In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five distinctnonisomorphic semigroups having two elements:
O2, the null semigroup of order two,
LO2 and RO2, the left zero semigroup of order two and right zero semigroup of order two, respectively,
({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,
(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.
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 and also inverse semigroups.
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}, ). 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}, ) 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. 113 of the 19683 binary operations determine 24 nonisomorphic semigroups, or 18 non-equivalent semigroups (with equivalence being isomorphism or anti-isomorphism).
[2] With the exception of the group with three elements, each of these has one (or more) of the above two-element semigroups as subsemigroups.[3] 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.[3][4][5] 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.
^Siniša Crvenkovič; Ivan Stojmenovic. "An algorithm for Cayley tables of algebras". 23 (2). Univ. u Novom Sadu, Zb. Rad. Prirod.-Mat. Fak. Review of Research, Faculty of Science: 221–231. Cite journal requires |journal= (help)[1] (Accessed on 9 May 2009)
^John A Hildebrant (2001). Handbook of Finite Semigroup Programs. (Preprint).[2]
