Nowhere commutative semigroup
Encyclopedia
In 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 semigroup S is nowhere commutative if and only if any two elements of S are inverse
s of each other.
in several different ways. If S is a semigroup then the following statements are equivalent
:
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 the semigroup. The approach via the definition of nowhere commutative semigroups rectifies the 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 of the Green classes Ra and La 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
defined by
is a bijection. If the Cartesian product (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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a nowhere commutative semigroup is a semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
S such that, for all a and b in S, if ab = ba then a = b. A semigroup S is nowhere commutative if and only if any two elements of S are inverse
Inverse semigroup
In mathematics, an inverse semigroup S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy...
s of each other.
Characterization of nowhere commutative semigroups
Nowhere commutative semigroups can be characterizedCharacterization (mathematics)
In mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in...
in several different ways. If S is a semigroup then the following statements are equivalent
Logical equivalence
In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...
:
- S is nowhere commutative.
- S is a rectangular band (in the sense in which the term is used by Howie).
- For all a and b in S, aba = a.
- For all a, b and c in S, a2 = 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 the semigroup. The approach via the definition of nowhere commutative semigroups rectifies the 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 of the Green classes Ra and La 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
- aψ = ( Ra, La )
is a bijection. If the Cartesian product (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.