Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

### The abstract of the Mizar article:

### Noetherian Lattices

**by****Christoph Schwarzweller**- Received June 9, 1999
- MML identifier: LATTICE6

- [ Mizar article, MML identifier index ]

environ vocabulary FINSET_1, LATTICES, LATTICE3, BOOLE, FINSEQ_1, RELAT_1, FUNCT_1, ORDERS_1, FILTER_1, RELAT_2, BHSP_3, WELLORD1, ARYTM_3, WAYBEL_6, ZF_LANG, REALSET1, BINOP_1, TARSKI, LATTICE6; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XREAL_0, RELAT_1, FUNCT_1, NAT_1, STRUCT_0, BINOP_1, LATTICES, LATTICE3, FINSET_1, WELLORD1, WAYBEL_6, GROUP_1, ORDERS_1, FINSEQ_1, WELLFND1, YELLOW_0, LATTICE2, REALSET1; constructors LATTICE2, WAYBEL_1, WAYBEL_6, NAT_1, GROUP_1, WELLFND1, WELLORD1, BINOP_1, REALSET1, MEMBERED; clusters SUBSET_1, STRUCT_0, LATTICE2, LATTICE3, YELLOW_1, FINSET_1, ORDERS_1, INT_1, FINSEQ_1, FINSEQ_6, WAYBEL_0, KNASTER, LATTICES; requirements NUMERALS, BOOLE, SUBSET, ARITHM; begin definition cluster finite Lattice; end; definition cluster finite -> complete Lattice; end; definition let L be Lattice; let D be Subset of L; func D% -> Subset of LattPOSet L equals :: LATTICE6:def 1 {d% where d is Element of L : d in D}; end; definition let L be Lattice; let D be Subset of LattPOSet L; func %D -> Subset of L equals :: LATTICE6:def 2 {%d where d is Element of LattPOSet L : d in D}; end; definition let L be finite Lattice; cluster LattPOSet L -> well_founded; end; definition let L be Lattice; attr L is noetherian means :: LATTICE6:def 3 LattPOSet L is well_founded; attr L is co-noetherian means :: LATTICE6:def 4 (LattPOSet L)~ is well_founded; end; definition cluster noetherian upper-bounded lower-bounded complete Lattice; end; definition cluster co-noetherian upper-bounded lower-bounded complete Lattice; end; theorem :: LATTICE6:1 for L being Lattice holds L is noetherian iff L.: is co-noetherian; definition cluster finite -> noetherian Lattice; cluster finite -> co-noetherian Lattice; end; definition let L be Lattice; let a,b be Element of L; pred a is-upper-neighbour-of b means :: LATTICE6:def 5 a <> b & b [= a & for c being Element of L holds (b [= c & c [= a) implies (c = a or c = b); synonym b is-lower-neighbour-of a; end; theorem :: LATTICE6:2 for L being Lattice for a being Element of L for b,c being Element of L st b <> c holds ((b is-upper-neighbour-of a & c is-upper-neighbour-of a) implies a = c "/\" b) & ((b is-lower-neighbour-of a & c is-lower-neighbour-of a) implies a = c "\/" b); theorem :: LATTICE6:3 for L being noetherian Lattice for a being Element of L for d being Element of L st a [= d & a <> d holds ex c being Element of L st c [= d & c is-upper-neighbour-of a; theorem :: LATTICE6:4 for L being co-noetherian Lattice for a being Element of L for d being Element of L st d [= a & a <> d holds ex c being Element of L st d [= c & c is-lower-neighbour-of a; theorem :: LATTICE6:5 for L being upper-bounded Lattice holds not(ex b being Element of L st b is-upper-neighbour-of Top L); theorem :: LATTICE6:6 for L being noetherian upper-bounded Lattice for a being Element of L holds a = Top L iff not(ex b being Element of L st b is-upper-neighbour-of a); theorem :: LATTICE6:7 for L being lower-bounded Lattice holds not(ex b being Element of L st b is-lower-neighbour-of Bottom L); theorem :: LATTICE6:8 for L being co-noetherian lower-bounded Lattice for a being Element of L holds a = Bottom L iff not(ex b being Element of L st b is-lower-neighbour-of a); definition let L be complete Lattice; let a be Element of L; func a*' -> Element of L equals :: LATTICE6:def 6 "/\"({d where d is Element of L : a [= d & d <> a},L); func *'a -> Element of L equals :: LATTICE6:def 7 "\/"({d where d is Element of L : d [= a & d <> a},L); end; definition let L be complete Lattice; let a be Element of L; attr a is completely-meet-irreducible means :: LATTICE6:def 8 a*' <> a; attr a is completely-join-irreducible means :: LATTICE6:def 9 *'a <> a; end; theorem :: LATTICE6:9 for L being complete Lattice for a being Element of L holds a [= a*' & *'a [= a; theorem :: LATTICE6:10 for L being complete Lattice holds Top L*' = Top L & (Top L)% is meet-irreducible; theorem :: LATTICE6:11 for L being complete Lattice holds *'Bottom L = Bottom L & (Bottom L)% is join-irreducible; theorem :: LATTICE6:12 for L being complete Lattice for a being Element of L st a is completely-meet-irreducible holds a*' is-upper-neighbour-of a & for c being Element of L holds c is-upper-neighbour-of a implies c = a*'; theorem :: LATTICE6:13 for L being complete Lattice for a being Element of L st a is completely-join-irreducible holds *'a is-lower-neighbour-of a & for c being Element of L holds c is-lower-neighbour-of a implies c = *'a; theorem :: LATTICE6:14 for L being noetherian complete Lattice for a being Element of L holds a is completely-meet-irreducible iff ex b being Element of L st b is-upper-neighbour-of a & for c being Element of L holds c is-upper-neighbour-of a implies c = b; theorem :: LATTICE6:15 for L being co-noetherian complete Lattice for a being Element of L holds a is completely-join-irreducible iff ex b being Element of L st b is-lower-neighbour-of a & for c being Element of L holds c is-lower-neighbour-of a implies c = b; theorem :: LATTICE6:16 for L being complete Lattice for a being Element of L holds a is completely-meet-irreducible implies a% is meet-irreducible; theorem :: LATTICE6:17 for L being complete noetherian Lattice for a being Element of L st a <> Top L holds a is completely-meet-irreducible iff a% is meet-irreducible; theorem :: LATTICE6:18 for L being complete Lattice for a being Element of L holds a is completely-join-irreducible implies a% is join-irreducible; theorem :: LATTICE6:19 for L being complete co-noetherian Lattice for a being Element of L st a <> Bottom L holds a is completely-join-irreducible iff a% is join-irreducible; theorem :: LATTICE6:20 for L being finite Lattice for a being Element of L st a <> Bottom L & a <> Top L holds (a is completely-meet-irreducible iff a% is meet-irreducible) & (a is completely-join-irreducible iff a% is join-irreducible); definition let L be Lattice; let a be Element of L; attr a is atomic means :: LATTICE6:def 10 a is-upper-neighbour-of Bottom L; attr a is co-atomic means :: LATTICE6:def 11 a is-lower-neighbour-of Top L; end; theorem :: LATTICE6:21 for L being complete Lattice for a being Element of L st a is atomic holds a is completely-join-irreducible; theorem :: LATTICE6:22 for L being complete Lattice for a being Element of L st a is co-atomic holds a is completely-meet-irreducible; definition let L be Lattice; attr L is atomic means :: LATTICE6:def 12 for a being Element of L holds ex X being Subset of L st (for x being Element of L st x in X holds x is atomic) & a = "\/"(X,L); end; definition cluster strict non empty trivial Lattice; end; definition cluster atomic complete Lattice; end; definition let L be complete Lattice; let D be Subset of L; attr D is supremum-dense means :: LATTICE6:def 13 for a being Element of L holds ex D' being Subset of D st a = "\/"(D',L); attr D is infimum-dense means :: LATTICE6:def 14 for a being Element of L holds ex D' being Subset of D st a = "/\"(D',L); end; theorem :: LATTICE6:23 for L being complete Lattice for D being Subset of L holds D is supremum-dense iff for a being Element of L holds a = "\/"({d where d is Element of L: d in D & d [= a},L); theorem :: LATTICE6:24 for L being complete Lattice for D being Subset of L holds D is infimum-dense iff for a being Element of L holds a = "/\"({d where d is Element of L : d in D & a [= d},L); theorem :: LATTICE6:25 for L being complete Lattice for D being Subset of L holds D is infimum-dense iff D% is order-generating; definition let L be complete Lattice; func MIRRS(L) -> Subset of L equals :: LATTICE6:def 15 {a where a is Element of L : a is completely-meet-irreducible}; func JIRRS(L) -> Subset of L equals :: LATTICE6:def 16 {a where a is Element of L : a is completely-join-irreducible }; end; theorem :: LATTICE6:26 for L being complete Lattice for D being Subset of L st D is supremum-dense holds JIRRS(L) c= D; theorem :: LATTICE6:27 for L being complete Lattice for D being Subset of L st D is infimum-dense holds MIRRS(L) c= D; definition let L be co-noetherian complete Lattice; cluster MIRRS(L) -> infimum-dense; end; definition let L be noetherian complete Lattice; cluster JIRRS(L) -> supremum-dense; end;

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