Isbell's zigzag theorem

Isbell's zigzag theorem, a theorem of abstract algebra characterizing the notion of a dominion, was introduced by American mathematician John R. Isbell in 1966. Dominion is a concept in semigroup theory, within the study of the properties of epimorphisms. For example, let Template:Mvar is a subsemigroup of Template:Mvar containing Template:Mvar, the inclusion map $U\hookrightarrow S$ is an epimorphism if and only if ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}\mathrm{=}\mathrm{S}$, furthermore, a map $\alpha :S\to T$ is an epimorphism if and only if ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{T}}\mathrm{(}\mathrm{i}\mathrm{m}\mathrm{\alpha }\mathrm{)}\mathrm{=}\mathrm{T}$.Template:R 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.Template:R Proofs of this theorem are topological in nature, beginning with Template:Harvtxt for semigroups, and continuing by Template:Harvtxt, completing Isbell's original proof.Template:RTemplate:RTemplate:R The pure algebraic proofs were given by Template:Harvtxt and Template:Harvtxt.Template:RTemplate:RTemplate:Refn

Statement

Zig-zag

The dashed line is the spine of the zig-zag.

Zig-zag:Template:RTemplate:RTemplate:Refn If Template:Mvar is a submonoid of a monoid (or a subsemigroup of a semigroup) Template:Mvar, then a system of equalities;

$\begin{array}{cccc}d& =x_1{u}_1,& u_1& =v_1{y}_1\\ x_{i-1}{v}_{i-1}& =x_i{u}_i,& u_i{y}_{i-1}& =v_i{y}_i(i=2,\dots ,m)\\ x_m{v}_m& =u_{m+1},& u_{m+1}{y}_m& =d\end{array}$

in which $u_1,\dots ,u_{m+1},v_1,\dots ,v_m\in U$ and $x_1,\dots ,x_m,y_1,\dots ,y_m\in S$, is called a zig-zag of length Template:Mvar in Template:Mvar over Template:Mvar with value Template:Mvar. By the spine of the zig-zag we mean the ordered Template:Mvar-tuple $(u_1,v_1,u_2,v_2,\dots ,u_m,v_m,u_{m+1})$.

Dominion

Dominion:Template:RTemplate:R Let Template:Mvar be a submonoid of a monoid (or a subsemigroup of a semigroup) Template:Mvar. The dominion ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}$ is the set of all elements $s\in S$ such that, for all homomorphisms $f,g:S\to T$ coinciding on Template:Mvar, $f(s)=g(s)$.

We call a subsemigroup Template:Mvar of a semigroup Template:Mvar closed if ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}\mathrm{=}\mathrm{U}$, and dense if ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}\mathrm{=}\mathrm{S}$.Template:R

Isbell's zigzag theorem

Isbell's zigzag theorem:

If Template:Mvar is a submonoid of a monoid Template:Mvar then $d\in {\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}$ if and only if either $d\in U$ or there exists a zig-zag in Template:Mvar over Template:Mvar with value Template:Mvar that is, there is a sequence of factorizations of Template:Mvar of the form

$d=x_1{u}_1=x_1{v}_1{y}_1=x_2{u}_2{y}_1=x_2{v}_2{y}_2=\cdots =x_m{v}_m{y}_m=u_{m+1}{y}_m$

This statement also holds for semigroups.Template:RTemplate:RTemplate:R

For monoids, this theorem can be written more concisely:Template:R

Let Template:Mvar be a monoid, let Template:Mvar be a submonoid of Template:Mvar, and let $d\in S$. Then $d\in {\mathrm{D}\mathrm{o}\mathrm{m}}_S(U)$ if and only if $d\otimes 1=1\otimes d$ in the tensor product $S{\otimes }_{U}S$.

Application

• Let Template:Mvar be a commutative subsemigroup of a semigroup Template:Mvar. Then ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}$ is commutative.
• Every epimorphism $\alpha :S\to T$ from a finite commutative semigroup Template:Mvar to another semigroup Template:Mvar is surjective.Template:R
• Inverse semigroups are absolutely closed.
• Example of non-surjective epimorphism in the category of rings: The inclusion $i:(\mathbb{Z},\cdot )\hookrightarrow (\mathbb{Q},\cdot )$ is an epimorphism in the category of all rings and ring homomorphisms by proving that any pair of ring homomorphisms $\beta ,\gamma :\mathbb{Q}\to ℝ$ which agree on $\mathbb{Z}$ are fact equal.

We show that: Let $\beta ,\gamma$ to be ring homomorphisms, and $n,m\in \mathbb{Z}$, $n\ne 0$. When $\beta (m)=\gamma (m)$ for all $m\in \mathbb{Z}$, then $\beta \left(\frac{m}{n}\right)=\gamma \left(\frac{m}{n}\right)$ for all $\frac{m}{n}\in \mathbb{Q}$.

$\begin{array}{cc}\beta \left(\frac{m}{n}\right)& =\beta (\frac{1}{n}\cdot m)=\beta \left(\frac{1}{n}\right)\cdot \beta (m)\\ & =\beta \left(\frac{1}{n}\right)\cdot \gamma (m)=\beta \left(\frac{1}{n}\right)\cdot \gamma (mn\cdot \frac{1}{n})\\ & =\beta \left(\frac{1}{n}\right)\cdot \gamma (mn)\cdot \gamma \left(\frac{1}{n}\right)=\beta \left(\frac{1}{n}\right)\cdot \beta (mn)\cdot \gamma \left(\frac{1}{n}\right)\\ & =\beta (\frac{1}{n}\cdot mn)\cdot \gamma \left(\frac{1}{n}\right)=\beta (m)\cdot \gamma \left(\frac{1}{n}\right)=\gamma (m)\cdot \gamma \left(\frac{1}{n}\right)\\ & =\gamma (m\cdot \frac{1}{n})=\gamma \left(\frac{m}{n}\right),\end{array}$

as required. Template:Cob

See also

• Epimorphisms

References

Bibliography

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Further reading

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External links

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