Isbell's zigzag theorem explained
is an epimorphism if and only if
, furthermore, a map
is an
epimorphism if and only if
\rm{Dom}T(\rm{im} \alpha)=T
. The categories of rings and semigroups are examples of categories with non-surjective epimorphism, and the Zig-zag theorem gives necessary and sufficient conditions for determining whether or not a given morphism is epi. Proofs of this theorem are topological in nature, beginning with for semigroups, and continuing by, completing Isbell's original proof. The pure algebraic proofs were given by and .
Statement
Zig-zag
Zig-zag: If is a submonoid of a monoid (or a subsemigroup of a semigroup), then a system of equalities;
\begin{align}
d&=x1u1,&u1&=v1y1\\
xivi&=xiui,&uiyi&=viyi (i=2,...,m)\\
xmvm&=um+1,&umym&=d
\end{align}
in which
and
, is called a zig-zag of length in over with value . By the spine of the zig-zag we mean the ordered -tuple
(u1,v1,u2,v2,...,um,vm,um+1)
.
Dominion
Dominion: Let be a submonoid of a monoid (or a subsemigroup of a semigroup) . The dominion
is the set of all elements
such that, for all homomorphisms
coinciding on,
.
We call a subsemigroup of a semigroup closed if
, and dense if
.
Isbell's zigzag theorem
Isbell's zigzag theorem:
If is a submonoid of a monoid then
if and only if either
or there exists a zig-zag in over with value that is, there is a sequence of factorizations of of the form
d=x1u1=x1v1y1=x2u2y1=x2v2y2= … =xmvmym=um+1ym
This statement also holds for semigroups.
For monoids, this theorem can be written more concisely:
Let be a monoid, let be a submonoid of, and let
. Then
if and only if
in the
tensor product
.
Application
- Let be a commutative subsemigroup of a semigroup . Then
is commutative.
from a finite commutative semigroup to another semigroup is surjective.
- Inverse semigroups are absolutely closed.
- Example of non-surjective epimorphism in the category of rings: The inclusion
i:(Z, ⋅ )\hookrightarrow(Q, ⋅ )
is an epimorphism in the category of all rings and ring homomorphisms by proving that any pair of
ring homomorphisms
which agree on
are fact equal.
We show that: Let
to be ring homomorphisms, and
,
. When
for all
, then
\right)=\gamma\left(
\right)
for all
.
\begin{align}
\beta\left( | m |
n |
\right)&=\beta\left(
⋅ m\right)=\beta\left(
\right) ⋅ \beta(m)\\
&=\beta\left(
\right) ⋅ \gamma(m)
=\beta\left(
\right) ⋅ \gamma\left(mn ⋅
\right)\\
&=\beta\left(
\right) ⋅ \gamma(mn) ⋅ \gamma\left(
\right)
=\beta\left(
\right) ⋅ \beta(mn) ⋅ \gamma\left(
\right)\\
&=\beta\left(
⋅ mn\right) ⋅ \gamma\left(
\right)
=\beta(m) ⋅ \gamma\left(
\right)
=\gamma(m) ⋅ \gamma\left(
\right)\\
&=\gamma\left(m ⋅
\right)
=\gamma\left(
\right),
\end{align}
as required.
See also
References
Bibliography
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Further reading
- Epimorphisms, dominions and
-commutative semigroups . 2020 . Alam . Noor . Higgins . Peter M. . Khan . Noor Mohammad . Semigroup Forum . 100 . 2 . 349–363 . 202133305. 10.1007/s00233-019-10050-z . 1908.01813 .
- 10.1007/s00233-019-10047-8 . Epimorphisms, dominions and varieties of bands . 2020 . Ahanger . Shabir Ahmad . Shah . Aftab Hussain . Semigroup Forum . 100 . 3 . 641–650 . 253772526 .
- 10.1017/S1446788700023041 . On saturated permutative varieties and consequences of permutation identities . 1985 . Khan . N. M. . Journal of the Australian Mathematical Society, Series A . 38 . 2 . 186–197 . 122979127 . free .
- Epimorphisms and Dominions, III . 2373286 . Isbell . John R. . American Journal of Mathematics . 1968 . 90 . 4 . 1025–1030 . 10.2307/2373286 .
- 10.1007/BF02945133 . Epimorphisms and dominions, V . 1973 . Isbell . J. R. . Algebra Universalis . 3 . 1 . 318–320 . 125292076 .
- On epics and dominions of bands . Semigroup Forum . 13 . 103–114 . Scheiblich . E. . 1976 . 1 . 10.1007/BF02194926 . 123580458 .
- 10.1016/0021-8693(74)90141-0 . A proof of Isbell's zig-zag theorem . 1974 . Philip . J.M . Journal of Algebra . 32 . 2 . 328–331 . free .
- 10.1007/BF01188058 . Epimorphisms, dominions and semigroups . 1985 . Higgins . Peter M. . Algebra Universalis . 21 . 2–3 . 225–233 . 121142819 .
- Book: 9780198535775 . Techniques of Semigroup Theory . Higgins . Peter M. . 1992 . Oxford University Press .
- Book: Howie, John M.. Fundamentals of Semigroup Theory. Semigroup amalgams. 1995. Clarendon Press. 0-19-851194-9. 0835.20077. 1455373.
- 26600306 . Dominions and Primitive Positive Functions . Campercholi . Miguel . The Journal of Symbolic Logic . 2018 . 83 . 1 . 40–54 . 10.1017/jsl.2017.18 . 11336/88474 . 19168037 . free .
External links