semigroup with two elements

Choosing the set {{nowrap|1=A = {{mset| 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

indicates a binary operation on A having the following Cayley table.

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|  Null semigroup O₂  

|  ≡ Semigroup {{nowrap|({{mset|0,1}}, $\backslash wedge$)}} 

|  {{nowrap|1=2·(1·2) = 2}}, {{nowrap|1=(2·1)·2 = 1}} 

|  Left zero semigroup LO₂ 

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|  {{nowrap|1=2·(1·2) = 1}}, {{nowrap|1=(2·1)·2 = 2}} 

|  Right zero semigroup RO₂ 

|  ≡ Group {{nowrap|(Z₂, ·₂)}} 

|  ≡ Semigroup {{nowrap|({{mset|0,1}}, $\backslash wedge$)}}

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|  {{nowrap|1=1·(1·2) = 2}}, {{nowrap|1=(1·1)·2 = 1}} 

|  ≡ Group {{nowrap|(Z₂, +₂)}} 

|  {{nowrap|1=1·(1·1) = 1}}, {{nowrap|1=(1·1)·1 = 2}} 

|  {{nowrap|1=1·(2·1) = 1}}, {{nowrap|1=(1·2)·1 = 2}} 

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|  {{nowrap|1=1·(1·1) = 2}}, {{nowrap|1=(1·1)·1 = 1}} 

|  {{nowrap|1=1·(2·1) = 2}}, {{nowrap|1=(1·2)·1 = 1}} 

|  {{nowrap|1=1·(1·2) = 2}}, {{nowrap|1=(1·1)·2 = 1}} 

|  Null semigroup O₂ 

|}

In this table:

• The semigroup {{nowrap|({0,1}, $\backslash wedge$)}} 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 {{nowrap|({0,1}, $\backslash wedge$)}}. 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 {{nowrap|(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 Cayley table for the semigroup ({0,1}, $\backslash wedge$) is given below:

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.

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

This group is isomorphic to the cyclic group Z₂ and the symmetric group S₂.

{{main|Semigroup with three elements}}

Let A be the three-element set {{nowrap|{{mset|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).

{{cite journal|last=Friðrik Diego|author2=Kristín Halla Jónsdóttir |title=Associative Operations on a Three-Element Set|journal=The Montana Mathematics Enthusiast|date=July 2008|volume=5|issue=2 & 3|pages=257–268|doi=10.54870/1551-3440.1106 |s2cid=118704099 |url=http://www.math.umt.edu/tmme/vol5no2and3/TMME_vol5nos2and3_a8_pp.257_268.pdf|accessdate=6 February 2014}} With the exception of the group with three elements, each of these has one (or more) of the above two-element semigroups as subsemigroups.Andreas Distler, [http://bcc2009.mcs.st-and.ac.uk/Theses/andreasDthesis.pdf Classification and enumeration of finite semigroups] {{webarchive|url=https://web.archive.org/web/20150402121217/http://bcc2009.mcs.st-and.ac.uk/Theses/andreasDthesis.pdf |date=2 April 2015 }}, PhD thesis, University of St. Andrews For example, the set {{nowrap|{{mset|−1, 0, 1}}}} under multiplication is a semigroup of order 3, and contains both {{nowrap|{{mset|0, 1}}}} and {{nowrap|{{mset|−1, 1}}}} as subsemigroups.

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.{{cite journal

| last1 = Crvenković | first1 = Siniša

| last2 = Stojmenović | first2 = Ivan

| issue = 2

| journal = Zbornik Radova Prirodno-Matematichkog Fakulteta. Serija za Matematiku. Review of Research. Faculty of Science. Mathematics Series

| mr = 1333549

| pages = 221–231

| title = An algorithm for Cayley tables of algebras

| url = https://www.emis.de/journals/NSJOM/Papers/23_2/NSJOM_23_2_221_231.pdf

| volume = 23

| year = 1993}}{{Cite book |last=John A Hildebrant |title=Handbook of Finite Semigroup Programs |publisher=(Preprint) |year=2001}}[http://www.math.lsu.edu/~preprint/2001/jah2001a.pdf] The number of nonisomorphic semigroups with n elements, for n a nonnegative integer, is listed under {{OEIS2C|A027851}} in the On-Line Encyclopedia of Integer Sequences. {{OEIS2C|A001423}} lists the number of non-equivalent semigroups, and {{OEIS2C|A023814}} the number of associative binary operations, out of a total of nⁿ², determining a semigroup.

• Empty semigroup
• Trivial semigroup (semigroup with one element)
• Semigroup with three elements
• Special classes of semigroups

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Category:Algebraic structures

Category:Semigroup theory