Monoid - Examples

Examples

• Every singleton set {x} gives rise to a particular one-element (trivial) monoid. The monoid axioms require that x*x = x in this case.
• Every group is a monoid and every abelian group a commutative monoid.
• 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 and Boolean algebra are endowed with these monoid structures.
• 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.
  • 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 natural numbers, N, form a commutative monoid under addition (identity element zero), or multiplication (identity element one). A submonoid of N under addition is called a numerical monoid.
• The positive integers, N − {0}, form a commutative monoid under multiplication (identity element one).
• The elements of any unital ring, with addition or multiplication as the operation.
  • The integers, rational numbers, real numbers or complex numbers, with addition or multiplication as operation.
  • 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.
• Given any monoid M, the opposite monoid Mop 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, M₁, ..., M_k), their cartesian product M × N is also a monoid (respectivelly, M₁ × ... × M_k). The associative operation and the identity element are defined pairwise.
• 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 in S and t in 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 nn elements.
• Generalizing the previous example, let C be a category and X an object in 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 the form c = na + mb where n is the integer ≥ 0 and m = 0, 1, or 2. We have 3b = a + b.
• Let be a cyclic monoid of order n, that is, . Then for some . In fact, 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 given by

or, equivalently

Multiplication of elements in is then given by function composition.

Note also that when then the function f is a permutation of and gives the unique cyclic group of order n.