Semigroup with two elements explained

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:

O₂, the null semigroup of order two.
LO₂, the left zero semigroup of order two.
RO₂, the right zero semigroup of order two.
(∧) (where "∧" is the logical connective "and"), or equivalently the set 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; this is also isomorphic to (Z₂, ·₂), the multiplicative group of modulo 2.
(Z₂, +₂) (where Z₂ = and "+₂" is "addition modulo 2"), or equivalently (⊕) (where "⊕" is the logical connective "xor"), or equivalently the set under multiplication: the only group of order two.

The semigroups LO₂ and RO₂ are antiisomorphic. O₂, and are commutative, and LO₂ and RO₂ are noncommutative. LO₂, RO₂ and are bands.

Determination of semigroups with two elements

Choosing the set 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	1
1	1

1	1
1	2

1	1
2	1

1	1
2	2

Null semigroup O₂

≡ Semigroup

Left zero semigroup LO₂

1	2
1	1

1	2
1	2

1	2
2	1

1	2
2	2

Right zero semigroup RO₂

≡ Group

≡ Semigroup

2	1
1	1

2	1
1	2

2	1
2	1

2	1
2	2

≡ Group

2	2
1	1

2	2
1	2

2	2
2	1

2	2
2	2

Null semigroup O₂

In this table:

The semigroup 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 . 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 .
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 (∧)

The Cayley table for the semigroup (

\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 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 two-element semigroup (Z₂, +₂)

The Cayley table for the semigroup 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

See main article: Semigroup with three elements. Let A be the three-element set . 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 under multiplication is a semigroup of order 3, and contains both and 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 in the On-Line Encyclopedia of Integer Sequences. lists the number of non-equivalent semigroups, and the number of associative binary operations, out of a total of nⁿ², determining a semigroup.

Notes and References

Friðrik Diego. Kristín Halla Jónsdóttir . Associative Operations on a Three-Element Set. The Montana Mathematics Enthusiast. July 2008. 5. 2 & 3. 257–268. 10.54870/1551-3440.1106 . 118704099 . 6 February 2014.
Andreas Distler, Classification and enumeration of finite semigroups, PhD thesis, University of St. Andrews
Crvenković . Siniša . Stojmenović . Ivan . 2 . Zbornik Radova Prirodno-Matematichkog Fakulteta. Serija za Matematiku. Review of Research. Faculty of Science. Mathematics Series . 1333549 . 221–231 . An algorithm for Cayley tables of algebras . 23 . 1993.
Book: John A Hildebrant . Handbook of Finite Semigroup Programs . (Preprint) . 2001. http://www.math.lsu.edu/~preprint/2001/jah2001a.pdf