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text: 'A monoid $M$ is a pair $(X, \\diamond)$ where $X$ is a [set_theory_set set] and $\\diamond$ is an [associative_function associative] binary [3h7 operator] with an identity. $\\diamond$ is often interpreted as concatenation; data structures that support concatenation and have an "empty element" (such as lists, strings, and the natural numbers under addition) are examples of monoids.\n\n Monoids are [3gx algebraic structures]. We write $x \\diamond y$ for the application of $\\diamond$ to $x, y \\in X$, which must be defined. $x \\diamond y$ is commonly abbreviated $xy$ when $\\diamond$ can be inferred from context. The monoid axioms (which govern the behavior of $\\diamond$) are as follows.\n\n1. (Closure) For all $x, y$ in $X$, $xy$ is also in $X$.\n1. (Associativity) For all $x, y, z$ in $X$, $x(yz) = (xy)z$.\n2. (Identity) There is an $e$ in $X$ such that, for all $x$ in $X$, $xe = ex = x.$\n\nThe axiom of closure says that $x \\diamond y \\in X$, i.e. that combining two elements of $X$ using $\\diamond$ yields another element of $X$. In other words, $X$ is [3gy closed] under $\\diamond$.\n\nThe axiom of associativity says that $\\diamond$ is an [3h4 associative] operation, which justifies omitting parenthesis when describing the application of $\\diamond$ to many elements in sequence..\n\nThe axiom of identity says that there is some element $e$ in $X$ that $\\diamond$ treats as "empty": If you apply $\\diamond$ to $e$ and $x$, then $\\diamond$ simply returns $x$. The identity is unique: Given two elements $e$ and $z$ that satisfy the axiom of identity, we have $ze = e = ez = z.$ Thus, we can speak of "the identity" $e$ of $M$. $e$ is often written $1$ or $1_M$.\n\n%%%knows-requisite([4c7]):\nEquivalently, a monoid is a category with exactly one object.\n%%%\n\nMonoids are [algebraic_semigroup semigroups] equipped with an identity. [3gd Groups] are monoids with inverses. For more on how monoids relate to other [algerbraic_structure algebraic structures], refer to the [algebraic_structure_tree tree of algebraic structures].\n\n# Notation\n\nGiven a monoid $M = (X, \\diamond)$, we say "$X$ forms a monoid under $\\diamond$." For example, the set of finite bitstrings forms a monoid under concatenation: The set of finite bitstrings is closed under concatenation; concatenation is an associative operation; and the empty bitstring is a finite bitstring that acts like an identity under concatenation. \n\n$X$ is called the [3gz underlying set] of $M$, and $\\diamond$ is called the _monoid operation_. $x \\diamond y$ is usually abbreviated $xy$. $M$ is generally allowed to substitute for $X$ when discussing the monoid. For example, we say that the elements $x, y \\in X$ are "in $M$," and sometimes write "$x, y \\in M$" or talk about the "elements of $M$."\n\n# Examples\n\nBitstrings form a monoid under concatenation, with the empty string as the identity.\n\nThe set of finite lists of elements drawn from $Y$ (for any set $Y$) form a monoid under concatenation, with the empty list as the identity.\n\nThe natural numbers [45h $\\mathbb N$] form a monid under addition, with $0$ as the identity.\n\nMonoids have found some use in functional programming languages such as [https://en.wikipedia.org/wiki/Haskell_(programming_language) Haskell] and [https://en.wikipedia.org/wiki/Scala_(programming_language) Scala], where they are used to generalize over data types in which values can be "combined" (by some operation $\\diamond$) and which include an "empty" value (the identity).\n',
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