abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, a

semigroup 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 (just notation, not necessarily th ...

with three elements is an object consisting of three elements and an

associative operation 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 ...

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 or morphism 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 the ...

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 In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

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 ...

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 (just notation, not necessarily the ...

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 Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

, so that the product of any element with itself is itself again. Two of these bands 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 ...

, 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 In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

to LO₂, the left zero semigroup with two elements (or, dually, to RO₂, the

right zero semigroup In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero. If every element of a semigroup is a left zero then the semigroup is called a l ...

). 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 semigroup In mathematics, a monogenic semigroup is a semigroup generated by a single element. Monogenic semigroups are also called cyclic semigroups. Structure The monogenic semigroup generated by the singleton set is denoted by \langle a \rangle. The set ...

s, one described by the equation ''x''⁴ = ''x''³, which has O₂, the

null semigroup In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero. If every element of a semigroup is a left zero then the semigroup is called a ...

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