Volume 8, 1996

University of Bialystok

Copyright (c) 1996 Association of Mizar Users

### The abstract of the Mizar article:

### Duality in Relation Structures

**by****Grzegorz Bancerek**- Received November 12, 1996
- MML identifier: YELLOW_7

- [ Mizar article, MML identifier index ]

environ vocabulary ORDERS_1, RELAT_1, OPPCAT_1, RELAT_2, LATTICE3, YELLOW_0, LATTICES, BHSP_3, WAYBEL_0, FINSET_1, QUANTAL1, BOOLE, ZF_LANG, PRE_TOPC, FUNCT_1, CAT_1, WELLORD1, SEQM_3, WAYBEL_1, PBOOLE, WAYBEL_5, FUNCT_6, FUNCOP_1, ZF_REFLE, FINSEQ_4, YELLOW_2, ORDINAL2, CARD_3, PRALG_1, REALSET1, YELLOW_7; notation TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, RELAT_2, RELSET_1, FUNCT_1, FUNCT_2, FINSET_1, CARD_3, FUNCT_6, REALSET1, PRE_TOPC, PRALG_1, PBOOLE, STRUCT_0, ORDERS_1, LATTICE3, YELLOW_0, YELLOW_2, WAYBEL_1, WAYBEL_0, WAYBEL_5; constructors LATTICE3, WAYBEL_1, ORDERS_3, WAYBEL_5, PRALG_2; clusters RELSET_1, STRUCT_0, PRALG_1, MSUALG_1, LATTICE3, YELLOW_0, WAYBEL_0, WAYBEL_1, WAYBEL_5, FINSET_1, PBOOLE, PRVECT_1, CANTOR_1, SUBSET_1, FRAENKEL, XBOOLE_0; requirements SUBSET, BOOLE; begin definition let L be RelStr; redefine func L~; synonym L opp; end; theorem :: YELLOW_7:1 for L being RelStr, x,y being Element of L opp holds x <= y iff ~x >= ~y; theorem :: YELLOW_7:2 for L being RelStr, x being Element of L, y being Element of L opp holds (x <= ~y iff x~ >= y) & (x >= ~y iff x~ <= y); theorem :: YELLOW_7:3 for L being RelStr holds L is empty iff L opp is empty; theorem :: YELLOW_7:4 for L being RelStr holds L is reflexive iff L opp is reflexive; theorem :: YELLOW_7:5 for L being RelStr holds L is antisymmetric iff L opp is antisymmetric; theorem :: YELLOW_7:6 for L being RelStr holds L is transitive iff L opp is transitive; theorem :: YELLOW_7:7 for L being non empty RelStr holds L is connected iff L opp is connected; definition let L be reflexive RelStr; cluster L opp -> reflexive; end; definition let L be transitive RelStr; cluster L opp -> transitive; end; definition let L be antisymmetric RelStr; cluster L opp -> antisymmetric; end; definition let L be connected (non empty RelStr); cluster L opp -> connected; end; theorem :: YELLOW_7:8 for L being RelStr, x being Element of L, X being set holds (x is_<=_than X iff x~ is_>=_than X) & (x is_>=_than X iff x~ is_<=_than X); theorem :: YELLOW_7:9 for L being RelStr, x being Element of L opp, X being set holds (x is_<=_than X iff ~x is_>=_than X) & (x is_>=_than X iff ~x is_<=_than X); theorem :: YELLOW_7:10 for L being RelStr, X being set holds ex_sup_of X,L iff ex_inf_of X,L opp; theorem :: YELLOW_7:11 for L being RelStr, X being set holds ex_sup_of X,L opp iff ex_inf_of X,L; theorem :: YELLOW_7:12 for L being non empty RelStr, X being set st ex_sup_of X,L or ex_inf_of X,L opp holds "\/"(X,L) = "/\"(X,L opp); theorem :: YELLOW_7:13 for L being non empty RelStr, X being set st ex_inf_of X,L or ex_sup_of X,L opp holds "/\"(X,L) = "\/"(X,L opp); theorem :: YELLOW_7:14 for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2 & L1 is with_infima holds L2 is with_infima; theorem :: YELLOW_7:15 for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2 & L1 is with_suprema holds L2 is with_suprema; theorem :: YELLOW_7:16 for L being RelStr holds L is with_infima iff L opp is with_suprema; :: LATTICE3:10 :: for L being RelStr holds L opp is with_infima iff L is with_suprema; theorem :: YELLOW_7:17 for L being non empty RelStr holds L is complete iff L opp is complete; definition let L be with_infima RelStr; cluster L opp -> with_suprema; end; definition let L be with_suprema RelStr; cluster L opp -> with_infima; end; definition let L be complete (non empty RelStr); cluster L opp -> complete; end; theorem :: YELLOW_7:18 for L being non empty RelStr for X being Subset of L, Y being Subset of L opp st X = Y holds fininfs X = finsups Y & finsups X = fininfs Y; theorem :: YELLOW_7:19 for L being RelStr for X being Subset of L, Y being Subset of L opp st X = Y holds downarrow X = uparrow Y & uparrow X = downarrow Y; theorem :: YELLOW_7:20 for L being non empty RelStr for x being Element of L, y being Element of L opp st x = y holds downarrow x = uparrow y & uparrow x = downarrow y; theorem :: YELLOW_7:21 for L being with_infima Poset, x,y being Element of L holds x"/\"y = (x~)"\/"(y~); theorem :: YELLOW_7:22 for L being with_infima Poset, x,y being Element of L opp holds (~x)"/\"(~y) = x"\/"y; theorem :: YELLOW_7:23 for L being with_suprema Poset, x,y being Element of L holds x"\/"y = (x~)"/\"(y~); theorem :: YELLOW_7:24 for L being with_suprema Poset, x,y being Element of L opp holds (~x)"\/"(~y) = x"/\"y; theorem :: YELLOW_7:25 for L being LATTICE holds L is distributive iff L opp is distributive; definition let L be distributive LATTICE; cluster L opp -> distributive; end; theorem :: YELLOW_7:26 for L being RelStr, x be set holds x is directed Subset of L iff x is filtered Subset of L opp; theorem :: YELLOW_7:27 for L being RelStr, x be set holds x is directed Subset of L opp iff x is filtered Subset of L; theorem :: YELLOW_7:28 for L being RelStr, x be set holds x is lower Subset of L iff x is upper Subset of L opp; theorem :: YELLOW_7:29 for L being RelStr, x be set holds x is lower Subset of L opp iff x is upper Subset of L; theorem :: YELLOW_7:30 for L being RelStr holds L is lower-bounded iff L opp is upper-bounded; theorem :: YELLOW_7:31 for L being RelStr holds L opp is lower-bounded iff L is upper-bounded; theorem :: YELLOW_7:32 for L being RelStr holds L is bounded iff L opp is bounded; theorem :: YELLOW_7:33 for L being lower-bounded antisymmetric non empty RelStr holds (Bottom L)~ = Top (L opp) & ~Top (L opp) = Bottom L; theorem :: YELLOW_7:34 for L being upper-bounded antisymmetric non empty RelStr holds (Top L)~ = Bottom (L opp) & ~Bottom (L opp) = Top L; theorem :: YELLOW_7:35 for L being bounded LATTICE, x,y being Element of L holds y is_a_complement_of x iff y~ is_a_complement_of x~; theorem :: YELLOW_7:36 for L being bounded LATTICE holds L is complemented iff L opp is complemented; definition let L be lower-bounded RelStr; cluster L opp -> upper-bounded; end; definition let L be upper-bounded RelStr; cluster L opp -> lower-bounded; end; definition let L be complemented (bounded LATTICE); cluster L opp -> complemented; end; :: Collorary: L is Boolean -> L opp is Boolean theorem :: YELLOW_7:37 for L being Boolean LATTICE, x being Element of L holds 'not'(x~) = 'not' x; definition let L be non empty RelStr; func ComplMap L -> map of L, L opp means :: YELLOW_7:def 1 for x being Element of L holds it.x = 'not' x; end; definition let L be Boolean LATTICE; cluster ComplMap L -> one-to-one; end; definition let L be Boolean LATTICE; cluster ComplMap L -> isomorphic; end; theorem :: YELLOW_7:38 for L being Boolean LATTICE holds L, L opp are_isomorphic; theorem :: YELLOW_7:39 for S,T being non empty RelStr, f be set holds (f is map of S,T iff f is map of S opp,T) & (f is map of S,T iff f is map of S,T opp) & (f is map of S,T iff f is map of S opp,T opp); theorem :: YELLOW_7:40 for S,T being non empty RelStr for f being map of S,T, g being map of S,T opp st f = g holds (f is monotone iff g is antitone) & (f is antitone iff g is monotone); theorem :: YELLOW_7:41 for S,T being non empty RelStr for f being map of S,T opp, g being map of S opp,T st f = g holds (f is monotone iff g is monotone) & (f is antitone iff g is antitone); theorem :: YELLOW_7:42 for S,T being non empty RelStr for f being map of S,T, g being map of S opp,T opp st f = g holds (f is monotone iff g is monotone) & (f is antitone iff g is antitone); theorem :: YELLOW_7:43 for S,T being non empty RelStr, f be set holds (f is Connection of S,T iff f is Connection of S~,T) & (f is Connection of S,T iff f is Connection of S,T~) & (f is Connection of S,T iff f is Connection of S~,T~); theorem :: YELLOW_7:44 for S,T being non empty Poset for f1 being map of S,T, g1 being map of T,S for f2 being map of S~,T~, g2 being map of T~,S~ st f1 = f2 & g1 = g2 holds [f1,g1] is Galois iff [g2,f2] is Galois; theorem :: YELLOW_7:45 for J being set, D being non empty set, K being ManySortedSet of J for F being DoubleIndexedSet of K,D holds doms F = K; definition let J, D be non empty set, K be non-empty ManySortedSet of J; let F be DoubleIndexedSet of K, D; let j be Element of J; let k be Element of K.j; redefine func F..(j,k) -> Element of D; end; theorem :: YELLOW_7:46 for L being non empty RelStr for J being set, K being ManySortedSet of J for x being set holds x is DoubleIndexedSet of K,L iff x is DoubleIndexedSet of K,L opp; theorem :: YELLOW_7:47 for L being complete LATTICE for J being non empty set, K being non-empty ManySortedSet of J for F being DoubleIndexedSet of K,L holds Sup Infs F <= Inf Sups Frege F; theorem :: YELLOW_7:48 for L being complete LATTICE holds L is completely-distributive iff for J being non empty set, K being non-empty ManySortedSet of J for F being DoubleIndexedSet of K,L holds Sup Infs F = Inf Sups Frege F; theorem :: YELLOW_7:49 for L being complete antisymmetric (non empty RelStr), F be Function holds \\/(F, L) = //\(F, L opp) & //\(F, L) = \\/(F, L opp); theorem :: YELLOW_7:50 for L being complete antisymmetric (non empty RelStr) for F be Function-yielding Function holds \//(F, L) = /\\(F, L opp) & /\\(F, L) = \//(F, L opp); definition cluster completely-distributive -> complete (non empty RelStr); end; definition cluster completely-distributive trivial strict (non empty Poset); end; theorem :: YELLOW_7:51 for L being non empty Poset holds L is completely-distributive iff L opp is completely-distributive;

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