, or equivalently , or equivalently the set under multiplication: the only group of order two.
The semigroups LO2 and RO2 are antiisomorphic. O2, and are commutative, and LO2 and RO2 are noncommutative. LO2, RO2 and are bands and also inverse semigroups.
Determination of semigroups with two elements
Choosing the set A = 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
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 rightzero 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.
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 is given below:
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 under multiplication. It is also isomorphic to the semigroup under matrix multiplication.
The two-element semigroup (Z2,+2)
The Cayley table for the semigroup is given below:
Let A be the three-element set. 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 the exception of the group with three elements, each of these has one of the above two-element semigroups as subsemigroups. For example, the set under multiplication is a semigroup of order 3, and contains both and 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 nn2, determining a semigroup.
