mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a semigroup with two elements is a semigroup for which the

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

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 In category theory, a branch of mathematics, an antiisomorphism (or anti-isomorphism) between structured sets ''A'' and ''B'' is an isomorphism from ''A'' to the opposite of ''B'' (or equivalently from the opposite of ''A'' to ''B''). If there ...

. O₂, and are

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

, 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 operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...

s 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 Named after the 19th-century United Kingdom, British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an additi ...

. 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 In the mathematics, mathematical discipline of order theory, a complemented lattice is a bounded lattice (order), lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' satisfyin ...

, 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

s defined by matrices in a white background are not

associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...

and hence none of them create a semigroup when paired with ''A''.

The two-element semigroup (, ∧)

The

for the semigroup (,

\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 In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in ...

" true" and 0 to be the

" false" and the operation to be the logical connective " and", we obtain this semigroup in

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

. It is isomorphic to the monoid under multiplication. It is also isomorphic to the semigroup :

S = \left\

under matrix multiplication.

The two-element semigroup (Z₂, +₂)

The

for the semigroup is given below: This group is isomorphic to the cyclic group Z₂ and the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

S₂.

Semigroups of order 3

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). 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
Classification and enumeration of finite semigroups
, PhD thesis, University of St. Andrews 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.
/ref> 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 ''n''^''n''², determining a semigroup.

References

{{DEFAULTSORT:Semigroup With Two Elements Algebraic structures Semigroup theory