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 ({0,1}, ) 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 under matrix multiplication. The two-element semigroup (Z2, +2) The Cayley table for the semigroup 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 . 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 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. See also Empty semigroup Trivial semigroup (semigroup with one element) Semigroup with three elements Special classes of