Volume 5, 1993

University of Bialystok

Copyright (c) 1993 Association of Mizar Users

### The abstract of the Mizar article:

### Representation Theorem for Boolean Algebras

**by****Jaroslaw Stanislaw Walijewski**- Received July 14, 1993
- MML identifier: LOPCLSET

- [ Mizar article, MML identifier index ]

environ vocabulary PRE_TOPC, SETFAM_1, BOOLE, BINOP_1, FUNCT_1, LATTICES, SUBSET_1, REALSET1, FILTER_0, RELAT_1, TARSKI, FINSUB_1, LATTICE2, SETWISEO, FINSET_1, RFINSEQ, CARD_1, GROUP_6, MOD_4, WELLORD1, LOPCLSET; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, BINOP_1, FINSUB_1, STRUCT_0, PRE_TOPC, LATTICES, LATTICE2, FILTER_0, REALSET1, FINSET_1, SETWISEO, LATTICE4, RFINSEQ, CARD_1, GRCAT_1; constructors BINOP_1, LATTICE2, REALSET1, SETWISEO, LATTICE4, RFINSEQ, FILTER_1, GRCAT_1, MEMBERED, PRE_TOPC; clusters FINSET_1, FINSUB_1, LATTICES, PRE_TOPC, RLSUB_2, STRUCT_0, RELSET_1, SUBSET_1, SETFAM_1, MEMBERED, ZFMISC_1, FUNCT_2, PARTFUN1; requirements BOOLE, SUBSET; begin reserve T for non empty TopSpace, X,Z for Subset of T; definition let T be non empty TopStruct; func OpenClosedSet(T) -> Subset-Family of T equals :: LOPCLSET:def 1 {x where x is Subset of T: x is open closed}; end; definition let T be non empty TopSpace; cluster OpenClosedSet(T) -> non empty; end; canceled; theorem :: LOPCLSET:2 X in OpenClosedSet(T) implies X is open; theorem :: LOPCLSET:3 X in OpenClosedSet(T) implies X is closed; theorem :: LOPCLSET:4 X is open closed implies X in OpenClosedSet(T); reserve x,y for Element of OpenClosedSet(T); definition let T;let C,D be Element of OpenClosedSet(T); redefine func C \/ D -> Element of OpenClosedSet(T); redefine func C /\ D -> Element of OpenClosedSet(T); end; definition let T; func T_join T -> BinOp of OpenClosedSet(T) means :: LOPCLSET:def 2 for A,B being Element of OpenClosedSet(T) holds it.(A,B) = A \/ B; func T_meet T -> BinOp of OpenClosedSet(T) means :: LOPCLSET:def 3 for A,B being Element of OpenClosedSet(T) holds it.(A,B) = A /\ B; end; theorem :: LOPCLSET:5 for x,y be Element of LattStr(#OpenClosedSet(T),T_join T,T_meet T#), x',y' be Element of OpenClosedSet(T) st x=x' & y=y' holds x"\/"y = x' \/ y'; theorem :: LOPCLSET:6 for x,y be Element of LattStr(#OpenClosedSet(T),T_join T,T_meet T#), x',y' be Element of OpenClosedSet(T) st x=x' & y=y' holds x"/\"y = x' /\ y'; theorem :: LOPCLSET:7 {} T is Element of OpenClosedSet(T); theorem :: LOPCLSET:8 [#] T is Element of OpenClosedSet(T); theorem :: LOPCLSET:9 x` is Element of OpenClosedSet(T); theorem :: LOPCLSET:10 LattStr(#OpenClosedSet(T),T_join T,T_meet T#) is Lattice; definition let T be non empty TopSpace; func OpenClosedSetLatt(T) -> Lattice equals :: LOPCLSET:def 4 LattStr(#OpenClosedSet(T),T_join T,T_meet T#); end; theorem :: LOPCLSET:11 for T being non empty TopSpace, x,y being Element of OpenClosedSetLatt(T) holds x"\/"y = x \/ y; theorem :: LOPCLSET:12 for T being non empty TopSpace , x,y being Element of OpenClosedSetLatt(T) holds x"/\"y = x /\ y; theorem :: LOPCLSET:13 the carrier of OpenClosedSetLatt(T) = OpenClosedSet(T); theorem :: LOPCLSET:14 OpenClosedSetLatt(T) is Boolean; theorem :: LOPCLSET:15 [#] T is Element of OpenClosedSetLatt(T); theorem :: LOPCLSET:16 {} T is Element of OpenClosedSetLatt(T); reserve x,y,X,Y for set; definition cluster non trivial B_Lattice; end; reserve BL for non trivial B_Lattice, a,b,c,p,q for Element of BL, UF,F,F0,F1,F2 for Filter of BL; definition let BL; func ultraset BL -> Subset of bool the carrier of BL equals :: LOPCLSET:def 5 {F : F is_ultrafilter}; end; definition let BL; cluster ultraset BL -> non empty; end; canceled; theorem :: LOPCLSET:18 x in ultraset BL iff (ex UF st UF = x & UF is_ultrafilter); theorem :: LOPCLSET:19 for a holds { F :F is_ultrafilter & a in F} c= ultraset BL; definition let BL; func UFilter BL -> Function means :: LOPCLSET:def 6 dom it = the carrier of BL & for a being Element of BL holds it.a = {UF: UF is_ultrafilter & a in UF }; end; theorem :: LOPCLSET:20 x in UFilter BL.a iff (ex F st F=x & F is_ultrafilter & a in F); theorem :: LOPCLSET:21 F in UFilter BL.a iff F is_ultrafilter & a in F; theorem :: LOPCLSET:22 for F st F is_ultrafilter holds a "\/" b in F iff a in F or b in F; theorem :: LOPCLSET:23 UFilter BL.(a "/\" b) = UFilter BL.a /\ UFilter BL.b; theorem :: LOPCLSET:24 UFilter BL.(a "\/" b) = UFilter BL.a \/ UFilter BL.b; definition let BL; redefine func UFilter BL -> Function of the carrier of BL, bool ultraset BL; end; definition let BL; func StoneR BL -> set equals :: LOPCLSET:def 7 rng UFilter BL; end; definition let BL; cluster StoneR BL -> non empty; end; theorem :: LOPCLSET:25 StoneR BL c= bool ultraset BL; theorem :: LOPCLSET:26 x in StoneR BL iff ( ex a st (UFilter BL).a =x); definition let BL; func StoneSpace BL -> strict TopSpace means :: LOPCLSET:def 8 the carrier of it =ultraset BL & the topology of it = {union A where A is Subset-Family of ultraset BL : A c= StoneR BL }; end; definition let BL; cluster StoneSpace BL -> non empty; end; theorem :: LOPCLSET:27 (F is_ultrafilter & not F in UFilter BL.a) implies not a in F; theorem :: LOPCLSET:28 ultraset BL \ UFilter BL.a = UFilter BL.a`; definition let BL; func StoneBLattice BL -> Lattice equals :: LOPCLSET:def 9 OpenClosedSetLatt(StoneSpace BL ); end; theorem :: LOPCLSET:29 UFilter BL is one-to-one; theorem :: LOPCLSET:30 union StoneR BL = ultraset BL; theorem :: LOPCLSET:31 for A,B,X being set holds (X c= union (A \/ B) & for Y being set st Y in B holds Y misses X ) implies X c= union A; theorem :: LOPCLSET:32 for X being non empty set ex Y being Finite_Subset of X st Y is non empty; definition let D be non empty set; cluster non empty Finite_Subset of D; end; canceled; theorem :: LOPCLSET:34 for L being non trivial B_Lattice, D being non empty Subset of L st Bottom L in <.D.) ex B being non empty Finite_Subset of the carrier of L st B c= D & FinMeet(B) = Bottom L; theorem :: LOPCLSET:35 for L being 0_Lattice holds not ex F being Filter of L st F is_ultrafilter & Bottom L in F; theorem :: LOPCLSET:36 UFilter BL.Bottom BL = {}; theorem :: LOPCLSET:37 UFilter BL.Top BL = ultraset BL; theorem :: LOPCLSET:38 ultraset BL = union X & X is Subset of StoneR BL implies ex Y being Finite_Subset of X st ultraset BL = union Y; canceled; theorem :: LOPCLSET:40 StoneR BL = OpenClosedSet(StoneSpace BL); definition let BL; redefine func UFilter BL -> Homomorphism of BL,StoneBLattice BL; end; theorem :: LOPCLSET:41 rng UFilter BL = the carrier of StoneBLattice BL; theorem :: LOPCLSET:42 UFilter BL is isomorphism; theorem :: LOPCLSET:43 BL,StoneBLattice BL are_isomorphic; theorem :: LOPCLSET:44 for BL being non trivial B_Lattice ex T being non empty TopSpace st BL, OpenClosedSetLatt(T) are_isomorphic;

Back to top