Volume 5, 1993

University of Bialystok

Copyright (c) 1993 Association of Mizar Users

### The abstract of the Mizar article:

### Representation Theorem for Heyting Lattices

**by****Jolanta Kamienska**- Received July 14, 1993
- MML identifier: OPENLATT

- [ Mizar article, MML identifier index ]

environ vocabulary LATTICE2, FILTER_0, LATTICES, PRE_TOPC, BOOLE, TOPS_1, SUBSET_1, SETFAM_1, BINOP_1, FUNCT_1, RELAT_1, ZFMISC_1, TARSKI, GROUP_6, MOD_4, WELLORD1, REALSET1, ZF_LANG, LATTICE4, OPENLATT, HAHNBAN; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, RELAT_1, FUNCT_1, FUNCT_2, BINOP_1, SETFAM_1, STRUCT_0, PRE_TOPC, TOPS_1, LATTICES, LATTICE2, FILTER_0, REALSET1, LATTICE4; constructors BINOP_1, TOPS_1, LATTICE2, REALSET1, FILTER_1, LATTICE4; clusters FILTER_0, PRE_TOPC, LATTICE2, RLSUB_2, STRUCT_0, RELSET_1, SUBSET_1, LATTICES, XBOOLE_0, ZFMISC_1; requirements BOOLE, SUBSET; begin definition cluster Heyting -> implicative 0_Lattice; cluster implicative -> upper-bounded Lattice; end; reserve T for TopSpace; reserve A,B for Subset of T; theorem :: OPENLATT:1 A /\ Int(A` \/ B) c= B; theorem :: OPENLATT:2 for C being Subset of T st C is open & A /\ C c= B holds C c= Int(A` \/ B); definition let T be TopStruct; func Topology_of T -> Subset-Family of T equals :: OPENLATT:def 1 the topology of T; end; definition let T; cluster Topology_of T -> non empty; end; theorem :: OPENLATT:3 for A being Subset of T holds A is open iff A in Topology_of T; definition let T be non empty TopSpace, P, Q be Element of Topology_of T; redefine func P \/ Q -> Element of Topology_of T; redefine func P /\ Q -> Element of Topology_of T; end; reserve T for non empty TopSpace; reserve P,Q for Element of Topology_of T; definition let T; func Top_Union T -> BinOp of Topology_of T means :: OPENLATT:def 2 it.(P,Q) = P \/ Q; func Top_Meet T -> BinOp of Topology_of T means :: OPENLATT:def 3 it.(P,Q) = P /\ Q; end; theorem :: OPENLATT:4 for T being non empty TopSpace holds LattStr(#Topology_of T,Top_Union T,Top_Meet T#) is Lattice; definition let T; func Open_setLatt(T) -> Lattice equals :: OPENLATT:def 4 LattStr(#Topology_of T,Top_Union T,Top_Meet T#); end; theorem :: OPENLATT:5 the carrier of Open_setLatt(T) = Topology_of T; reserve p,q for Element of Open_setLatt(T); theorem :: OPENLATT:6 p "\/" q = p \/ q & p "/\" q = p /\ q; theorem :: OPENLATT:7 p [= q iff p c= q; theorem :: OPENLATT:8 for p',q' being Element of Topology_of T st p=p' & q=q' holds p [= q iff p' c= q'; theorem :: OPENLATT:9 Open_setLatt(T) is implicative; theorem :: OPENLATT:10 Open_setLatt(T) is lower-bounded & Bottom Open_setLatt(T) = {}; definition let T; cluster Open_setLatt(T) -> Heyting; end; theorem :: OPENLATT:11 Top Open_setLatt(T) = the carrier of T; reserve L for D_Lattice; reserve F for Filter of L; reserve a,b for Element of L; reserve x,X,X1,X2,Y,Z for set; definition let L; func F_primeSet(L) -> set equals :: OPENLATT:def 5 { F: F <> the carrier of L & F is prime}; end; theorem :: OPENLATT:12 F in F_primeSet(L) iff F <> the carrier of L & F is prime; definition let L; func StoneH(L) -> Function means :: OPENLATT:def 6 dom it = the carrier of L & it.a = { F :F in F_primeSet(L) & a in F}; end; theorem :: OPENLATT:13 F in StoneH(L).a iff F in F_primeSet(L) & a in F; theorem :: OPENLATT:14 x in StoneH(L).a iff (ex F st F=x & F <> the carrier of L & F is prime & a in F); definition let L; func StoneS(L) -> set equals :: OPENLATT:def 7 rng StoneH(L); end; definition let L; cluster StoneS(L) -> non empty; end; theorem :: OPENLATT:15 x in StoneS(L) iff ex a st x=StoneH(L).a; theorem :: OPENLATT:16 StoneH(L).(a "\/" b) = StoneH(L).a \/ StoneH(L).b; theorem :: OPENLATT:17 StoneH(L).(a "/\" b) = StoneH(L).a /\ StoneH(L).b; definition let L, a; func SF_have a -> Subset-Family of L equals :: OPENLATT:def 8 { F : a in F }; end; definition let L;let a; cluster SF_have a -> non empty; end; theorem :: OPENLATT:18 x in SF_have a iff x is Filter of L & a in x; theorem :: OPENLATT:19 x in SF_have b \ SF_have a implies x is Filter of L & b in x & not a in x; theorem :: OPENLATT:20 for Z st Z <> {} & Z c= SF_have b \ SF_have a & Z is c=-linear ex Y st Y in SF_have b \ SF_have a & for X1 st X1 in Z holds X1 c= Y ; theorem :: OPENLATT:21 not b [= a implies <.b.) in SF_have b \ SF_have a; theorem :: OPENLATT:22 not b [= a implies ex F st (F in F_primeSet(L) & not a in F & b in F ); theorem :: OPENLATT:23 a <> b implies ex F st F in F_primeSet(L); theorem :: OPENLATT:24 a <> b implies StoneH(L).a <> StoneH(L).b; theorem :: OPENLATT:25 StoneH(L) is one-to-one; definition let L;let A,B be Element of StoneS(L); redefine func A \/ B -> Element of StoneS(L); redefine func A /\ B -> Element of StoneS(L); end; definition let L; func Set_Union L -> BinOp of StoneS(L) means :: OPENLATT:def 9 for A,B being Element of StoneS(L) holds it.(A,B) = A \/ B; func Set_Meet L -> BinOp of StoneS(L) means :: OPENLATT:def 10 for A,B being Element of StoneS(L) holds it.(A,B) = A /\ B; end; theorem :: OPENLATT:26 LattStr(#StoneS(L),Set_Union L,Set_Meet L#) is Lattice; definition let L; func StoneLatt L -> Lattice equals :: OPENLATT:def 11 LattStr(#StoneS(L),Set_Union L,Set_Meet L#); end; reserve p,q for Element of StoneLatt(L); theorem :: OPENLATT:27 for L holds the carrier of StoneLatt(L) = StoneS(L); theorem :: OPENLATT:28 p "\/" q = p \/ q & p "/\" q = p /\ q; theorem :: OPENLATT:29 p [= q iff p c= q; definition let L; redefine func StoneH(L) -> Homomorphism of L,StoneLatt L; end; theorem :: OPENLATT:30 StoneH(L) is isomorphism; theorem :: OPENLATT:31 StoneLatt(L) is distributive; theorem :: OPENLATT:32 L,StoneLatt L are_isomorphic; definition cluster non trivial H_Lattice; end; reserve H for non trivial H_Lattice; reserve p',q' for Element of H; theorem :: OPENLATT:33 StoneH(H).(Top H) = F_primeSet(H); theorem :: OPENLATT:34 StoneH(H).(Bottom H) = {}; theorem :: OPENLATT:35 StoneS(H) c= bool F_primeSet(H); definition let H; cluster F_primeSet(H) -> non empty; end; definition let H; func HTopSpace H -> strict TopStruct means :: OPENLATT:def 12 the carrier of it = F_primeSet(H) & the topology of it ={union A where A is Subset of StoneS(H):not contradiction}; end; definition let H; cluster HTopSpace H -> non empty TopSpace-like; end; theorem :: OPENLATT:36 the carrier of Open_setLatt(HTopSpace H) = {union A where A is Subset of StoneS(H):not contradiction}; theorem :: OPENLATT:37 StoneS(H) c= the carrier of Open_setLatt(HTopSpace H); definition let H; redefine func StoneH(H) -> Homomorphism of H,Open_setLatt(HTopSpace H); end; theorem :: OPENLATT:38 StoneH(H) is monomorphism; theorem :: OPENLATT:39 StoneH(H).(p' => q') = (StoneH(H).p') => (StoneH(H).q'); theorem :: OPENLATT:40 StoneH(H) preserves_implication; theorem :: OPENLATT:41 StoneH(H) preserves_top; theorem :: OPENLATT:42 StoneH(H) preserves_bottom;

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