Notation set for Boolean algebras

The representation theorem for Boolean algebras (also: representation theorem of Stone or Stone shear representation theorem ) is a set of the lattice theory , which in 1936 by the American mathematician Marshall Harvey Stone was discovered. It says that every Boolean algebra is isomorphic to a set algebra, namely to the Boolean algebra of closed and at the same time open sets in a so-called Stone space .

statement

Be a Boolean algebra. Then there is a set and an injective mapping such that the following applies to all : ${\displaystyle \langle B,\wedge ,\lor ,\neg ,0,1\rangle }$ ${\displaystyle M}$ ${\displaystyle h\colon \,B\to {\mathcal {P}}(M)}$ ${\displaystyle b,c\in B}$

• ${\displaystyle h(0)=\emptyset }$, ${\displaystyle h(1)=M}$
• ${\displaystyle h(b\wedge c)=h(b)\cap h(c)}$
• ${\displaystyle h(b\lor c)=h(b)\cup h(c)}$
• ${\displaystyle h(\neg b)=M\setminus h(b)}$

The Boolean algebra is isomorphic to the set algebra on . ${\displaystyle h(B)}$

proof

Be the set of all ultrafilters on . For define . Then: ${\displaystyle M}$ ${\displaystyle B}$ ${\displaystyle b\in B}$ ${\displaystyle h(b)=\lbrace U\in M\mid b\in U\rbrace }$

• Injectivity: Be , so or . Applies without restriction . Therefore it can be expanded to an ultrafilter. But this does not contain , so ${\displaystyle b\neq c}$ ${\displaystyle b\not \leq c}$ ${\displaystyle c\not \leq b}$ ${\displaystyle b\not \leq c}$ ${\displaystyle \lbrace b\wedge (\neg c)\rbrace \neq \lbrace \emptyset \rbrace }$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle h(b)\neq h(c)}$
• ${\displaystyle h(0)=\emptyset }$and , because no ultrafilter contains the and every ultrafilter contains the ${\displaystyle h(1)=M}$ ${\displaystyle 0}$ ${\displaystyle 1}$
• ${\displaystyle h(b\wedge c)=h(b)\cap h(c)}$, because the following applies to every filter : ${\displaystyle U}$ ${\displaystyle b\wedge c\in U\Leftrightarrow \lbrace b,c\rbrace \subseteq U}$
• ${\displaystyle h(b\lor c)=h(b)\cup h(c)}$
  • " ": Let ultrafilter with , suppose , well , and therefore , this contradicts the fact that ultrafilter is. ${\displaystyle \subseteq }$ ${\displaystyle U}$ ${\displaystyle b\lor c=\neg (\neg b\wedge \neg c)\in U}$ ${\displaystyle b,c\notin U}$ ${\displaystyle \lbrace \neg b,\neg c\rbrace \subseteq U}$ ${\displaystyle \neg b\wedge \neg c\in U}$ ${\displaystyle U}$
  • " ": Be ultrafilter with , then is so and ${\displaystyle \supseteq }$ ${\displaystyle U}$ ${\displaystyle b\lor c\notin U}$ ${\displaystyle \neg b\wedge \neg c=\neg (b\lor c)\in U}$ ${\displaystyle b\notin U}$ ${\displaystyle c\notin U}$
• ${\displaystyle h(\neg b)=M\setminus h(b)}$, because ${\displaystyle \neg b\in U\Leftrightarrow b\notin U}$

Duality theory

Stone's representation theorem actually makes an even more precise statement and can be expanded into a duality theory, as explained in the textbook by Paul Halmos given below .

If a Boolean algebra stands for the two-element Boolean algebra, then let the space of homomorphisms be . This space is a closed set in , whereby the latter is provided with the product topology . Therefore a so-called Stone room or Boolean room is a totally incoherent , compact Hausdorff room ; it is called the too dual space. For this reason, totally incoherent, compact Hausdorff rooms are also called Boolean rooms . ${\displaystyle B}$ ${\displaystyle 2:=\{0,1\}}$ ${\displaystyle X}$ ${\displaystyle B\rightarrow 2}$ ${\displaystyle 2^{B}=\{0,1\}^{B}}$ ${\displaystyle X}$ ${\displaystyle B}$

Conversely, if a Stone space is, then let the Boolean algebra of open-ended sets in ; this is called too- dual Boolean algebra. ${\displaystyle X}$ ${\displaystyle B}$ ${\displaystyle X}$ ${\displaystyle X}$

Stone's representation theorem now states that every Boolean algebra is isomorphic to its bidual, that is, to the dual algebra of its dual space. Therefore one can say more precisely that every Boolean algebra is isomorphic to a set algebra, where the sets are exactly the open-closed sets of a Stone space.

The duality also applies to the Stone spaces: every Boolean space is homeomorphic to its bidual, that is, to the dual space of its dual Boolean algebra.

In addition, the homomorphisms from Boolean algebra to Boolean algebra naturally correspond to the continuous mappings from the dual space of to the dual space of , that is, the mapping to the dual space can naturally be converted into a contravariant equivalence between the Continue the category of Boolean algebras and the category of Stone spaces. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle X}$

literature

• Paul R. Halmos : Lectures on Boolean Algebra. Springer, Berlin et al. 1974, ISBN 0-387-90094-2 .
• Sabine Koppelberg: Handbook of Boolean Algebras. Volume 1. North-Holland Publishing Co., Amsterdam et al. 1989, ISBN 0-444-70261-X .
• Marshall Harvey Stone : The theory of representations for Boolean algebras. In: Transactions of the American Mathematical Society. Vol. 40, 1936, ISSN  0002-9947 , pp. 37-111, doi : 10.1090 / S0002-9947-1936-1501865-8 .