Subdirect product

In universal algebra the problem arises that not all (universal) algebras can be represented as a direct product of directly irreducible algebras. The so-called subdirect product , a certain type of sub-algebra of a direct product, offers a solution . Garrett Birkhoff's first theorem then states that every algebra can be written as a subdirect product of subdirectly irreducible algebras.

definition

Let algebras be of the same type, that is, of the same algebraic structure , and an index family. A subalgebra is called a subdirect product which , if holds for all , denotes the canonical projection . ${\displaystyle A_{i}(i\in I)}$ ${\displaystyle I}$ ${\displaystyle B\subseteq \prod _{i\in I}A_{i}}$ ${\displaystyle A_{i}}$ ${\displaystyle \alpha _{j}(B)=A_{j}}$ ${\displaystyle j\in I}$ ${\displaystyle \alpha _{j}:\prod _{i\in I}A_{i}\rightarrow A_{j}}$

Subdirect irreducibility

An embedding is called subdirect representation of , if subdirect product is the. ${\displaystyle \varphi :A\rightarrow \prod _{i\in I}A_{i}}$ ${\displaystyle A}$ ${\displaystyle \varphi (A)}$ ${\displaystyle A_{i}}$

${\displaystyle A}$is called subdirectly irreducible if for every subdirect representation there exists such that it is an isomorphism. ${\displaystyle j\in I}$ ${\displaystyle \alpha _{j}\circ \varphi :A\rightarrow A_{j}}$

motivation

The following example shows that an algebra cannot normally be represented as a direct product of directly irreducible algebras: A Boolean algebra is directly or subdirectly irreducible if and only if applies. A countably infinite Boolean algebra is given by with carrier set . This cannot possibly be a direct product of two-element algebras, since such a product is either finite or uncountable. ${\displaystyle B}$ ${\displaystyle \mathrm {card} \,B\leq 2}$ ${\displaystyle B=(X,\cup ,\cap ,',\emptyset ,\mathbb {N} )}$ ${\displaystyle X:=\left\{M\subseteq \mathbb {N} \,:\,M{\mbox{ endlich}}{\mbox{ oder }}\mathbb {N} \setminus M{\mbox{ endlich}}\right\}}$

Representation set by Birkhoff

Every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras of the same type. The representation as a subdirect product is not clear.

example

For example, the Boolean algebra mentioned above has the following subdirect representation:

${\displaystyle \varphi :X\rightarrow \left\{0,1\right\}^{\mathbb {N} }}$ With ${\displaystyle (\varphi (x))_{j}={\begin{cases}1,&{\mbox{falls }}j\in x\\0,&{\mbox{falls }}j\not \in x\end{cases}}}$

literature

• Thomas Ihringer: General Algebra . Berlin study series on mathematics. Volume 10. Heldermann Verlag , 2003 Lemgo