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noids edit 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 1 b symbolically n is a submonoid of m if e n m and x y n whenever x y n in this case n is a monoid under the binary operation inherited from m on the other hand if n is a subset of a monoid that is closed under the monoid operation and is a monoid for this inherited operation then n is not always a submonoid since the identity elements may differ for example the singleton set 0 is closed under multiplication and is not a submonoid of the multiplicative monoid of the nonnegative integers generators edit a subset s of m is said to generate m if the smallest submonoid of m containing s is m if there is a finite set that generates m then m is said to be a finitely generated monoid commutative monoid edit 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 2 an order unit of a commutative monoid m is an element u of m such that for any element x of m there exists v in the set generated by u such that x v 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 partially commutative monoid edit 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 examples edit out of the 16 possible binary boolean operators four have a two sided identity that is also commutative and associative these four each make the set false true a commutative monoid under the standard definitions and and xnor have the identity true while xor and or have the identity false the monoids from and and or are also idempotent while those from xor and xnor are not the set of natural numbers n 0 1 2 is a commutative monoid under addition identity element 0 or multiplication identity element 1 a submonoid of n under addition is called a numerical monoid the set of positive integers n 0 is a commutative monoid under multiplication identity element 1 given a set a the set of subsets of a is a commutative monoid under intersection identity element is a itself given a set a the set of subsets of a is a commutative monoid under union identity element is the empty set generalizing the previous example every bounded semilattice is an idempotent commutative monoid in particular any bounded lattice can be endowed with both a meet and a join monoid structure the identity elements are the lattice s top and its bottom respectively being lattices heyting algebras and boolean algebras are endowed with these monoid structures every singleton set x closed under a binary operation forms the trivial one element monoid which is also the trivial group every group is a monoid and every abelian group a commutative monoid any semigroup s may be turned into a monoid simply by adjoining an element e not in s and defining e s s s e for all s s this conversion of any semigroup to the monoid is done by the free functor between the category of semigroups and the category of monoids 3 thus an idempotent monoid sometimes known as find first may be formed by adjoining an identity element e to the left zero semigroup over a set s the opposite monoid sometimes called find last is formed from the right zero semigroup over s adjoin an identity e to the left zero semigroup with two elements lt gt then the resulting idempotent monoid lt e gt models the lexicographical order of a sequence given the orders of its elements with e representing equality the underlying set of any ring with addition or multiplication as the operation by definition a ring has a multiplicative identity 1 the integers rational numbers real numbers or complex numbers with addition or multiplication as operation 4 the set of all n by n matrices over a given ring with matrix addition or matrix multiplication as the operation the set of all finite strings over some fixed alphabet σ forms a monoid with string concatenation as the operation the empty string serves as the identity element this monoid is denoted σ and is called the free monoid over σ it is not commutative if σ has at least two elements given any monoid m the opposite monoid m op has the same carrier set and identity element as m and its operation is defined by x op y y x any commutative monoid is the opposite monoid of itself given two sets m and n endowed with monoid structure or in general any finite number of monoids m 1 m k their cartesian product m n with the binary operation and identity element defined on corresponding coordinates called the direct product is also a monoid respectively m 1 m k 5 fix a monoid m the set of all functions from a given set to m is also a monoid the identity element is a constant function mapping any value to the identity of m the associative operation is defined pointwise fix a monoid m with the operation and identity element e and consider its power set p m consisting of all subsets of m a binary operation for such subsets can be defined by s t s t s s t t this turns p m into a monoid with identity element e in the same way the power set of a group g is a monoid under the product of group subsets let s be a set the set of all functions s s forms a monoid under function composition the identity is just the identity function it is also called the full transformation monoid of s if s is finite with n elements the monoid of functions on s is finite with n n elements generalizing the previous example let c be a category and x an object of c the set of all endomorphisms of x denoted end c x forms a monoid under composition of morphisms for more on the relationship between category theory and monoids see below the set of homeomorphism classes of compact surfaces with the connected sum its unit element is the class of the ordinary 2 sphere furthermore if a denotes the class of the torus and b denotes the class of the projective plane then every element c of the monoid has a unique expression in the form c na mb where n is a positive integer and m 0 1 or 2 we have 3 b a b let f be a cyclic monoid of order n that is f f 0 f 1 f n 1 then f n f k for some 0 k n each such k gives a distinct monoid of order n and every cyclic monoid is isomorphic to one of these moreover f can be considered as a function on the points 0 1 2 n 1 given by 0 1 2 n 2 n 1 1 2 3 n 1 k displaystyle begin bmatrix 0 1 2 cdots n 2 n 1 1 2 3 cdots n 1 k end bmatrix or equivalently f i i 1 if 0 i n 1 k if i n 1 displaystyle f i begin cases i 1 text if 0 leq i n 1 k text if i n 1 end cases multiplication of elements in f is then given by function composition when k 0 then the function f is a permutation of 0 1 2 n 1 and gives the unique cyclic group of order n properties edit by the monoid axioms the identity element e is unique since if e and f were identity elements of a monoid then e ef f products and powers edit for each nonnegative integer n one can define the product p n i 1 n a i displaystyle p_ n textstyle prod _ i 1 n a_ i of any sequence a 1 a n of n elements of a monoid recursively let p 0 e and let p m p m 1 a m for 1 m n as a special case one can define nonnegative integer powers of an element x of a monoid x 0 1 and x n x n 1 x for n 1 then x m n x m x n for all m n 0 invertible elements edit an element x is called invertible if there exists an element y such that x y e and y x e the element y is called the inverse of x inverses if they exist are unique if y and z are inverses of x then by associativity y ey zx y z xy ze z 6 if x is invertible say with inverse y then one can define negative powers of x by setting x n y n for each n 1 this makes the equation x m n x m x n hold for all m n z the set of all invertible elements in a monoid together with the operation forms a group grothendieck group edit main article grothendieck group not every monoid sits inside a group for instance it is perfectly possible to have a monoid in which two elements a and b exist such that a b a holds even though b is not the identity element for example take a 0 and b 5 in the multiplicative monoid of nonnegative integers such a monoid cannot be embedded in a group because in the group multiplying both sides with the inverse of a would get that b e which is not true a monoid m has the cancellation property or is cancellative if for all a b and c in m the equality a b a c implies b c and the equality b a c a implies b c a commutative monoid with the cancellation property can always be embedded in a group via the grothendieck group construction that is how the additive group of the integers a group with operation is constructed from the additive monoid of natural numbers a commutative monoid with operation and cancellation property however a non commutative cancellative monoid need not be embeddable in a group if a monoid has the cancellation property and is finite then it is in fact a group c the right and left cancellative elements of a monoid each in turn form a submonoid i e are closed under the operation and obviously include the identity this means that the cancellative elements of any commutative monoid can be extended to a group the cancellative property in a monoid is not necessary to perform the grothendieck construction commutativity is sufficient however if a commutative monoid does not have the cancellation property the homomorphism of the monoid into its grothendieck group is not injective more precisely if a b a c then b and c have the same image in the grothendieck group even if b c in particular if the monoid has an absorbing element then its grothendieck group is the trivial group types of monoids edit an inverse monoid is a monoid where for every a in m there exists a unique a 1 in m such that a a a 1 a and a 1 a 1 a a 1 if an inverse monoid is cancellative then it is a group in the opposite direction a zerosumfree monoid is an additively written monoid in which a b 0 implies that a 0 and b 0 7 equivalently that no element other than zero has an additive inverse acts and operator monoids edit main article monoid act let m be a monoid with the binary operation denoted by and the identity element denoted by e then a left m act or left act over m is a set x together with an operation m x x which is compatible with the monoid structure as follows for all x in x e x x for all a b in m and x in x a b x a b x this is the analogue in monoid theory of a left group action right m acts are defined in a similar way a monoid with an act is also known as an operator monoid important examples include transition systems of semiautomata a transformation semigroup can be made into an operator monoid by adjoining the identity transformation monoid homomorphisms edit example monoid homomorphism x 2 x from n 0 to n 1 it is injective but not surjective a homomorphism between two monoids m and n is a function f m n such that f x y f x f y for all x y in m f e m e n where e m and e n are the identities on m and n respectively monoid homomorphisms are sometimes simply called monoid morphisms not every semigroup homomorphism between monoids is a monoid homomorphism since it may not map the identity to the identity of the target monoid even though the identity is the identity of the image of the homomorphism d for example consider z n the set of residue classes modulo n equipped with multiplication in particular 1 n is the identity element function f z 3 z 6 given by k 3 3 k 6 is a semigroup homomorphism since 3 k 3 l 6 9 kl 6 3 kl 6 however f 1 3 3 6 1 6 so a monoid homomorphism is a semigroup homomorphism between monoids that maps the identity of the first monoid to the identity of the second monoid and the latter condition cannot be omitted in contrast a semigroup homomorphism between groups is always a group homomorphism as it necessarily preserves the identity because in the target group of the homomorphism the identity element is the only element x such that x x x a bijective monoid homomorphism is called a monoid isomorphism two monoids are said to be isomorphic if there is a monoid isomorphism between them equational presentation edit main article presentation of a monoid monoids may be given a presentation much in the same way that groups can be specified by means of a group presentation one does this by specifying a set of generators σ and a set of relations on the free monoid σ one does this by extending finite binary relations on σ to monoid congruences and then constructing the quotient monoid as above given a binary relation r σ σ one defines its symmetric closure as r r 1 this can be extended to a symmetric relation e σ σ by defining x e y if and only if x sut and y svt for some strings u v s t σ with u v r r 1 finally one takes the reflexive and transitive closure of e which is then a monoid congruence in the typical situation the relation r is simply given as a set of equations so that r u 1 v 1 u n v n thus for example p q p q 1 displaystyle langle p q vert pq 1 rangle is the equational presentation for the bicyclic monoid and a b a b a b a a b b a b a b displaystyle langle a b vert aba baa bba bab rangle is the plactic monoid of degree 2 it has infinite order elements of this plactic monoid may be written as a i b j b a k displaystyle a i b j ba k for integers i j k as the relations show that ba commutes with both a and b relation to category theory edit group like structures total associative identity divisible partial magma unneeded unneeded unneeded unneeded multiplicative unneeded unneeded required unneeded semigroupoid unneeded required unneeded unneeded small category unneeded required required unneeded groupoid unneeded required required required magma required unneeded unneeded unneeded quasigroup required unneeded unneeded required unital magma required unneeded required unneeded loop required unneeded required required semigroup required required unneeded unneeded associative quasigroup required required unneeded required monoid required required required unneeded group required required required required 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 8 that is a monoid is essentially the same thing as a category with a single object 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 just functors between single object categories 8 so this construction gives an equivalence between the category of small monoids mon and a full subcategory of the category o...