In abstract algebra, a semigroup with three elements is an object consisting of three elements and an associative operation defined on them. The basic example would be the three integers 0, 1, and −1, together with the operation of multiplication. Multiplication of integers is associative, and the product of any two of these three integers is again one of these three integers. There are 18 inequivalent ways to define an associative operation on three elements: while there are, altogether, a total of 3⁹ = 19683 different binary operations that can be defined, only 113 of these are associative, and many of these are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

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 ...

so that there are essentially only 18 possibilities.Andreas Distler
Classification and enumeration of finite semigroups
, PhD thesis, University of St. Andrews One of these is C₃, the cyclic group with three elements. The others all have a

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 nonisomorphic semigroups having two elements: * O2, the null semigroup of order two, * LO2, the left zero ...

subsemigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

s. In the example above, the set under multiplication contains both and as subsemigroups (the latter is a sub''group'', C₂). Six of these are bands, meaning that all three elements are idempotent, so that the product of any element with itself is itself again. Two of these bands are commutative, therefore semilattices (one of them is the three-element totally ordered set, and the other is a three-element semilattice that is not a lattice). The other four come in anti-isomorphic pairs. One of these non-commutative bands results from adjoining an identity element to LO₂, the left zero semigroup with two elements (or, dually, to RO₂, the right zero semigroup). It is sometimes called the flip-flop monoid, referring to flip-flop circuits used in electronics: the three elements can be described as "set", "reset", and "do nothing". This semigroup occurs in the Krohn–Rhodes decomposition of finite semigroups."This innocuous three-element semigroup plays an important role in what follows..." �
Applications of Automata Theory and Algebra
by John L. Rhodes. The irreducible elements in this decomposition are the finite simple groups plus this three-element semigroup, and its subsemigroups. There are two cyclic semigroups, one described by the equation ''x''⁴ = ''x''³, which has O₂, the null semigroup with two elements, as a subsemigroup. The other is described by ''x''⁴ = ''x''² and has C₂, the group with two elements, as a subgroup. (The equation ''x''⁴ = ''x'' describes C₃, the group with three elements, already mentioned.) There are seven other non-cyclic non-band commutative semigroups, including the initial example of , and O₃, the null semigroup with three elements. There are also two other anti-isomorphic pairs of non-commutative non-band semigroups.

References

{{Reflist Algebraic structures Semigroup theory

See also

References