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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are semigroups with identity. Monoids occur in several branches of mathematics; for instance, they can be regarded as categories with a single object. Thus, they capture the idea of function composition within a set. Monoids are also commonly used in computer science, both in its foundational aspects and in practical programming. The set of strings built from a given set of characters is a free monoid. Thalgebra)|magma]] with associativity and identity. The identity element of a monoid is unique.^{[1]} A monoid in which each element has an inverse is a group.
Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by juxtaposition; for example, the monoid axioms may be written (ab)c = a(bc) and ea=ae=a. This notation does not imply that it is numbers being multiplied.
A submonoid of a monoid (M, •) is a subset N of M that is closed under the monoid operation and contains the identity element e of M.^{[2]}^{[3]} Symbolically, N is a submonoid of M if N ⊆ M, x • y ∈ N whenever x, y ∈ N, and e ∈ N. N is thus a monoid under the binary operation inherited from M.
A subset S of M is said to be a generator of M if M is the smallest set containing S that is closed under the monoid operation, or equivalently M is the result of applying the finitary closure operator to S. If there is a generator of M that has finite cardinality, then M is said to be finitely generated. Not every set S will generate a monoid, as the generated structure may lack an identity element.
A monoid whose operation is commutative is called a commutative monoid (or, less commonly, an abelian monoid). Commutative monoids are often written additively. Any commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y if there exists z such that x + z = y.^{[4]} An order-unit of a commutative monoid M is an element u of M such that for any element x of M, there exists a positive integer n such that x ≤ nu. This is often used in case M is the positive cone of a partially ordered abelian group G, in which case we say that u is an order-unit of G.
A monoid for which the operation is commutative for some, but not all elements is a trace monoid; trace monoids commonly occur in the theory of concurrent computation.
is the equational presentation for the bicyclic monoid, and
is the plactic monoid of degree 2 (it has infinite order). Elements of this plactic monoid may be written as a^ib^j(ba)^k for integers i, j, k, as the relations show that ba commutes with both a and b.
Monoids can be viewed as a special class of categories. Indeed, the axioms required of a monoid operation are exactly those required of morphism composition when restricted to the set of all morphisms whose source and target is a given object.^{[9]} That is,
More precisely, given a monoid (M, •), one can construct a small category with only one object and whose morphisms are the elements of M. The composition of morphisms is given by the monoid operation •.
Likewise, monoid homomorphisms are sures that the operation can be parallelized by employing a prefix sum or similar algorithm, in order to utilize multiple cores or processors efficiently.
Given a sequence of values of type M with identity element \varepsilon and associative operation *, the fold operation is defined as follows:
In addition, any data structure can be 'folded' in a similar way, given a serialization of its elements. For instance, the result of "folding" a binary tree might differ depending on pre-order vs. post-order tree traversal.
A complete monoid is a commutative monoid equipped with an infinitary sum operation \Sigma_I for any index set I such that:^{[10]}^{[11]}^{[12]}^{[13]}
\sum_{i \in \emptyset}{m_i} =0;\quad \sum_{i \in \{j\}}{m_i} = m_j;\quad \sum_{i \in \{j, k\}}{m_i} = m_j+m_k \quad \textrm{for}\; j\neq k
and
\sum_{j \in J} In , a branch of , a '''monoid''' is an with a single and an . Monoids are studied in theory as they are semigroups with identity. Monoids occur in several branches of mathematics; for instance, they can be regarded as with a single . Thus, they capture the idea of within a set. Monoids are also commonly used in , both in its foundational aspects and in practical programming. The set of built from a given set of is a . The and are used in describing , whereas s and s provide a foundation for and . Some of the more important results in the study of monoids are the and the . The history of monoids, as well as a discussion of additional general properties, are found in the article on s. == Definition == Suppose that ''S'' is a and • is some , then ''S'' with • is a '''monoid''' if it satisfies the following two axioms: ;Associativity: For all ''a'', ''b'' and ''c'' in ''S'', the equation holds. ;Identity element: There exists an element ''e'' in ''S'' such that for every element ''a'' in ''S'', the equations hold. In other words, a monoid is a with an . It can also be thought of as a with associativity and identity. The identity element of a monoid is unique.If both ''e''_{1} and ''e''_{2} satisfy the above equations, then ''e''_{1} = ''e''_{1} • ''e''_{2} = ''e''_{2}. A monoid in which each element has an is a . Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by juxtaposition; for example, the monoid axioms may be written $(ab)c\; =\; a(bc)$ and ea=ae=a. This notation does not imply that it is numbers being multiplied.
A submonoid of a monoid (M, •) is a subset N of M that is closed under the monoid operation and contains the identity element e of M.^{[14]}^{[15]} Symbolically, N is a submonoid of M if N ⊆ M, x • y ∈ N whenever x, y ∈ N, and e ∈ N. N is thus a monoid under the binary operation inherited from M.
A monoid whose operation is commutative is called a commutative monoid (or, less commonly, an abelian monoid). Commutative monoids are often written additively. Any commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y if there exists z such that x + z = y.^{[16]} An order-unit of a commutative monoid M is an element u of M such that for any element x of M, there exists a positive integer n such that x ≤ nu. This is often used in case M is the positive cone of a partially ordered abelian group G, in which case we say that u is an order-unit of G.
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