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:

  • 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.

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}, ). 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

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
Main article: Semigroup with three elements

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.

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See also

References
  1. "Semigroup with two elements". PlanetMath.
  2. Friðrik Diego; Kristín Halla Jónsdóttir (July 2008). "Associative Operations on a Three-Element Set" (PDF). The Montana Mathematics Enthusiast. 5 (2 & 3): 257–268. Retrieved 6 February 2014.
  3. Andreas Distler, Classification and enumeration of finite semigroups Archived 2 April 2015 at the Wayback Machine, 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: 221–231. Cite journal requires |journal= (help) (Accessed on 9 May 2009)
  5. John A Hildebrant (2001). Handbook of Finite Semigroup Programs. (Preprint).

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