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For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function. In the study of several complex variables, the definition of a domain is extended to include any connected open subset of C n.
In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be ...
The set of points in the red oval X is the domain of f. Graph of the real-valued square root function, f(x) = √ x, whose domain consists of all nonnegative real numbers. In mathematics, the domain of a function is the set of inputs accepted by the function.
and what we obtain is an extension by definitions ′ of . Then in T ′ {\displaystyle T'} we can prove that for every x , there exists a unique y such that x × y = y × x = e . Consequently, the first-order theory T ″ {\displaystyle T''} obtained from T ′ {\displaystyle T'} by adding a unary function symbol f {\displaystyle f} and the axiom
Extension of a function, defined on a larger domain; Extension of a polyhedron, in geometry; Extension of a line segment (finite) into an infinite line (e.g., extended base) Exterior algebra, Grassmann's theory of extension, in geometry; Field extension, in Galois theory; Group extension, in abstract algebra and homological algebra
The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether 0 belongs to its domain of definition (see Halting problem).
An Ore extension of a domain is a domain. An Ore extension of a skew field is a non-commutative principal ideal domain. If σ is an automorphism and R is a left Noetherian ring then the Ore extension R[ λ; σ, δ ] is also left Noetherian.
the domain of A is a subset of the domain of B, f A = f B | A n for every n-ary function symbol f in σ, and; R A R B A n for every n-ary relation symbol R in σ. A is said to be a substructure of B, or a subalgebra of B, if A is a weak subalgebra of B and, moreover,