelian Groups**: These are groups in which the group operation is commutative; that is, for any two elements \( a \) and \( b \) in the group, \( a \cdot b = b \cdot a \).

A **commutative ring** is a fundamental algebraic structure in mathematics, particularly in abstract algebra. It is defined as a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication.

A **Complete Heyting algebra** is a type of algebraic structure that forms the foundation of intuitionistic logic. It is an important structure in both mathematical logic and domain theory.

A **completely regular semigroup** is an important structure in the theory of semigroups, which are algebraic structures consisting of a set equipped with an associative binary operation. Specifically, a completely regular semigroup has properties that relate to its elements and the existence of certain types of idempotent elements.

In mathematics, a **composition ring** is an algebraic structure related to the study of quadratic forms and their interactions with certain types of fields. Specifically, a composition ring is a commutative ring with identity that has the property that every element can be expressed in terms of the "composition" of two other elements in a specific way. This concept is often encountered in the context of quadratic forms and modules over rings.

In mathematics, particularly in the field of functional analysis and convex analysis, a **convex space** (or **convex set**) refers to a set of points in which, for any two points within the set, the line segment connecting those two points also lies entirely within the set. This concept is foundational in various areas of optimization, economics, and geometry.

Damm algorithm is a checksum algorithm designed to provide a way to detect errors in numerical data. It operates through a specific method of encoding and decoding numerical data, particularly useful for validating data integrity in applications like digital communication, credit card numbers, and other identification systems. The Damm algorithm uses a particular modulo operation, based on a predetermined finite state machine (FSM) that is defined by a specific set of rules.

In ring theory, a **domain** is a specific type of ring that satisfies certain properties. More formally, a domain refers to an integral domain, which is defined as a commutative ring \( R \) with the following characteristics: 1. **Commutative**: The ring is commutative under multiplication, meaning for any \( a, b \in R \), \( ab = ba \).

A double groupoid is a mathematical structure that generalizes the concept of a groupoid. To understand what a double groupoid is, it helps to first clarify what a groupoid is. ### Groupoid A **groupoid** consists of a set of objects and a set of morphisms (arrows) between these objects satisfying certain axioms. Specifically: - Each morphism has a source and target object.

In the context of semigroup theory, an **E-dense semigroup** relates to a specific type of dense semigroup. A semigroup is a set equipped with an associative binary operation. The term "E-dense" generally refers to certain properties of the semigroup concerning its embeddings and the way it interacts with a certain subset or structure designated as \( E \).

An *E-semigroup* is a specific type of algebraic structure that can be understood within the context of semigroup theory, which in turn is a branch of abstract algebra. Although there isn't a universally accepted definition for E-semigroup because the terminology can vary, it often refers to a semigroup equipped with additional properties or operations related to particular contexts, such as in semigroups associated with certain algebraic identities or functional operations.

Effect algebra is a mathematical structure that originates from the study of quantum mechanics and the foundations of probability theory. It provides a framework for discussing the concepts of effects, states, and observables in a generalized manner that captures certain features of quantum systems without requiring a full Hilbert space representation. ### Key Concepts in Effect Algebra: 1. **Effects**: In the context of quantum mechanics, an effect can be understood as a positive operator that corresponds to the outcome of a measurement.

Elliptic algebra is a concept in mathematics that arises in the study of algebraic structures known as elliptic curves, along with their associated functions and symmetries. Elliptic algebras can be seen as extensions of traditional algebraic concepts, incorporating properties of elliptic functions, which are complex functions defined on elliptic curves.

An empty semigroup is a mathematical structure that consists of an empty set equipped with a binary operation that is associative. A semigroup is defined as a set accompanied by a binary operation that satisfies two conditions: 1. **Associativity:** For any elements \( a, b, c \) in the semigroup, the equation \( (a * b) * c = a * (b * c) \) holds, where \( * \) is the binary operation.

As of my last update in October 2023, "Epigroup" does not refer to a widely recognized term, company, or concept in major fields such as business, technology, or science. It’s possible that it could be a specific brand, organization, or a term used in a niche context that hasn't gained significant recognition or coverage.

In mathematics, the concept of **essential dimension** is a notion in algebraic geometry and representation theory, primarily related to the study of algebraic structures and their invariant properties under field extensions. It provides a way to quantify the "complexity" of objects, such as algebraic varieties or algebraic groups, in terms of the dimensions of the fields needed to define them.

"Exponential field" can refer to different concepts depending on the context in which it is used. Below are a couple of interpretations: 1. **Mathematics**: In a mathematical context, an exponential field could refer to a field in which the exponential function plays a significant role. For example, in fields of algebra, one might study exponential equations or growth models that describe exponential behavior, such as in calculus with respect to exponential functions and their properties.

The Finite Lattice Representation Problem is a concept in the field of lattice theory, which deals with partially ordered sets that have specific algebraic properties. In particular, this problem pertains to determining whether a given finite partially ordered set (poset) can be represented as a lattice.

A finitely generated abelian group is a specific type of group in abstract algebra that has some important properties. 1. **Group**: An abelian group is a set equipped with an operation (often called addition) that satisfies four properties: closure, associativity, identity, and inverses. Additionally, an abelian group is commutative, meaning that the order in which you combine elements does not matter (i.e.

A generic matrix ring is a mathematical structure that is used in algebra, particularly in the study of algebras and representations. It is typically denoted as \( M_n(R) \), where \( R \) is a commutative ring and \( n \) is a positive integer. The generic matrix ring can also be defined in a more abstract setting where elements of the ring are not necessarily evaluated at specific entries but can be treated as formal matrices with entries from the ring \( R \).

The Grothendieck group is an important concept in abstract algebra, particularly in the areas of algebraic topology, algebraic geometry, and category theory. It is used to construct a group from a given commutative monoid, allowing the extension of operations and structures in a way that respects the original monoid's properties.

In mathematics, a **group** is a fundamental algebraic structure that consists of a set of elements combined with a binary operation. This binary operation must satisfy four specific properties known as the group axioms: 1. **Closure**: For any two elements \( a \) and \( b \) in the group, the result of the operation \( a * b \) is also in the group.

A **groupoid** is a concept in mathematics that generalizes the notion of a group. While a group consists of a single set with a binary operation that combines two elements to produce a third, a groupoid consists of a category in which every morphism (arrows connecting objects) has an inverse, and morphisms can be thought of as symmetries or transformations between objects.

A Hardy field is a type of mathematical structure used in the field of real analysis and model theory, specifically in the study of asymptotic behaviors of functions. It is named after the mathematician G. H. Hardy. A Hardy field is essentially a field of functions that satisfies certain algebraic and order properties.

In ring theory, which is a branch of abstract algebra, an *ideal* is a special subset of a ring that allows for the construction of quotient rings and provides a way to generalize certain properties of numbers to more complex algebraic structures. Formally, let \( R \) be a ring (with or without identity).

The integral closure of an ideal is a concept from commutative algebra and algebraic geometry that has to do with the properties of rings and ideals.

An integral element is a term used in various fields, primarily in mathematics and abstract algebra, as well as in related contexts like computer science and physics. However, without a specific context, the meaning can vary. 1. **Mathematics/Abstract Algebra**: In ring theory, an integral element refers to an element of an integral domain (a type of commutative ring) that satisfies a monic polynomial equation with coefficients from that domain.

Interior algebra is a branch of mathematics that deals with the study of certain algebraic structures that arise in the context of topology, particularly in relation to topological spaces and their properties. Its primary focus is on the algebraic operations defined on sets of open and closed sets in a topological space. In more detail, interior algebra typically involves concepts like: 1. **Interior and Closure**: The operations of taking the interior and closure of sets within a topological space.

An **inverse semigroup** is a specific type of algebraic structure that combines the properties of semigroups and the concept of invertibility of elements. A semigroup is an algebraic structure consisting of a set equipped with an associative binary operation, while an inverse semigroup has additional properties related to inverses.

The term "J-structure" can refer to different concepts depending on the context in which it is used. Here are a few possible interpretations: 1. **Mathematics and Geometry**: In the context of mathematics, particularly in algebraic topology or manifold theory, J-structure can refer to a specific type of geometric or topological structure associated with a mathematical object. It might relate to an almost complex structure or a similar concept depending on the area of study.

A Jónsson–Tarski algebra is a specific type of algebraic structure related to Boolean algebras and described by properties connected to the concept of free algebras. The notion is named after the mathematicians Bjarni Jónsson and Alfred Tarski, who made significant contributions to the fields of mathematical logic and algebra.

The Kasch ring is a geometric structure used in the field of differential geometry and topology, particularly in relation to the study of manifolds and their properties. Specifically, the Kasch ring is associated with the concept of the curvature of a Riemannian manifold, and it may also arise in the context of algebraic topology.

Kleene algebra is a mathematical structure used in theoretical computer science, formal language theory, and algebra. It is named after the mathematician Stephen Kleene, who made significant contributions to the foundations of automata theory and formal languages. Kleene algebra consists of a set equipped with certain operations and axioms that support reasoning about the properties of regular languages and automata.

In mathematics, particularly in order theory, a **lattice** is a specific type of algebraic structure that is a partially ordered set (poset) with unique least upper bounds (suprema or joins) and greatest lower bounds (infima or meets) for any two elements.

Lindenbaum–Tarski algebra is a structure in mathematical logic and model theory that arises from the study of formal systems, particularly those dealing with propositional or predicate logic. It is named after the mathematicians Adolf Lindenbaum and Alfred Tarski, who contributed significantly to the foundations of mathematical logic. In essence, a Lindenbaum–Tarski algebra is a specific type of Boolean algebra that is constructed from the collection of all consistent sets of formulas in a given formal system.

MV-algebra, or many-valued algebra, is a mathematical structure used in the study of many-valued logics, particularly those that generalize classical propositional logic. The concept was introduced in the context of Lukasiewicz logic, which allows for truth values beyond just "true" and "false.

In the field of algebra, a **magma** is a very basic algebraic structure. It is defined as a set \( M \) equipped with a binary operation \( * \) that combines two elements of the set to produce another element in the set. Formally, a magma is defined as follows: - A **magma** is a pair \( (M, *) \) where: - \( M \) is a non-empty set.

A "matrix field" can refer to different concepts depending on the context in which it is used. Here are a few interpretations based on various disciplines: 1. **Mathematics and Linear Algebra**: In mathematics, particularly in linear algebra, a matrix field often refers to an array of numbers (or functions) organized in rows and columns that can represent linear transformations or systems of equations. However, “matrix field” might not be a standard term, as fields themselves are mathematical structures.

A matrix ring is a specific type of ring constructed from matrices over a ring. Formally, if \( R \) is a ring (which can be, for example, a field or another ring), then the set of \( n \times n \) matrices with entries from \( R \) forms a ring, denoted by \( M_n(R) \). The operations defined in this ring are matrix addition and matrix multiplication.

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.

A **monoid** is an algebraic structure that consists of a set equipped with a binary operation that satisfies two key properties: associativity and identity. More formally, a monoid is defined as a tuple \((M, \cdot, e)\), where: 1. **Set \(M\)**: This is a non-empty set of elements.

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.

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 **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.

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 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.

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 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.

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 **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.

The term "Right Group" can refer to different organizations or movements depending on the context, such as political or ideological groups that advocate for conservative or right-leaning policies. However, it is not a widely recognized or specific organization without additional context. If you're referring to a particular group, organization, or movement (e.g.

In mathematics, specifically in abstract algebra, a **ring** is a set equipped with two binary operations that generalize the arithmetic of integers. Specifically, a ring consists of a set \( R \) together with two operations: addition (+) and multiplication (·). The structure must satisfy the following properties: 1. **Additive Closure**: For any \( a, b \in R \), the sum \( a + b \) is also in \( R \).

In algebra, particularly in the context of ring theory, the term "rng" (pronounced "ring") is an abbreviation that refers to a mathematical structure that is similar to a ring but does not necessarily require the existence of a multiplicative identity (i.e., an element that acts as 1 in multiplication).

The term "semifield" can refer to different concepts depending on the context in which it is used. In mathematics, particularly in abstract algebra, a semifield is a generalization of a field. ### Semifield in Algebra: 1. **Definition**: A semifield is a set equipped with two operations (typically addition and multiplication) that satisfy some but not all of the field axioms.

In abstract algebra, a **semigroup** is a fundamental algebraic structure consisting of a set equipped with an associative binary operation. Formally, a semigroup is defined as follows: 1. **Set**: Let \( S \) be a non-empty set.

A **semigroup with involution** is an algebraic structure that combines the properties of a semigroup with the concept of an involution. ### Components of a Semigroup with Involution 1. **Semigroup**: A semigroup is a set \( S \) equipped with a binary operation (let's denote it as \( \cdot \)) that satisfies the associative property.

A **semigroup** is an algebraic structure consisting of a set equipped with an associative binary operation.

A semigroup is an algebraic structure consisting of a set equipped with an associative binary operation. Specifically, a set \( S \) with a binary operation \( * \) is a semigroup if it satisfies two conditions: 1. **Closure**: For any \( a, b \in S \), the result of the operation \( a * b \) is also in \( S \).

A **semigroupoid** is an algebraic structure that generalizes the notion of a semigroup to a situation where the elements can be thought of as processes or mappings rather than simple algebraic objects. More formally, a semigroupoid can be defined as a category in which every morphism (or arrow) is invertible, but it has a single object, or it can be thought of as a partially defined operation among elements.

A **semilattice** is an algebraic structure that is a specific type of partially ordered set (poset).

In mathematics, particularly in the field of abstract algebra, a **semimodule** is a generalization of the concept of a module, specifically over a semiring instead of a ring. ### Definitions 1. **Semiring**: A semiring is an algebraic structure consisting of a set equipped with two binary operations: addition (+) and multiplication (×). These operations must satisfy certain properties: - The set is closed under addition and multiplication.

A **semiring** is an algebraic structure that is a generalization of both a ring and a monoid. It consists of a set equipped with two binary operations that generalize addition and multiplication. A semiring is defined by the following properties: 1. **Set**: Let \( S \) be a non-empty set.

A simplicial commutative ring is a mathematical structure that combines concepts from algebra and topology, specifically within the realm of simplicial sets and commutative rings. To understand simplicial commutative rings, we first need to clarify two important concepts: 1. **Simplicial Set**: A simplicial set is a construction in algebraic topology that encodes a topological space in terms of its simplicial complex structure.

In the field of algebra, semigroups are algebraic structures consisting of a set equipped with an associative binary operation. Special classes of semigroups refer to particular types of semigroups that possess additional properties or structures, leading to interesting applications and deeper insights. Here are some notable special classes of semigroups: 1. **Monoids**: A monoid is a semigroup that has an identity element.