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Concept
Kernel (set theory)
Summary
In set theory, the kernel of a function (or equivalence kernel) may be taken to be either
the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function can tell", or
the corresponding partition of the domain.
An unrelated notion is that of the kernel of a non-empty family of sets which by definition is the intersection of all its elements:
This definition is used in the theory of filters to classify them as being free or principal.
For the formal definition, let be a function between two sets.
Elements are equivalent if and are equal, that is, are the same element of
The kernel of is the equivalence relation thus defined.
The is
The kernel of is also sometimes denoted by The kernel of the empty set, is typically left undefined.
A family is called and is said to have if its is not empty.
A family is said to be if it is not fixed; that is, if its kernel is the empty set.
Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition:
This quotient set is called the of the function and denoted (or a variation).
The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the , specifically, the equivalence class of in (which is an element of ) corresponds to in (which is an element of ).
Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product
In this guise, the kernel may be denoted (or a variation) and may be defined symbolically as
The study of the properties of this subset can shed light on
Kernel (algebra)
If and are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function is a homomorphism, then is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of is a quotient of
The bijection between the coimage and the image of is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem.
Official source
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