(Z2, +2) (where Z2 = {0,1} and "+2" is "addition modulo 2"): the only group of order two.
The semigroups LO2 and RO2 are antiisomorphic. O2, ({0,1}, ∧) and (Z2, +2) are commutative, LO2 and RO2 are noncommutative. LO2, RO2 and ({0,1}, ∧) are bands and also inverse semigroups.
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 O2
≡ Semigroup ({0,1}, )
2·(1·2) = 2, (2·1)·2 = 1
Left zero semigroup LO2
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 RO2
≡ Group (Z2, +2)
≡ Semigroup ({0,1}, )
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 (Z2, +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 O2
In this table:
The semigroup ({0,1}, ) 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}, ).
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 semigroupO2 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 semigroupLO2.
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 semigroupRO2.
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}, ) 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 {0,1} under multiplication. It is also isomorphic to the semigroup
Let A be the three-element set { 1, 2, 3 }. Altogether, a total of 39 = 19683 different binary operations can be defined on A. It is impractical to prepare a list of all these binary operations, and determine those that make A nonisomorphic semigroups.
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 order three or higher.[3][4] In fact, 113 of the 19683 binary operations determine 24 nonisomorphic semigroups, or 18 non-equivalent semigroups (with equivalence being isomorphism or anti-isomorphism). 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 nn2, determining a semigroup.
^Friðrik Diego (July 2008). "Associative Operations on a Three-Element Set"(PDF). The Montana Mathematics Enthusiast. Vol.5, Nos.2&3 (July 2008) pp. 167-462 (2 & 3): 257–268. Retrieved 6 February 2014. {{cite journal}}: |volume= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
