# Semigroup with two elements

In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five nonisomorphic semigroups having two elements:
* O$_2$, the null semigroup of order two.
* LO$_2$, the left zero semigroup of order two.
* RO$_2$, the right zero semigroup of order two.
* ({0,1}, ∧) (where "∧" is the logical connective "and"), or equivalently the set {0,1} under multiplication: the only semilattice with two elements and the only non-null semigroup with zero of order two, also a monoid, and ultimately the two-element Boolean algebra; this is also isomorphic to (Z$_2$, ·$_2$), the multiplicative group of {0,1} modulo 2.
* (Z$_2$, +$_2$) (where Z$_2$ = {0,1} and "+$_2$" is "addition modulo 2"), or equivalently ({0,1}, ⊕) (where "⊕" is the logical connective "xor"), or equivalently the set {−1,1} under multiplication: the only group of order two.
The semigroups LO$_2$ and RO$_2$ are antiisomorphic. O$_2$, ({0,1}, ∧) and (Z$_2$, +$_2$) are commutative, and LO$_2$ and RO$_2$ are noncommutative. LO$_2$, RO$_2$ and ({0,1}, ∧) are bands.

## Determination of semigroups with two elements

Choosing the set A = { 1, 2 } as the underlying set having two elements, sixteen binary operations can be defined in A. These operations are shown in the table below. In the table, a matrix of the form
| x | y |
| z | t |
indicates a binary operation on A having the following Cayley table.
| | 1 | 2 |
| 1 | x | y |
| 2 | z | t |
List of binary operations in { 1, 2 } :
| 1   1   1   1 | 1   1   1   2 | 1   1   2   1 | 1   1   2   2 |
| Null semigroup O$_2$ | ≡ Semigroup ({0,1}, $\wedge$) | 2·(1·2) = 2, (2·1)·2 = 1 | Left zero semigroup LO$_2$ |
| 1   2   1   1 | 1   2   1   2 | 1   2   2   1 | 1   2   2   2 |
| 2·(1·2) = 1, (2·1)·2 = 2 | Right zero semigroup RO$_2$ | ≡ Group (Z$_2$, ·$_2$) | ≡ Semigroup ({0,1}, $\wedge$) |
| 2   1   1   1 | 2   1   1   2 | 2   1   2   1 | 2   1   2   2 |
| 1·(1·2) = 2, (1·1)·2 = 1 | ≡ Group (Z$_2$, +$_2$) | 1·(1·1) = 1, (1·1)·1 = 2 | 1·(2·1) = 1, (1·2)·1 = 2 |
| 2   2   1   1 | 2   2   1   2 | 2   2   2   1 | 2   2   2   2 |
| 1·(1·1) = 2, (1·1)·1 = 1 | 1·(2·1) = 2, (1·2)·1 = 1 | 1·(1·2) = 2, (1·1)·2 = 1 | Null semigroup O$_2$ |
In this table:
* The semigroup ({0,1}, $\wedge$) denotes the two-element semigroup containing the zero element 0 and the unit element 1. The two binary operations defined by matrices in a green background are associative and pairing either with A creates a semigroup isomorphic to the semigroup ({0,1}, $\wedge$). Every element is idempotent in this semigroup, so it is a band. Furthermore, it is commutative (abelian) and thus a semilattice. The order induced is a linear order, and so it is in fact a lattice and it is also a distributive and complemented lattice, i.e. it is actually the two-element Boolean algebra.
* The two binary operations defined by matrices in a blue background are associative and pairing either with A creates a semigroup isomorphic to the null semigroup O$_2$ with two elements.
* The binary operation defined by the matrix in an orange background is associative and pairing it with A creates a semigroup. This is the left zero semigroup LO$_2$. It is not commutative.
* The binary operation defined by the matrix in a purple background is associative and pairing it with A creates a semigroup. This is the right zero semigroup RO$_2$. It is also not commutative.
* The two binary operations defined by matrices in a red background are associative and pairing either with A creates a semigroup isomorphic to the group (Z$_2$, +$_2$).
* The remaining eight binary operations defined by matrices in a white background are not associative and hence none of them create a semigroup when paired with A.

## The two-element semigroup ({0,1}, ∧)

The Cayley table for the semigroup ({0,1}, $\wedge$) is given below:
| $\wedge$ | 0 | 1 |
| 0 | 0 | 0 |
| 1 | 0 | 1 |
This is the simplest non-trivial example of a semigroup that is not a group. This semigroup has an identity element, 1, making it a monoid. It is also commutative. It is not a group because the element 0 does not have an inverse, and is not even a cancellative semigroup because we cannot cancel the 0 in the equation 1·0 = 0·0.
This semigroup arises in various contexts. For instance, if we choose 1 to be the truth value "true" and 0 to be the truth value "false" and the operation to be the logical connective "and", we obtain this semigroup in logic. It is isomorphic to the monoid {0,1} under multiplication. It is also isomorphic to the semigroup
    $S=\left\{{\begin{pmatrix}1&0\\0&1\end{pmatrix}},{\begin{pmatrix}1&0\\0&0\end{pmatrix}}\right\}$
under matrix multiplication.

## The two-element semigroup (Z$_2$, +$_2$)

The Cayley table for the semigroup (Z$_2$, +$_2$) is given below:
| +$_2$ | 0 | 1 |
| 0 | 0 | 1 |
| 1 | 1 | 0 |
This group is isomorphic to the cyclic group Z$_2$ and the symmetric group S$_2$.

## Semigroups of order 3

Let A be the three-element set {1, 2, 3}. Altogether, a total of 3$^9$ = 19683 different binary operations can be defined on A. 113 of the 19683 binary operations determine 24 nonisomorphic semigroups, or 18 non-equivalent semigroups (with equivalence being isomorphism or anti-isomorphism). With the exception of the group with three elements, each of these has one (or more) of the above two-element semigroups as subsemigroups. For example, the set {−1, 0, 1} under multiplication is a semigroup of order 3, and contains both {0, 1} and {−1, 1} as subsemigroups.

## Finite semigroups of higher orders

Algorithms and computer programs have been developed for determining nonisomorphic finite semigroups of a given order. These have been applied to determine the nonisomorphic semigroups of small order. The number of nonisomorphic semigroups with n elements, for n a nonnegative integer, is listed under OEIS: A027851 in the On-Line Encyclopedia of Integer Sequences. OEIS: A001423 lists the number of non-equivalent semigroups, and OEIS: A023814 the number of associative binary operations, out of a total of n$^{n2}$, determining a semigroup.