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

{\displaystyle {\rm {{Dom}_{S}(U)=S}}}

α :

{\displaystyle {\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] 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;

{\displaystyle {\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}}}

, 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

Dominion:[5][6] Let U be a submonoid of a monoid (or a subsemigroup of a semigroup) S. The dominion

is the set of all elements

such that, for all homomorphisms

We call a subsemigroup U of a semigroup U closed if

[2][12] Isbell's zigzag theorem:[13] If U is a submonoid of a monoid S then

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

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

in the tensor product

We show that: Let

β , γ

to be ring homomorphisms, and

β ( m ) = γ ( m )

⋅ γ ( m ) = β

⋅ β ( m n ) ⋅ γ

= β ( m ) ⋅ γ

{\displaystyle {\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}}}

as required.