Invariant (mathematics)

An invariant is a property or quantity preserved under a specified transformation or operation. Invariants help classify mathematical objects and reveal what changes can and cannot alter.

Graph invariant: A quantity or property of a graph unchanged by specified graph transformations or preserved under graph isomorphism. Graph invariants distinguish graph structures without depending on how vertices are labeled.
Group action: A group acting on a set through transformations that satisfy the group's composition rules. Group actions specify the transformations against which many invariants are defined.
Transformation (mathematics): A function that maps elements of a mathematical space to elements of the same or another space. An invariant is defined relative to a chosen family of transformations.
Variant (mathematics): A mathematical quantity or property that changes under a specified transformation or operation. A variant changes where an invariant remains fixed under the same operations.
Topological invariant: A property of a topological space preserved by homeomorphisms or other specified continuous transformations. Topological invariants identify features that survive stretching or bending a space.
Orbit (mathematics): The set of points obtainable from one point by applying every transformation in a group action. An invariant must take the same value throughout each orbit.
Function (mathematics): A rule assigning each input in a domain exactly one output in a codomain. Many invariants are functions whose values remain constant under transformations.
Covariant (mathematics): A mathematical object whose transformation follows a specified rule when coordinates or inputs change. Covariants may transform systematically rather than remain unchanged.
Polynomial invariant: A polynomial-valued quantity that remains unchanged under a specified group action or equivalence. Polynomial invariants encode structural information in algebraic form.
Fixed point (mathematics): A point left unchanged by a function or transformation. A fixed point is a special case where the object itself, not merely a property, is preserved.

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