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.^[1] Dominion is a concept in semigroup theory, within the study of the properties of epimorphisms. For example, let $U$ is a subsemigroup of $S$ containing $U$ , the inclusion map $U\hookrightarrow S$ is an epimorphism if and only if ${\rm {{Dom}_{S}(U)=S}}$ , furthermore, a map $\alpha \colon S\to T$ is an epimorphism if and only if ${\rm {{Dom}_{T}({\rm {{im}\;\alpha )=T}}}}$ .^[2] 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.^[3] Proofs of this theorem are topological in nature, beginning with Isbell (1966) for semigroups, and continuing by Philip (1974), completing Isbell's original proof.^[3]^[4]^[5] The pure algebraic proofs were given by Howie (1976) and Storrer (1976).^[3]^[4]^{[note 1]}

Statement

Zig-zag

Zig-zag:^[7]^[2]^[8]^[9]^[10]^{[note 2]} If $U$ is a submonoid of a monoid (or a subsemigroup of a semigroup) $S$ , then a system of equalities;

${\begin{aligned}d&=x_{1}u_{1},&u_{1}&=v_{1}y_{1}\\x_{i-1}v_{i-1}&=x_{i}u_{i},&u_{i}y_{i-1}&=v_{i}y_{i}\;(i=2,\dots ,m)\\x_{m}v_{m}&=u_{m+1},&u_{m+1}y_{m}&=d\end{aligned}}$

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 $m$ in $S$ over $U$ with value $d$ . By the spine of the zig-zag we mean the ordered $(2m + 1)$ -tuple $(u_{1},v_{1},u_{2},v_{2},\dots ,u_{m},v_{m},u_{m+1})$ .

Dominion

Dominion:^[5]^[6] Let $U$ be a submonoid of a monoid (or a subsemigroup of a semigroup) $S$ . The dominion ${\rm {{Dom}_{S}(U)}}$ is the set of all elements $s\in S$ such that, for all homomorphisms $f,g:S\to T$ coinciding on $U$ , $f(s)=g(s)$ .

We call a subsemigroup $U$ of a semigroup $U$ closed if ${\rm {{Dom}_{S}(U)=U}}$ , and dense if ${\rm {{Dom}_{S}(U)=S}}$ .^[2]^[12]

Isbell's zigzag theorem

Isbell's zigzag theorem:^[13]

If $U$ is a submonoid of a monoid $S$ then $d\in {\rm {{Dom}_{S}(U)}}$ if and only if either $d\in U$ or there exists a zig-zag in $S$ over $U$ with value $d$ that is, there is a sequence of factorizations of $d$ 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.^[7]^[14]^[9]^[4]^[10]

For monoids, this theorem can be written more concisely:^[15]^[2]^[16]

Let $S$ be a monoid, let $U$ be a submonoid of $S$ , and let $d\in S$ . Then $d\in \mathrm {Dom} _{S}(U)$ if and only if $d\otimes 1=1\otimes d$ in the tensor product $S\otimes _{U}S$ .

Application

Let $U$ be a commutative subsemigroup of a semigroup $S$ . Then ${\rm {{Dom}_{S}(U)}}$ is commutative.^[10]
Every epimorphism $\alpha \colon S\to T$ from a finite commutative semigroup $S$ to another semigroup $T$ is surjective.^[10]
Inverse semigroups are absolutely closed.^[7]
Example of non-surjective epimorphism in the category of rings:^[3] 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 ${\displaystyle \beta ,\gamma$ which agree on $\mathbb {Z}$ are fact equal.

More information We show that: Let

...

A proof sketch for example of non-surjective epimorphism in the category of rings by using zig-zag

We show that: Let $\beta ,\gamma$ to be ring homomorphisms, and $n,m\in \mathbb {Z}$ , $n\neq 0$ . 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 ${\frac {m}{n}}\in \mathbb {Q}$ .

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

Isbell's zigzag theorem

Statement

Zig-zag

Dominion

Isbell's zigzag theorem

Application

See also

References

Citations

Bibliography

Further reading

Footnote

External links

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