Monoid and Pointwise

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Difference between Monoid and Pointwise

Monoid vs. Pointwise

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. 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 — operations defined on functions by applying the operations to function values separately for each point in the domain of definition.

Algebraic structure

In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.

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Associative property

In mathematics, the associative property is a property of some binary operations.

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Closure operator

In mathematics, a closure operator on a set S is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of S to itself which satisfies the following conditions for all sets X,Y\subseteq S |- | X \subseteq \operatorname(X) | (cl is extensive) |- | X\subseteq Y \Rightarrow \operatorname(X) \subseteq \operatorname(Y) | (cl is increasing) |- | \operatorname(\operatorname(X)).

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Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

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Idempotence

Idempotence is the property of certain operations in mathematics and computer science that they can be applied multiple times without changing the result beyond the initial application.

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Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.

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Lattice (order)

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.

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Mathematics

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

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Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

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Partially ordered set

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

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Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

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