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 distinct nonisomorphic semigroups having two elements:

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}, \wedge)     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}, \wedge)
  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 two-element semigroup ({0,1}, ∧)

The Cayley table for the semigroup ({0,1}, \wedge) is given below:

 \wedge  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

 
S = \left\{
\begin{pmatrix}
  1 & 0 \\
  0 & 1 
\end{pmatrix}, 
\begin{pmatrix}
  1 & 0 \\
  0 & 0 
\end{pmatrix}
\right\}

under matrix multiplication.[1]

The two-element semigroup (Z2,+2)

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

+2  0   1 
  0    0    1 
  1    1    0 

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, 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 A027851 in the On-Line Encyclopedia of Integer Sequences. A001423 lists the number of non-equivalent semigroups, and A023814 the number of associative binary operations, out of a total of nn2, determining a semigroup.

See also

References

  1. Semigroup with two elements at PlanetMath.org.
  2. Friðrik Diego; Kristín Halla Jónsdóttir (July 2008). "Associative Operations on a Three-Element Set". The Montana Mathematics Enthusiast 5 (2 & 3): 257–268. Retrieved 6 February 2014.
  3. 3.0 3.1 Andreas Distler, Classification and enumeration of finite semigroups, PhD thesis, University of St. Andrews
  4. 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. pp. 221–231. (Accessed on 9 May 2009)
  5. John A Hildebrant (2001). Handbook of Finite Semigroup Programs. (Preprint).