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: The semigroups LO2 and RO2 are antiisomorphic. O2, ({0,1}, ∧) and (Z2, +2) are commutative, and LO2 and RO2 are noncommutative. LO2, RO2 and ({0,1}, ∧) are bands.
 * O2, the null semigroup of order two.
 * LO2, the left zero semigroup of order two.
 * RO2, 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 (Z2, ·2), the multiplicative group of {0,1} modulo 2.
 * (Z2, +2) (where Z2 = {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.

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 indicates a binary operation on A having the following Cayley table.

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 O2 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 LO2. 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 RO2. 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 (Z2, +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: 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 (Z2, +2)
The Cayley table for the semigroup (Z2, +2) is given below:

This group is isomorphic to the cyclic group Z2 and the symmetric group S2.

Semigroups of order 3
Let A be the three-element set $\{ 1, 2 \}$. Altogether, a total of 39 = 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 $\{0,1\}$ under multiplication is a semigroup of order 3, and contains both $\{0,1\}$ and $\{1, 2, 3\}$ 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 in the On-Line Encyclopedia of Integer Sequences. lists the number of non-equivalent semigroups, and the number of associative binary operations, out of a total of nn 2, determining a semigroup.