Английская Википедия: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 Шаблон:Mvar is a subsemigroup of Шаблон:Mvar containing Шаблон:Mvar, the inclusion map <math>U \hookrightarrow S</math> is an epimorphism if and only if <math>\rm{Dom}_S (U) = S</math>, furthermore, a map <math>\alpha \colon S \to T</math> is an epimorphism if and only if <math>\rm{Dom}_T (\rm{im} \; \alpha) = T</math>.Шаблон: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.Шаблон:R Proofs of this theorem are topological in nature, beginning with Шаблон:Harvtxt for semigroups, and continuing by Шаблон:Harvtxt, completing Isbell's original proof.Шаблон:RШаблон:RШаблон:R The pure algebraic proofs were given by Шаблон:Harvtxt and Шаблон:Harvtxt.Шаблон:RШаблон:RШаблон:Refn
Statement
Zig-zag
Zig-zag:Шаблон:R[2][3][4]Шаблон:RШаблон:Refn If Шаблон:Mvar is a submonoid of a monoid (or a subsemigroup of a semigroup) Шаблон:Mvar, then a system of equalities;
<math>\begin{align}
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{align}</math>
in which <math>u_1, \dots , u_{m + 1}, v_1, \dots , v_{m} \in U</math> and <math>x_1, \dots , x_{m}, y_1, \dots , y_{m} \in S</math>, is called a zig-zag of length Шаблон:Mvar in Шаблон:Mvar over Шаблон:Mvar with value Шаблон:Mvar. By the spine of the zig-zag we mean the ordered Шаблон:Mvar-tuple <math>(u_1,v_1,u_2,v_2,\dots,u_{m},v_{m}, u_{m+1})</math>.
Dominion
Dominion:Шаблон:RШаблон:R Let Шаблон:Mvar be a submonoid of a monoid (or a subsemigroup of a semigroup) Шаблон:Mvar. The dominion <math>\rm{Dom}_S (U)</math> is the set of all elements <math>s \in S</math> such that, for all homomorphisms <math>f, g : S \to T</math> coinciding on Шаблон:Mvar, <math>f(s) = g(s)</math>.
We call a subsemigroup Шаблон:Mvar of a semigroup Шаблон:Mvar closed if <math>\rm{Dom}_S (U) = U</math>, and dense if <math>\rm{Dom}_S (U) = S</math>.Шаблон:R[5]
Isbell's zigzag theorem
Isbell's zigzag theorem:[6]
If Шаблон:Mvar is a submonoid of a monoid Шаблон:Mvar then <math>d \in \rm{Dom}_S (U)</math> if and only if either <math>d \in U</math> or there exists a zig-zag in Шаблон:Mvar over Шаблон:Mvar with value Шаблон:Mvar that is, there is a sequence of factorizations of Шаблон:Mvar of the form
<math>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}</math>
This statement also holds for semigroups.Шаблон:R[7]Шаблон:R[8]Шаблон:R
For monoids, this theorem can be written more concisely:[9]Шаблон:R[10]
Let Шаблон:Mvar be a monoid, let Шаблон:Mvar be a submonoid of Шаблон:Mvar, and let <math>d \in S</math>. Then <math>d \in \mathrm{Dom}_{S} (U)</math> if and only if <math>d \otimes 1 = 1 \otimes d</math> in the tensor product <math>S \otimes_{U} S</math>.
Application
- Let Шаблон:Mvar be a commutative subsemigroup of a semigroup Шаблон:Mvar. Then <math>\rm{Dom}_S (U)</math> is commutative.[11]
- Every epimorphism <math>\alpha \colon S \to T</math> from a finite commutative semigroup Шаблон:Mvar to another semigroup Шаблон:Mvar is surjective.Шаблон:R
- Inverse semigroups are absolutely closed.[12]
- Example of non-surjective epimorphism in the category of rings:[13] The inclusion <math>i: (\mathbb{Z},\cdot)\hookrightarrow (\mathbb{Q},\cdot)</math> is an epimorphism in the category of all rings and ring homomorphisms by proving that any pair of ring homomorphisms <math>\beta, \gamma: \mathbb{Q} \to \mathbb{R}</math> which agree on <math>\mathbb{Z}</math> are fact equal.
We show that: Let <math>\beta, \gamma</math> to be ring homomorphisms, and <math>n,m \in \mathbb{Z}</math>, <math>n \neq 0</math>. When <math>\beta(m) = \gamma(m)</math> for all <math>m \in \mathbb{Z}</math>, then <math>\beta\left(\frac{m}{n}\right) = \gamma\left(\frac{m}{n}\right)</math> for all <math>\frac{m}{n} \in \mathbb{Q}</math>.
<math> \begin{align} \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{align} </math>
as required. Шаблон:Cob
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