mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a nowhere commutative semigroup is a semigroup ''S'' such that, for all ''a'' and ''b'' in ''S'', if ''ab'' = ''ba'' then ''a'' = ''b''.

A. H. Clifford Alfred Hoblitzelle Clifford (July 11, 1908 – December 27, 1992) was an American mathematician born in St. Louis, Missouri who is known for Clifford theory and for his work on semigroups. He did his undergraduate studies at Yale University, Yal ...

G. B. Preston Gordon Bamford Preston (28 April 1925 – 14 April 2015) was an English mathematician best known for his work on semigroups. He received his D.Phil. in mathematics in 1954 from Magdalen College, Oxford. He was born in Workington and broug ...

(1964). ''The Algebraic Theory of Semigroups Vol. I'' (Second Edition). American Mathematical Society (p.26). A semigroup ''S'' is nowhere commutative if and only if any two elements of ''S'' are inverses of each other.

Characterization of nowhere commutative semigroups

Nowhere commutative semigroups can be characterized in several different ways. If ''S'' is a semigroup then the following statements are equivalent: *''S'' is nowhere commutative. *''S'' is a

rectangular band In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square). Bands were first studied and named by ; the lattice of varieties of bands was described indepe ...

(in the sense in which the term is used by John Howie ). *For all ''a'' and ''b'' in ''S'', ''aba'' = ''a''. *For all ''a'', ''b'' and ''c'' in ''S'', ''a''² = ''a'' and ''abc'' = ''ac''. Even though, by definition, the rectangular bands are concrete semigroups, they have the defect that their definition is formulated not in terms of the basic

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, an internal binary op ...

in the semigroup. The approach via the definition of nowhere commutative semigroups rectifies this defect. To see that a nowhere commutative semigroup is a rectangular band, let ''S'' be a nowhere commutative semigroup. Using the defining properties of a nowhere commutative semigroup, one can see that for every ''a'' in ''S'' the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

of the Green classes ''R''_''a'' and ''L''_''a'' contains the unique element ''a''. Let ''S''/''L'' be the family of ''L''-classes in ''S'' and ''S''/''R'' be the family of ''R''-classes in ''S''. The mapping :ψ : ''S '' → (''S''/''R'') × (''S''/''L'') defined by :''a''ψ = (''R''_''a'', ''L''_''a'') is a

bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

. If the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

(''S''/''R'') × (''S''/''L'') is made into a semigroup by furnishing it with the rectangular band multiplication, the map ψ becomes an isomorphism. So ''S'' is isomorphic to a rectangular band. Other claims of equivalences follow directly from the relevant definitions.

References

{{reflist Algebraic structures Semigroup theory

Characterization of nowhere commutative semigroups

See also

References