Semigroup with two elements

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In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five nonisomorphic semigroups having two elements:

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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.

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 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

under matrix multiplication.

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). [1] With the exception of the group with three elements, each of these has one (or more) of the above two-element semigroups as subsemigroups. [2] 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. [2] [3] [4] 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.

See also

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References

  1. ↑ Friðrik Diego; Kristín Halla Jónsdóttir (July 2008). "Associative Operations on a Three-Element Set" (PDF). The Montana Mathematics Enthusiast. 5 (2 & 3): 257–268. doi:10.54870/1551-3440.1106. S2CID   118704099 . Retrieved 6 February 2014.
  2. 1 2 Andreas Distler, Classification and enumeration of finite semigroups Archived 2 April 2015 at the Wayback Machine , PhD thesis, University of St. Andrews
  3. ↑ Crvenković, Siniša; Stojmenović, Ivan (1993). "An algorithm for Cayley tables of algebras" (PDF). Zbornik Radova Prirodno-Matematichkog Fakulteta. Serija za Matematiku. Review of Research. Faculty of Science. Mathematics Series. 23 (2): 221–231. MR   1333549.
  4. ↑ John A Hildebrant (2001). Handbook of Finite Semigroup Programs. (Preprint).

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