Monoid

A monoid is a set equipped with an associative binary operation and an identity element. The operation combines any two elements, and the identity leaves every element unchanged.

Semigroup: A set with an associative binary operation. A monoid is precisely a semigroup equipped with an identity element.
Monoid homomorphism: A function between monoids that preserves the operation and identity element. It carries monoid computations into another monoid without changing their algebraic structure.
Natural numbers under addition: The natural numbers with addition as operation and zero as identity. Repeated addition gives a familiar monoid whose elements need not have additive inverses.
Group: A set with an associative operation, an identity, and an inverse for every element. Every group is a monoid, but monoid elements need not have inverses.
Binary operation: A rule that assigns an element of a set to each ordered pair of elements from that set. It is the operation whose associativity and identity determine a monoid.
Monoid action: A monoid acting on a set through transformations that respect multiplication and identity. It turns monoid elements into composable transformations of other objects.
Strings under concatenation: Finite strings over an alphabet with concatenation and the empty string as identity. Concatenation is associative, while the empty string supplies the identity.
Commutative monoid: A monoid whose operation gives the same result when its two inputs are exchanged. It adds commutativity, an axiom absent from the general definition of monoid.
Associative property: A property of a binary operation where grouping three inputs in either order gives the same result. Associativity lets products of monoid elements be regrouped without changing their value.
Power of an element: The repeated product of an element with itself, including the zeroth power when an identity is available. Associativity makes powers in a monoid unambiguous.

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