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# Zero element

In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. [1]

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written (the axiom of commutativity).

## Absorbing element

In mathematics, an absorbing element is a special type of element of a set with respect to a binary operation on that set.

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

## Addition

Addition (often signified by the plus symbol "+") is one of the four elementary, mathematical operations of arithmetic, with the others being subtraction, multiplication and division.

## Additive identity

In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

## Algebraic structure

In mathematics, and more specifically in abstract algebra, the term algebraic structure generally refers to a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a some list of axioms.

## Cartesian product

In mathematics, a Cartesian product is a mathematical operation which returns a set (or product set or simply product) from multiple sets.

## Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows".

## Category of groups

In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms.

## Coproduct

In category theory, the coproduct, or categorical sum, is a category-theoretic construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces.

## Empty set

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## Empty sum

In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero.

## Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction and can be added to other vectors according to vector algebra.

## Field (mathematics)

In abstract algebra, a field is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication.

## Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

## Greatest element

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if Hence, the greatest element of S is an upper bound of S that is contained within this subset.

## Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

## Identity element

In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set.

## Initial and terminal objects

In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists a single morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first, literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Lattice (order)

In mathematics, a lattice is a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).

## Linear algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change.

## Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array—of numbers, symbols, or expressions, arranged in rows and columns—that is interpreted and manipulated in certain prescribed ways.

## Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring (with identity).

## Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

## Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "·" or by the absence of symbol) is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division.

## Partially ordered set

In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

## Pointwise

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the pointwise operations &mdash; operations defined on functions by applying the operations to function values separately for each point in the domain of definition.

## Principal ideal

A principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R.

## Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces.

## Ring (mathematics)

In mathematics, and more specifically in algebra, a ring is an algebraic structure with operations that generalize the arithmetic operations of addition and multiplication.

## Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.

## Semiring

In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.

## Tensor

Tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.

## Tensor product

In mathematics, the tensor product, denoted by, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects.

## Trivial group

In mathematics, a trivial group is a group consisting of a single element.

## Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all distinct elements in the collection.

## Zero divisor

In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that, or equivalently if the map from to that sends to is not injective.

## Zero element

In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures.

## Zero morphism

In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object.

## Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation In other words, a "zero" of a function is an input value that produces an output of zero (0).

## Zerosumfree monoid

In abstract algebra, an additive monoid (M, 0, +) is said to be zerosumfree, conical, centerless or positive if nonzero elements do not sum to zero.

## 0 (number)

0 (zero; BrE: or AmE) is both a number and the numerical digit used to represent that number in numerals.

## References

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