In mathematics, inclusion is a partial order on sets. Under this order, A ≤ B if A is a subset of B. Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...

When A is a subset of B, the inclusion function (inclusion map, or canonical injection) is the function i that sends each element of A to the same element in B: In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

i:A → B, i(x) = x.

This and other analogous injective functions from substructures are sometimes called natural injections. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... This article needs to be wikified. ...

Inclusion as partial order

The order on ordinal numbers is given by inclusion. Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

For the power set of a set X, the inclusion partial order is (up to isomorphism) the direct product of |X| copies of the partial order on {0,1} for which 0 < 1. In mathematics, a set S, the power set of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ...

Inclusion maps

Inclusion maps tend to be homomorphisms of algebraic structures; more precisely, given a sub-structure closed under some operations, the inclusion map will be a homomorphism for tautological reasons, given the very definition by restriction of what one checks. For example, for a binary operation @, to require that In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. ... In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ...

i(x@y) = i(x)@i(y)

is simply to say that @ is consistently computed in the sub-structure and the large structure. The case of a unary operation is similar; but one should also look at nullary operations, which pick out a constant element. Here the point is that closure means such constants must already be given in the substructure. In mathematics, a unary operation is an operation with only one operand. ... In mathematics, the arity of a function or an operator is the number of arguments or operands it takes, A function or operator can thus be described as unary, binary, ternary,etc. ... In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...

Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds. Contravariant objects such as differential forms restrict to submanifolds, giving a mapping in the other direction. Another example, more sophisticated, is that of affine schemes, for which the inclusions Geometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... This is a glossary of terms specific to differential geometry and differential topology. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...

Spec(R/I) → Spec(R)

and

Spec(R/I²) → Spec(R)

may be different morphisms, where R is a commutative ring and I a ideal. In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...