:: Properties of Relational Structures, Posets, Lattices and Maps :: by Mariusz \.Zynel and Czes{\l}aw Byli\'nski :: :: Received September 20, 1996 :: Copyright (c) 1996-2018 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies XBOOLE_0, ORDERS_2, SUBSET_1, TARSKI, WAYBEL_0, LATTICE3, XXREAL_0, RELAT_2, YELLOW_0, EQREL_1, LATTICES, RELAT_1, WELLORD1, CAT_1, STRUCT_0, FUNCT_1, GROUP_6, SEQM_3, ORDINAL2, BINOP_1, REWRITE1, SETFAM_1, CARD_FIL, YELLOW_1, ZFMISC_1, WELLORD2, YELLOW_2; notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, SETFAM_1, LATTICE3, WELLORD1, DOMAIN_1, ORDERS_1, STRUCT_0, ORDERS_2, QUANTAL1, ORDERS_3, YELLOW_0, YELLOW_1, WAYBEL_0; constructors SETFAM_1, DOMAIN_1, TOLER_1, QUANTAL1, ORDERS_3, WAYBEL_0, YELLOW_1, RELSET_1; registrations XBOOLE_0, SUBSET_1, RELSET_1, FUNCT_2, STRUCT_0, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1; requirements SUBSET, BOOLE; definitions TARSKI, LATTICE3, QUANTAL1, ORDERS_3, WAYBEL_0; equalities WAYBEL_0, STRUCT_0; expansions TARSKI, LATTICE3, ORDERS_3, WAYBEL_0; theorems TARSKI, RELAT_1, FUNCT_1, FUNCT_2, SETFAM_1, WELLORD1, ORDERS_2, YELLOW_0, YELLOW_1, WAYBEL_0, XBOOLE_0, XBOOLE_1; schemes FUNCT_2, DOMAIN_1; begin :: Basic Facts reserve x, X, Y for set; theorem for L being non empty RelStr for x being Element of L for X being Subset of L holds X c= downarrow x iff X is_<=_than x by WAYBEL_0:17; theorem for L being non empty RelStr for x being Element of L for X being Subset of L holds X c= uparrow x iff x is_<=_than X by WAYBEL_0:18; theorem for L being antisymmetric transitive with_suprema RelStr for X, Y being set st ex_sup_of X,L & ex_sup_of Y,L holds ex_sup_of (X \/ Y),L & "\/"(X \/ Y, L) = "\/"(X, L) "\/" "\/"(Y, L) proof let L be antisymmetric transitive with_suprema RelStr; let X, Y be set such that A1: ex_sup_of X,L and A2: ex_sup_of Y,L; set a = "\/"(X, L) "\/" "\/"(Y, L); A3: X \/ Y is_<=_than a proof let x be Element of L; assume A4: x in X \/ Y; per cases by A4,XBOOLE_0:def 3; suppose A5: x in X; X is_<=_than "\/"(X, L) by A1,YELLOW_0:30; then A6: x <= "\/"(X, L) by A5; "\/"(X, L) <= a by YELLOW_0:22; hence thesis by A6,ORDERS_2:3; end; suppose A7: x in Y; Y is_<=_than "\/"(Y, L) by A2,YELLOW_0:30; then A8: x <= "\/"(Y, L) by A7; "\/"(Y, L) <= a by YELLOW_0:22; hence thesis by A8,ORDERS_2:3; end; end; for b being Element of L st X \/ Y is_<=_than b holds a <= b proof let b be Element of L; assume A9: X \/ Y is_<=_than b; Y c= X \/ Y by XBOOLE_1:7; then Y is_<=_than b by A9; then A10: "\/"(Y, L) <= b by A2,YELLOW_0:30; X c= X \/ Y by XBOOLE_1:7; then X is_<=_than b by A9; then "\/"(X, L) <= b by A1,YELLOW_0:30; hence thesis by A10,YELLOW_0:22; end; hence thesis by A3,YELLOW_0:30; end; theorem for L being antisymmetric transitive with_infima RelStr for X, Y being set st ex_inf_of X,L & ex_inf_of Y,L holds ex_inf_of (X \/ Y),L & "/\"(X \/ Y, L) = "/\"(X, L) "/\" "/\"(Y, L) proof let L be antisymmetric transitive with_infima RelStr; let X, Y be set such that A1: ex_inf_of X,L and A2: ex_inf_of Y,L; set a = "/\"(X, L) "/\" "/\"(Y, L); A3: X \/ Y is_>=_than a proof let x be Element of L; assume A4: x in X \/ Y; per cases by A4,XBOOLE_0:def 3; suppose A5: x in X; X is_>=_than "/\"(X, L) by A1,YELLOW_0:31; then A6: x >= "/\"(X, L) by A5; "/\"(X, L) >= a by YELLOW_0:23; hence thesis by A6,ORDERS_2:3; end; suppose A7: x in Y; Y is_>=_than "/\"(Y, L) by A2,YELLOW_0:31; then A8: x >= "/\"(Y, L) by A7; "/\"(Y, L) >= a by YELLOW_0:23; hence thesis by A8,ORDERS_2:3; end; end; for b being Element of L st X \/ Y is_>=_than b holds a >= b proof let b be Element of L; assume A9: X \/ Y is_>=_than b; Y c= X \/ Y by XBOOLE_1:7; then Y is_>=_than b by A9; then A10: "/\"(Y, L) >= b by A2,YELLOW_0:31; X c= X \/ Y by XBOOLE_1:7; then X is_>=_than b by A9; then "/\"(X, L) >= b by A1,YELLOW_0:31; hence thesis by A10,YELLOW_0:23; end; hence thesis by A3,YELLOW_0:31; end; begin :: Relational Substructures theorem Th5: for R being Relation for X, Y being set holds X c= Y implies R |_2 X c= R |_2 Y proof let R be Relation, X,Y be set; assume A1: X c= Y; then X|`R c= Y|`R by RELAT_1:100; then A2: X|`R|X c= Y|`R|X by RELAT_1:76; Y|`R|X c= Y|`R|Y by A1,RELAT_1:75; then X|`R|X c= Y|`R|Y by A2; then R|_2 X c= Y|`R|Y by WELLORD1:10; hence thesis by WELLORD1:10; end; theorem for L being RelStr for S, T being full SubRelStr of L st the carrier of S c= the carrier of T holds the InternalRel of S c= the InternalRel of T proof let L be RelStr, S1,S2 be full SubRelStr of L; the InternalRel of S1 = (the InternalRel of L)|_2 the carrier of S1 & the InternalRel of S2 = (the InternalRel of L)|_2 the carrier of S2 by YELLOW_0:def 14; hence thesis by Th5; end; theorem Th7: for L being non empty RelStr for S being non empty SubRelStr of L holds (X is directed Subset of S implies X is directed Subset of L) & (X is filtered Subset of S implies X is filtered Subset of L) proof let L be non empty RelStr; let S be non empty SubRelStr of L; thus X is directed Subset of S implies X is directed Subset of L proof assume A1: X is directed Subset of S; the carrier of S c= the carrier of L by YELLOW_0:def 13; then reconsider X as Subset of L by A1,XBOOLE_1:1; for x, y being Element of L st x in X & y in X ex z being Element of L st z in X & x <= z & y <= z proof let x, y be Element of L; assume A2: x in X & y in X; then reconsider x9= x, y9= y as Element of S by A1; consider z being Element of S such that A3: z in X & x9 <= z & y9 <= z by A1,A2,WAYBEL_0:def 1; reconsider z as Element of L by YELLOW_0:58; take z; thus thesis by A3,YELLOW_0:59; end; hence thesis by WAYBEL_0:def 1; end; thus X is filtered Subset of S implies X is filtered Subset of L proof assume A4: X is filtered Subset of S; the carrier of S c= the carrier of L by YELLOW_0:def 13; then reconsider X as Subset of L by A4,XBOOLE_1:1; for x, y being Element of L st x in X & y in X ex z being Element of L st z in X & z <= x & z <= y proof let x, y be Element of L; assume A5: x in X & y in X; then reconsider x9= x, y9= y as Element of S by A4; consider z being Element of S such that A6: z in X & z <= x9 & z <= y9 by A4,A5,WAYBEL_0:def 2; reconsider z as Element of L by YELLOW_0:58; take z; thus thesis by A6,YELLOW_0:59; end; hence thesis by WAYBEL_0:def 2; end; end; theorem for L being non empty RelStr for S, T being non empty full SubRelStr of L st the carrier of S c= the carrier of T for X being Subset of S holds X is Subset of T & for Y being Subset of T st X = Y holds (X is filtered implies Y is filtered) & (X is directed implies Y is directed) proof let L be non empty RelStr, S1,S2 be non empty full SubRelStr of L; assume A1: the carrier of S1 c= the carrier of S2; let X be Subset of S1; thus X is Subset of S2 by A1,XBOOLE_1:1; let X2 be Subset of S2 such that A2: X = X2; hereby assume A3: X is filtered; thus X2 is filtered proof let x, y be Element of S2; assume A4: x in X2 & y in X2; then reconsider x9= x, y9= y as Element of S1 by A2; consider z being Element of S1 such that A5: z in X and A6: z <= x9 & z <= y9 by A2,A3,A4; reconsider x1 = x, y1 = y, z1 = z as Element of L by YELLOW_0:58; reconsider z as Element of S2 by A2,A5; take z; z1 <= x1 & z1 <= y1 by A6,YELLOW_0:59; hence thesis by A2,A5,YELLOW_0:60; end; end; assume A7: X is directed; let x, y be Element of S2; assume A8: x in X2 & y in X2; then reconsider x9= x, y9= y as Element of S1 by A2; consider z being Element of S1 such that A9: z in X and A10: x9 <= z & y9 <= z by A2,A7,A8; reconsider x1 = x, y1 = y, z1 = z as Element of L by YELLOW_0:58; reconsider z as Element of S2 by A2,A9; take z; x1 <= z1 & y1 <= z1 by A10,YELLOW_0:59; hence thesis by A2,A9,YELLOW_0:60; end; begin :: Maps definition let J be set, L be RelStr; let f, g be Function of J, the carrier of L; pred f <= g means for j being set st j in J ex a, b being Element of L st a = f.j & b = g.j & a <= b; end; notation let J be set, L be RelStr; let f, g be Function of J, the carrier of L; synonym g >= f for f <= g; end; theorem for L, M being non empty RelStr, f,g being Function of L, M holds f <= g iff for x being Element of L holds f.x <= g.x proof let L, M be non empty RelStr, f,g be Function of L, M; hereby assume A1: f <= g; let x be Element of L; ex m1,m2 being Element of M st m1 = f.x & m2 = g.x & m1 <= m2 by A1; hence f.x <= g.x; end; assume A2: for x being Element of L holds f.x <= g.x; let x be set; assume x in the carrier of L; then reconsider x as Element of L; take f.x, g.x; thus thesis by A2; end; begin :: The Image of a Map definition let L, M be non empty RelStr; let f be Function of L, M; func Image f -> strict full SubRelStr of M equals subrelstr rng f; correctness; end; theorem for L, M being non empty RelStr for f being Function of L, M for y being Element of Image f ex x being Element of L st f.x = y proof let L1, L2 be non empty RelStr, g be Function of L1,L2; let s be Element of Image g; dom g = the carrier of L1 by FUNCT_2:def 1; then A1: rng g is non empty by RELAT_1:42; rng g = the carrier of Image g by YELLOW_0:def 15; then consider l being object such that A2: l in dom g and A3: s = g.l by A1,FUNCT_1:def 3; reconsider l as Element of L1 by A2; take l; thus thesis by A3; end; registration let L be non empty RelStr; let X be non empty Subset of L; cluster subrelstr X -> non empty; coherence by YELLOW_0:def 15; end; registration let L, M be non empty RelStr; let f be Function of L, M; cluster Image f -> non empty; coherence proof dom f = the carrier of L by FUNCT_2:def 1; then rng f is non empty by RELAT_1:42; hence thesis; end; end; begin :: Monotone Maps theorem for L being non empty RelStr holds id L is monotone proof let L be non empty RelStr; let l1,l2 be Element of L; assume l1 <= l2; then l1 <= (id L).l2 by FUNCT_1:18; hence thesis by FUNCT_1:18; end; theorem for L, M, N being non empty RelStr for f being Function of L, M, g being Function of M, N holds f is monotone & g is monotone implies g*f is monotone proof let L1,L2,L3 be non empty RelStr; let g1 be Function of L1,L2, g2 be Function of L2,L3 such that A1: g1 is monotone and A2: g2 is monotone; let s1,s2 be Element of L1; assume s1 <= s2; then g1.s1 <= g1.s2 by A1; then g2.(g1.s1) <= g2.(g1.s2) by A2; then (g2*g1).s1 <= g2.(g1.s2) by FUNCT_2:15; hence thesis by FUNCT_2:15; end; theorem for L, M being non empty RelStr for f being Function of L, M for X being Subset of L, x being Element of L holds f is monotone & x is_<=_than X implies f.x is_<=_than f.:X proof let L1,L2 be non empty RelStr, g be Function of L1,L2; let X be Subset of L1, x be Element of L1 such that A1: g is monotone and A2: x is_<=_than X; let y2 be Element of L2; assume y2 in g.:X; then consider x2 being Element of L1 such that A3: x2 in X and A4: y2 = g.x2 by FUNCT_2:65; reconsider x2 as Element of L1; x <= x2 by A2,A3; hence thesis by A1,A4; end; theorem for L, M being non empty RelStr for f being Function of L, M for X being Subset of L, x being Element of L holds f is monotone & X is_<=_than x implies f.:X is_<=_than f.x proof let L1,L2 be non empty RelStr, g be Function of L1,L2; let X be Subset of L1, x be Element of L1 such that A1: g is monotone and A2: x is_>=_than X; let y2 be Element of L2; assume y2 in g.:X; then consider x2 being Element of L1 such that A3: x2 in X and A4: y2 = g.x2 by FUNCT_2:65; reconsider x2 as Element of L1; x >= x2 by A2,A3; hence thesis by A1,A4; end; theorem for S, T being non empty RelStr for f being Function of S, T for X being directed Subset of S holds f is monotone implies f.:X is directed proof let S, T be non empty RelStr; let f be Function of S, T; let X be directed Subset of S; set Y = f.:X; assume A1: f is monotone; for y1, y2 being Element of T st y1 in Y & y2 in Y ex z being Element of T st z in Y & y1 <= z & y2 <= z proof let y1, y2 be Element of T; assume that A2: y1 in Y and A3: y2 in Y; consider x1 being object such that x1 in dom f and A4: x1 in X and A5: y1 = f.x1 by A2,FUNCT_1:def 6; consider x2 being object such that x2 in dom f and A6: x2 in X and A7: y2 = f.x2 by A3,FUNCT_1:def 6; reconsider x1, x2 as Element of S by A4,A6; consider z being Element of S such that A8: z in X and A9: x1 <= z & x2 <= z by A4,A6,WAYBEL_0:def 1; take f.z; thus f.z in Y by A8,FUNCT_2:35; thus thesis by A1,A5,A7,A9; end; hence thesis; end; theorem Th16: for L being with_suprema Poset for f being Function of L, L holds f is directed-sups-preserving implies f is monotone proof let L be with_suprema Poset; let f be Function of L, L; assume A1: f is directed-sups-preserving; let x, y be Element of L such that A2: x <= y; A3: y = y"\/"x by A2,YELLOW_0:24; for a, b being Element of L st a in {x, y} & b in {x, y} ex z being Element of L st z in {x, y} & a <= z & b <= z proof let a, b be Element of L such that A4: a in {x, y} & b in {x, y}; take y; thus y in {x, y} by TARSKI:def 2; thus thesis by A2,A4,TARSKI:def 2; end; then {x, y} is directed non empty; then A5: f preserves_sup_of {x, y} by A1; A6: dom f = the carrier of L by FUNCT_2:def 1; y <= y; then A7: {x, y} is_<=_than y by A2,YELLOW_0:8; for b being Element of L st {x, y} is_<=_than b holds y <= b by YELLOW_0:8; then ex_sup_of {x, y},L by A7,YELLOW_0:30; then sup(f.:{x, y}) = f.sup{x, y} by A5 .= f.y by A3,YELLOW_0:41; then A8: f.y = sup{f.x, f.y} by A6,FUNCT_1:60 .= f.y"\/"f.x by YELLOW_0:41; let afx, bfy be Element of L; assume afx = f.x & bfy = f.y; hence afx <= bfy by A8,YELLOW_0:22; end; theorem for L being with_infima Poset for f being Function of L, L holds f is filtered-infs-preserving implies f is monotone proof let L be with_infima Poset; let f be Function of L, L; assume A1: f is filtered-infs-preserving; let x, y be Element of L such that A2: x <= y; A3: x = x"/\"y by A2,YELLOW_0:25; for c, d being Element of L st c in {x, y} & d in {x, y} ex z being Element of L st z in {x, y} & z <= c & z <= d proof let c, d be Element of L such that A4: c in {x, y} & d in {x, y}; take x; thus x in {x, y} by TARSKI:def 2; thus thesis by A2,A4,TARSKI:def 2; end; then {x, y} is filtered non empty; then A5: f preserves_inf_of {x, y} by A1; A6: dom f = the carrier of L by FUNCT_2:def 1; x <= x; then A7: x is_<=_than {x, y} by A2,YELLOW_0:8; for c being Element of L st c is_<=_than {x, y} holds c <= x by YELLOW_0:8; then ex_inf_of {x, y},L by A7,YELLOW_0:31; then inf(f.:{x, y}) = f.inf{x, y} by A5 .= f.x by A3,YELLOW_0:40; then A8: f.x = inf{f.x, f.y} by A6,FUNCT_1:60 .= f.x"/\"f.y by YELLOW_0:40; let a, b be Element of L; assume a = f.x & b = f.y; hence a <= b by A8,YELLOW_0:23; end; begin :: Idempotent Maps theorem for S being non empty 1-sorted for f being Function of S, S holds f is idempotent implies for x being Element of S holds f.(f.x) = f.x proof let L be non empty 1-sorted, p be Function of L,L; assume A1: p*p = p; let l be Element of L; thus thesis by A1,FUNCT_2:15; end; theorem Th19: for S being non empty 1-sorted for f being Function of S, S holds f is idempotent implies rng f = {x where x is Element of S: x = f.x} proof let S be non empty 1-sorted; let f be Function of S, S; set M = {x where x is Element of S: x = f.x}; assume A1: f = f*f; A2: rng f c= M proof let y be object; assume A3: y in rng f; then reconsider y9= y as Element of S; ex x being object st x in dom f & y = f.x by A3,FUNCT_1:def 3; then y9= f.y9 by A1,FUNCT_1:13; hence thesis; end; M c= rng f proof let y be object; assume y in M; then A4: ex y9 being Element of S st y9 = y & y9 = f.y9; dom f = the carrier of S by FUNCT_2:def 1; hence thesis by A4,FUNCT_1:def 3; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th20: for S being non empty 1-sorted for f being Function of S, S st f is idempotent holds X c= rng f implies f.:X = X proof let S be non empty 1-sorted; let f be Function of S, S such that A1: f is idempotent; set M = {x where x is Element of S: x = f.x}; assume X c= rng f; then A2: X c= M by A1,Th19; A3: f.:X c= X proof let y be object; assume y in f.:X; then consider x being object such that x in dom f and A4: x in X and A5: y = f.x by FUNCT_1:def 6; x in M by A2,A4; then ex x9 being Element of S st x9 = x & x9 = f.x9; hence thesis by A4,A5; end; X c= f.:X proof let y be object; assume A6: y in X; then y in M by A2; then ex x being Element of S st x = y & x = f.x; hence thesis by A6,FUNCT_2:35; end; hence thesis by A3,XBOOLE_0:def 10; end; theorem for L being non empty RelStr holds id L is idempotent proof let L be non empty RelStr; thus (id L)*(id L) = id ((the carrier of L) /\ (the carrier of L)) by FUNCT_1:22 .= (id L); end; begin :: Complete Lattices (CCL Chapter 0, Section 2, pp. 8 -- 9) reserve L for complete LATTICE, a for Element of L; theorem Th22: a in X implies a <= "\/"(X, L) & "/\"(X, L) <= a proof assume A1: a in X; X is_<=_than "\/"(X, L) by YELLOW_0:32; hence a <= "\/"(X, L) by A1; X is_>=_than "/\"(X, L) by YELLOW_0:33; hence thesis by A1; end; theorem Th23: for L being non empty RelStr holds (for X holds ex_sup_of X,L) iff (for Y holds ex_inf_of Y,L) proof let L be non empty RelStr; hereby assume A1: for X holds ex_sup_of X,L; let Y; set X = {x where x is Element of L: x is_<=_than Y}; ex_sup_of X,L by A1; then consider a being Element of L such that A2: X is_<=_than a and A3: for b being Element of L st X is_<=_than b holds b >= a and A4: for c being Element of L st X is_<=_than c & for b being Element of L st X is_<=_than b holds b >= c holds c = a by YELLOW_0:def 7; A5: a is_<=_than Y proof let b be Element of L; assume A6: b in Y; b is_>=_than X proof let c be Element of L; assume c in X; then ex x being Element of L st c = x & x is_<=_than Y; hence thesis by A6; end; hence thesis by A3; end; A7: for c being Element of L st Y is_>=_than c & for b being Element of L st Y is_>=_than b holds b <= c holds c = a proof let c be Element of L such that A8: Y is_>=_than c and A9: for b being Element of L st Y is_>=_than b holds b <= c; A10: for b being Element of L st X is_<=_than b holds b >= c proof let b be Element of L; assume A11: X is_<=_than b; c in X by A8; hence thesis by A11; end; X is_<=_than c proof let b be Element of L; assume b in X; then ex x being Element of L st b = x & x is_<=_than Y; hence thesis by A9; end; hence thesis by A4,A10; end; for b being Element of L st Y is_>=_than b holds a >= b proof let b be Element of L; assume b is_<=_than Y; then b in X; hence thesis by A2; end; hence ex_inf_of Y,L by A5,A7,YELLOW_0:def 8; end; assume A12: for Y holds ex_inf_of Y,L; let X; set Y = {y where y is Element of L: X is_<=_than y}; ex_inf_of Y,L by A12; then consider a being Element of L such that A13: Y is_>=_than a and A14: for b being Element of L st Y is_>=_than b holds b <= a and A15: for c being Element of L st Y is_>=_than c & for b being Element of L st Y is_>=_than b holds b <= c holds c = a by YELLOW_0:def 8; A16: X is_<=_than a proof let b be Element of L; assume A17: b in X; b is_<=_than Y proof let c be Element of L; assume c in Y; then ex y being Element of L st c = y & X is_<=_than y; hence thesis by A17; end; hence thesis by A14; end; A18: for c being Element of L st X is_<=_than c & for b being Element of L st X is_<=_than b holds b >= c holds c = a proof let c be Element of L such that A19: X is_<=_than c and A20: for b being Element of L st X is_<=_than b holds b >= c; A21: for b being Element of L st Y is_>=_than b holds b <= c proof let b be Element of L; assume A22: Y is_>=_than b; c in Y by A19; hence thesis by A22; end; Y is_>=_than c proof let b be Element of L; assume b in Y; then ex x being Element of L st b = x & x is_>=_than X; hence thesis by A20; end; hence thesis by A15,A21; end; for b being Element of L st X is_<=_than b holds a <= b proof let b be Element of L; assume X is_<=_than b; then b in Y; hence thesis by A13; end; hence thesis by A16,A18,YELLOW_0:def 7; end; theorem Th24: ::Proposition 2.2 (i) (variant 1) cf YELLOW_0 for L being non empty RelStr holds (for X holds ex_sup_of X,L) implies L is complete proof let L be non empty RelStr; assume A1: for X holds ex_sup_of X,L; L is complete proof let X be set; take a = "\/"(X, L); A2: ex_sup_of X,L by A1; hence X is_<=_than a by YELLOW_0:def 9; thus thesis by A2,YELLOW_0:def 9; end; hence thesis; end; theorem Th25: ::Proposition 2.2 (i) (variant 2) cf variant 3 for L being non empty RelStr holds (for X holds ex_inf_of X,L) implies L is complete proof let L be non empty RelStr; assume for X holds ex_inf_of X,L; then for X holds ex_sup_of X,L by Th23; hence thesis by Th24; end; theorem Th26: for L being non empty RelStr holds (for A being Subset of L holds ex_inf_of A, L) implies for X holds ex_inf_of X,L & "/\"(X, L) = "/\"(X /\ the carrier of L, L) proof let L be non empty RelStr; assume A1: for A being Subset of L holds ex_inf_of A, L; let X; set Y = X /\ the carrier of L; set a = "/\"(Y, L); reconsider Y as Subset of L by XBOOLE_1:17; A2: ex_inf_of Y,L by A1; A3: a is_<=_than X proof let x be Element of L; assume x in X; then A4: x in Y by XBOOLE_0:def 4; a is_<=_than Y by A2,YELLOW_0:def 10; hence thesis by A4; end; A5: for b being Element of L st b is_<=_than X holds b <= a proof let b be Element of L; A6: Y c= X by XBOOLE_1:17; assume b is_<=_than X; then b is_<=_than Y by A6; hence thesis by A2,YELLOW_0:def 10; end; ex_inf_of X,L by A2,YELLOW_0:50; hence thesis by A3,A5,YELLOW_0:def 10; end; theorem for L being non empty RelStr holds (for A being Subset of L holds ex_sup_of A, L) implies for X holds ex_sup_of X,L & "\/"(X, L) = "\/"(X /\ the carrier of L, L) proof let L be non empty RelStr; assume A1: for A being Subset of L holds ex_sup_of A, L; let X; set Y = X /\ the carrier of L; set a = "\/"(Y, L); reconsider Y as Subset of L by XBOOLE_1:17; A2: ex_sup_of Y,L by A1; A3: X is_<=_than a proof let x be Element of L; assume x in X; then A4: x in Y by XBOOLE_0:def 4; Y is_<=_than a by A2,YELLOW_0:def 9; hence thesis by A4; end; A5: for b being Element of L st X is_<=_than b holds a <= b proof let b be Element of L; A6: Y c= X by XBOOLE_1:17; assume X is_<=_than b; then Y is_<=_than b by A6; hence thesis by A2,YELLOW_0:def 9; end; ex_sup_of X,L by A2,YELLOW_0:50; hence thesis by A3,A5,YELLOW_0:def 9; end; theorem Th28: ::Proposition 2.2 (i) (variant 3) for L being non empty RelStr holds (for A being Subset of L holds ex_inf_of A,L) implies L is complete proof let L be non empty RelStr; assume for A being Subset of L holds ex_inf_of A,L; then for X holds ex_inf_of X, L by Th26; hence thesis by Th25; end; registration cluster up-complete /\-complete upper-bounded -> complete for non empty Poset; correctness; end; theorem Th29: :: Theorem 2.3 (The Fixed-Point Theorem) for f being Function of L, L st f is monotone for M being Subset of L st M = {x where x is Element of L: x = f.x} holds subrelstr M is complete LATTICE proof let f be Function of L, L such that A1: f is monotone; let M be Subset of L such that A2: M = {x where x is Element of L: x = f.x}; set S = subrelstr M; A3: for X being Subset of S for Y being Subset of L st Y = {y where y is Element of L: X is_<=_than f.y & f.y <= y} holds inf Y in M proof let X be Subset of S; let Y be Subset of L such that A4: Y = {y where y is Element of L: X is_<=_than f.y & f.y <= y}; set z = inf Y; A5: f.z is_<=_than Y proof let u be Element of L; assume A6: u in Y; then consider y being Element of L such that A7: y = u and X is_<=_than f.y and A8: f.y <= y by A4; z <= u by A6,Th22; then f.z <= f.y by A1,A7; hence f.z <= u by A7,A8,ORDERS_2:3; end; A9: X is_<=_than f.(f.z) proof let m be Element of L; assume A10: m in X; m is_<=_than Y proof let u be Element of L; assume u in Y; then consider y being Element of L such that A11: y = u and A12: X is_<=_than f.y and A13: f.y <= y by A4; m <= f.y by A10,A12; hence m <= u by A11,A13,ORDERS_2:3; end; then m <= z by YELLOW_0:33; then A14: f.m <= f.z by A1; X c= the carrier of S; then X c= M by YELLOW_0:def 15; then m in M by A10; then ex x being Element of L st x = m & x = f.x by A2; hence m <= f.(f.z) by A1,A14; end; ex_inf_of Y,L by YELLOW_0:17; then A15: f.z <= z by A5,YELLOW_0:31; then f.(f.z) <= f.z by A1; then f.z in Y by A4,A9; then z <= f.z by Th22; then z = f.z by A15,ORDERS_2:2; hence thesis by A2; end; M c= the carrier of S by YELLOW_0:def 15; then reconsider M as Subset of S; defpred P[Element of L] means M is_<=_than f.$1 & f.$1 <= $1; reconsider Y = {y where y is Element of L: P[y]} as Subset of L from DOMAIN_1:sch 7; inf Y in M by A3; then reconsider S as non empty Poset; for X being Subset of S holds ex_sup_of X, S proof let X be Subset of S; defpred Q[Element of L] means X is_<=_than f.$1 & f.$1 <= $1; reconsider Y = {y where y is Element of L: Q[y]} as Subset of L from DOMAIN_1:sch 7; set z = inf Y; z in M by A3; then reconsider z as Element of S; A16: X is_<=_than z proof let x be Element of S; x in the carrier of S; then x in M by YELLOW_0:def 15; then consider m being Element of L such that A17: x = m and m = f.m by A2; assume A18: x in X; m is_<=_than Y proof let u be Element of L; assume u in Y; then consider y being Element of L such that A19: y = u and A20: X is_<=_than f.y and A21: f.y <= y; m <= f.y by A18,A17,A20; hence m <= u by A19,A21,ORDERS_2:3; end; then m <= inf Y by YELLOW_0:33; hence x <= z by A17,YELLOW_0:60; end; for b being Element of S st X is_<=_than b holds z <= b proof let b be Element of S; b in the carrier of S; then b in M by YELLOW_0:def 15; then consider x being Element of L such that A22: x = b and A23: x = f.x by A2; assume X is_<=_than b; then X is_<=_than f.x by A22,A23,YELLOW_0:62; then x in Y by A23; hence thesis by A22,Th22,YELLOW_0:60; end; hence thesis by A16,YELLOW_0:30; end; then reconsider S as complete non empty Poset by YELLOW_0:53; S is complete LATTICE; hence thesis; end; theorem Th30: for S being infs-inheriting non empty full SubRelStr of L holds S is complete LATTICE proof let S be infs-inheriting non empty full SubRelStr of L; for X being Subset of S holds ex_inf_of X,S proof let X be Subset of S; A1: ex_inf_of X,L by YELLOW_0:17; then "/\"(X,L) in the carrier of S by YELLOW_0:def 18; hence thesis by A1,YELLOW_0:63; end; then S is complete by Th28; hence thesis; end; theorem Th31: for S being sups-inheriting non empty full SubRelStr of L holds S is complete LATTICE proof let S be sups-inheriting non empty full SubRelStr of L; for X being Subset of S holds ex_sup_of X, S proof let X be Subset of S; A1: ex_sup_of X,L by YELLOW_0:17; then "\/"(X,L) in the carrier of S by YELLOW_0:def 19; hence thesis by A1,YELLOW_0:64; end; then S is complete by YELLOW_0:53; hence thesis; end; theorem Th32: ::Remark 2.4 (Part I, variant 1) for M being non empty RelStr for f being Function of L, M st f is sups-preserving holds Image f is sups-inheriting proof let M be non empty RelStr; let f be Function of L, M such that A1: f is sups-preserving; set S = subrelstr(rng f); for Y being Subset of S st ex_sup_of Y,M holds "\/"(Y, M) in the carrier of S proof let Y be Subset of S; assume ex_sup_of Y,M; A2: f preserves_sup_of (f"Y) & ex_sup_of f"Y,L by A1,YELLOW_0:17; Y c= the carrier of S; then Y c= rng f by YELLOW_0:def 15; then "\/"(Y, M) = sup(f.:(f"Y)) by FUNCT_1:77 .= f.sup(f"Y) by A2; then "\/"(Y, M) in rng f by FUNCT_2:4; hence thesis by YELLOW_0:def 15; end; hence thesis by YELLOW_0:def 19; end; theorem Th33: ::Remark 2.4 (Part I, variant 2) for M being non empty RelStr for f being Function of L, M st f is infs-preserving holds Image f is infs-inheriting proof let M be non empty RelStr; let f be Function of L, M such that A1: f is infs-preserving; set S = subrelstr(rng f); for Y being Subset of S st ex_inf_of Y,M holds "/\"(Y, M) in the carrier of S proof let Y be Subset of S; assume ex_inf_of Y,M; A2: f preserves_inf_of (f"Y) & ex_inf_of f"Y,L by A1,YELLOW_0:17; Y c= the carrier of S; then Y c= rng f by YELLOW_0:def 15; then "/\"(Y, M) = inf(f.:(f"Y)) by FUNCT_1:77 .= f.inf(f"Y) by A2; then "/\"(Y, M) in rng f by FUNCT_2:4; hence thesis by YELLOW_0:def 15; end; hence thesis by YELLOW_0:def 18; end; theorem ::Remark 2.4 (Part II) for L, M being complete LATTICE for f being Function of L, M st f is sups-preserving or f is infs-preserving holds Image f is complete LATTICE proof let L, M be complete LATTICE; let f be Function of L, M such that A1: f is sups-preserving or f is infs-preserving; per cases by A1; suppose f is sups-preserving; then Image f is sups-inheriting by Th32; hence thesis by Th31; end; suppose f is infs-preserving; then Image f is infs-inheriting by Th33; hence thesis by Th30; end; end; theorem ::Remark 2.5 for f being Function of L, L st f is idempotent directed-sups-preserving holds Image f is directed-sups-inheriting & Image f is complete LATTICE proof let f be Function of L, L; set S = subrelstr(rng f); set a = the Element of L; dom f = the carrier of L by FUNCT_2:def 1; then f.a in rng f by FUNCT_1:def 3; then reconsider S9= S as non empty full SubRelStr of L; assume A1: f is idempotent directed-sups-preserving; for X being directed Subset of S st X <> {} & ex_sup_of X,L holds "\/"(X ,L) in the carrier of S proof let X be directed Subset of S; X c= the carrier of S; then A2: X c= rng f by YELLOW_0:def 15; assume X <> {}; then X is non empty directed Subset of S9; then reconsider X9= X as non empty directed Subset of L by Th7; assume A3: ex_sup_of X,L; f preserves_sup_of X9 by A1; then sup(f.:X9) = f.sup X9 by A3; then sup X9 = f.sup X9 by A1,A2,Th20; then "\/"(X, L) in rng f by FUNCT_2:4; hence thesis by YELLOW_0:def 15; end; hence Image f is directed-sups-inheriting; rng f = {x where x is Element of L: x = f.x} by A1,Th19; hence thesis by A1,Th16,Th29; end; begin :: A Lattice of Ideals theorem Th36: for L being RelStr for F being Subset-Family of L st for X being Subset of L st X in F holds X is lower holds meet F is lower Subset of L proof let L be RelStr; let F be Subset-Family of L; reconsider F9 = F as Subset-Family of L; assume A1: for X being Subset of L st X in F holds X is lower; reconsider M = meet F9 as Subset of L; per cases; suppose F = {}; then for x, y being Element of L st x in M & y <= x holds y in M by SETFAM_1:def 1; hence thesis by WAYBEL_0:def 19; end; suppose A2: F <> {}; for x, y being Element of L st x in M & y <= x holds y in M proof let x, y be Element of L; assume that A3: x in M and A4: y <= x; for Y being set st Y in F holds y in Y proof let Y be set; assume A5: Y in F; then reconsider X = Y as Subset of L; X is lower & x in X by A1,A3,A5,SETFAM_1:def 1; hence thesis by A4; end; hence thesis by A2,SETFAM_1:def 1; end; hence thesis by WAYBEL_0:def 19; end; end; theorem for L being RelStr for F being Subset-Family of L st for X being Subset of L st X in F holds X is upper holds meet F is upper Subset of L proof let L be RelStr; let F be Subset-Family of L; reconsider F9 = F as Subset-Family of L; assume A1: for X being Subset of L st X in F holds X is upper; reconsider M = meet F9 as Subset of L; per cases; suppose F = {}; then for x, y being Element of L st x in M & x <= y holds y in M by SETFAM_1:def 1; hence thesis by WAYBEL_0:def 20; end; suppose A2: F <> {}; for x, y being Element of L st x in M & x <= y holds y in M proof let x, y be Element of L; assume that A3: x in M and A4: x <= y; for Y being set st Y in F holds y in Y proof let Y be set; assume A5: Y in F; then reconsider X = Y as Subset of L; X is upper & x in X by A1,A3,A5,SETFAM_1:def 1; hence thesis by A4; end; hence thesis by A2,SETFAM_1:def 1; end; hence thesis by WAYBEL_0:def 20; end; end; theorem Th38: for L being with_suprema antisymmetric RelStr for F being Subset-Family of L st for X being Subset of L st X in F holds X is lower directed holds meet F is directed Subset of L proof let L be with_suprema antisymmetric RelStr; let F be Subset-Family of L; assume A1: for X being Subset of L st X in F holds X is lower directed; reconsider F9 = F as Subset-Family of L; reconsider M = meet F9 as Subset of L; per cases; suppose A2: F = {}; M is directed by A2,SETFAM_1:def 1; hence thesis; end; suppose A3: F <> {}; M is directed proof let x, y be Element of L such that A4: x in M and A5: y in M; take z = x"\/"y; for Y being set st Y in F holds z in Y proof let Y be set; assume A6: Y in F; then reconsider X = Y as Subset of L; A7: y in X by A5,A6,SETFAM_1:def 1; X is lower directed & x in X by A1,A4,A6,SETFAM_1:def 1; hence thesis by A7,WAYBEL_0:40; end; hence z in M by A3,SETFAM_1:def 1; thus thesis by YELLOW_0:22; end; hence thesis; end; end; theorem for L being with_infima antisymmetric RelStr for F being Subset-Family of L st for X being Subset of L st X in F holds X is upper filtered holds meet F is filtered Subset of L proof let L be with_infima antisymmetric RelStr; let F be Subset-Family of L; assume A1: for X being Subset of L st X in F holds X is upper filtered; reconsider F9 = F as Subset-Family of L; reconsider M = meet F9 as Subset of L; per cases; suppose A2: F = {}; M is filtered by A2,SETFAM_1:def 1; hence thesis; end; suppose A3: F <> {}; M is filtered proof let x, y be Element of L such that A4: x in M and A5: y in M; take z = x"/\"y; for Y being set st Y in F holds z in Y proof let Y be set; assume A6: Y in F; then reconsider X = Y as Subset of L; A7: y in X by A5,A6,SETFAM_1:def 1; X is upper filtered & x in X by A1,A4,A6,SETFAM_1:def 1; hence thesis by A7,WAYBEL_0:41; end; hence z in M by A3,SETFAM_1:def 1; thus thesis by YELLOW_0:23; end; hence thesis; end; end; theorem Th40: for L being with_infima Poset for I, J being Ideal of L holds I /\ J is Ideal of L proof let L be with_infima Poset; let I, J be Ideal of L; set i = the Element of I,j = the Element of J; set a = i"/\"j; a <= j by YELLOW_0:23; then A1: a in J by WAYBEL_0:def 19; a <= i by YELLOW_0:23; then a in I by WAYBEL_0:def 19; hence thesis by A1,WAYBEL_0:27,44,XBOOLE_0:def 4; end; registration let L be non empty reflexive transitive RelStr; cluster Ids L -> non empty; correctness proof set x = the Element of L; downarrow x in the set of all Y where Y is Ideal of L; hence thesis; end; end; theorem Th41: for L being non empty reflexive transitive RelStr holds (x is Element of InclPoset(Ids L) iff x is Ideal of L) proof let L be non empty reflexive transitive RelStr; hereby assume x is Element of InclPoset(Ids L); then x in the carrier of InclPoset(Ids L); then x in Ids L by YELLOW_1:1; then ex J being Ideal of L st J = x; hence x is Ideal of L; end; assume x is Ideal of L; then x in the set of all Y where Y is Ideal of L; hence thesis by YELLOW_1:1; end; theorem Th42: for L being non empty reflexive transitive RelStr for I being Element of InclPoset(Ids L) holds x in I implies x is Element of L proof let L be non empty reflexive transitive RelStr; let I be Element of InclPoset(Ids L); reconsider I9= I as non empty Subset of L by Th41; assume x in I; then reconsider x9= x as Element of I9; x9 in the carrier of L; hence thesis; end; theorem for L being with_infima Poset for x, y being Element of InclPoset(Ids L) holds x "/\" y = x /\ y proof let L be with_infima Poset; let x, y be Element of InclPoset(Ids L); reconsider x9= x, y9= y as Ideal of L by Th41; x9 /\ y9 is Ideal of L by Th40; then x9 /\ y9 in the set of all X where X is Ideal of L; hence thesis by YELLOW_1:9; end; registration let L be with_infima Poset; cluster InclPoset(Ids L) -> with_infima; correctness proof for x, y be set st x in Ids L & y in Ids L holds x /\ y in Ids L proof let x, y be set; assume that A1: x in Ids L and A2: y in Ids L; consider I being Ideal of L such that A3: I = x by A1; consider J being Ideal of L such that A4: J = y by A2; I /\ J is Ideal of L by Th40; hence thesis by A3,A4; end; hence thesis by YELLOW_1:12; end; end; theorem Th44: for L being with_suprema Poset for x, y being Element of InclPoset(Ids L) holds ex Z being Subset of L st Z = {z where z is Element of L : z in x or z in y or ex a, b being Element of L st a in x & b in y & z = a "\/" b} & ex_sup_of {x, y},InclPoset(Ids L) & x "\/" y = downarrow Z proof let L be with_suprema Poset; set P = InclPoset(Ids L); let x, y be Element of P; defpred P[Element of L] means $1 in x or $1 in y or ex a, b being Element of L st a in x & b in y & $1 = a "\/" b; reconsider Z = {z where z is Element of L: P[z]} as Subset of L from DOMAIN_1:sch 7; take Z; reconsider x9= x, y9= y as Ideal of L by Th41; set z = the Element of x9; z in Z; then reconsider Z as non empty Subset of L; set DZ = downarrow Z; for u, v being Element of L st u in Z & v in Z ex z being Element of L st z in Z & u <= z & v <= z proof A1: for p, q being Element of L st p in y & ex a, b being Element of L st a in x & b in y & q = a "\/" b holds ex z being Element of L st z in Z & p <= z & q <= z proof let p, q be Element of L such that A2: p in y and A3: ex a, b being Element of L st a in x & b in y & q = a "\/" b; consider a, b being Element of L such that A4: a in x and A5: b in y and A6: q = a "\/" b by A3; reconsider c = b "\/" p as Element of L; take z = a "\/" c; c in y9 by A2,A5,WAYBEL_0:40; hence z in Z by A4; A7: c <= z by YELLOW_0:22; A8: p <= c & a <= z by YELLOW_0:22; b <= c by YELLOW_0:22; then b <= z by A7,ORDERS_2:3; hence thesis by A6,A7,A8,ORDERS_2:3,YELLOW_0:22; end; A9: for p, q being Element of L st p in x & ex a, b being Element of L st a in x & b in y & q = a "\/" b holds ex z being Element of L st z in Z & p <= z & q <= z proof let p, q be Element of L such that A10: p in x and A11: ex a, b being Element of L st a in x & b in y & q = a "\/" b; consider a, b being Element of L such that A12: a in x and A13: b in y and A14: q = a "\/" b by A11; reconsider c = a "\/" p as Element of L; take z = c "\/" b; c in x9 by A10,A12,WAYBEL_0:40; hence z in Z by A13; A15: c <= z by YELLOW_0:22; A16: p <= c & b <= z by YELLOW_0:22; a <= c by YELLOW_0:22; then a <= z by A15,ORDERS_2:3; hence thesis by A14,A15,A16,ORDERS_2:3,YELLOW_0:22; end; let u, v be Element of L such that A17: u in Z and A18: v in Z; consider p being Element of L such that A19: p = u and A20: p in x or p in y or ex a, b being Element of L st a in x & b in y & p = a "\/" b by A17; consider q being Element of L such that A21: q = v and A22: q in x or q in y or ex a, b being Element of L st a in x & b in y & q = a "\/" b by A18; per cases by A20,A22; suppose p in x & q in x; then consider z being Element of L such that A23: z in x9 & p <= z & q <= z by WAYBEL_0:def 1; take z; thus thesis by A19,A21,A23; end; suppose A24: p in x & q in y; take p "\/" q; thus thesis by A19,A21,A24,YELLOW_0:22; end; suppose p in x & ex a, b being Element of L st a in x & b in y & q = a "\/" b; then consider z being Element of L such that A25: z in Z & p <= z & q <= z by A9; take z; thus thesis by A19,A21,A25; end; suppose A26: p in y & q in x; take q "\/" p; thus thesis by A19,A21,A26,YELLOW_0:22; end; suppose p in y & q in y; then consider z being Element of L such that A27: z in y9 & p <= z & q <= z by WAYBEL_0:def 1; take z; thus thesis by A19,A21,A27; end; suppose p in y & ex a, b being Element of L st a in x & b in y & q = a "\/" b; then consider z being Element of L such that A28: z in Z & p <= z & q <= z by A1; take z; thus thesis by A19,A21,A28; end; suppose q in x & ex a, b being Element of L st a in x & b in y & p = a "\/" b; then consider z being Element of L such that A29: z in Z & q <= z & p <= z by A9; take z; thus thesis by A19,A21,A29; end; suppose q in y & ex a, b being Element of L st a in x & b in y & p = a "\/" b; then consider z being Element of L such that A30: z in Z & q <= z & p <= z by A1; take z; thus thesis by A19,A21,A30; end; suppose (ex a, b being Element of L st a in x & b in y & p = a "\/" b) & ex a, b being Element of L st a in x & b in y & q = a "\/" b; then consider a, b, c, d being Element of L such that A31: a in x & b in y and A32: p = a "\/" b and A33: c in x & d in y and A34: q = c "\/" d; reconsider ac = a "\/" c, bd = b "\/" d as Element of L; take z = ac "\/" bd; ac in x9 & bd in y9 by A31,A33,WAYBEL_0:40; hence z in Z; A35: ac <= z by YELLOW_0:22; A36: bd <= z by YELLOW_0:22; b <= bd by YELLOW_0:22; then A37: b <= z by A36,ORDERS_2:3; a <= ac by YELLOW_0:22; then a <= z by A35,ORDERS_2:3; hence u <= z by A19,A32,A37,YELLOW_0:22; d <= bd by YELLOW_0:22; then A38: d <= z by A36,ORDERS_2:3; c <= ac by YELLOW_0:22; then c <= z by A35,ORDERS_2:3; hence thesis by A21,A34,A38,YELLOW_0:22; end; end; then Z is directed; then reconsider DZ as Element of P by Th41; A39: for d being Element of P st d >= x & d >= y holds DZ <= d proof let d be Element of P; assume that A40: x <= d and A41: y <= d; A42: y c= d by A41,YELLOW_1:3; A43: x c= d by A40,YELLOW_1:3; DZ c= d proof let p be object; assume p in DZ; then p in {q where q is Element of L: ex u being Element of L st q <= u & u in Z} by WAYBEL_0:14; then consider p9 being Element of L such that A44: p9 = p and A45: ex u being Element of L st p9 <= u & u in Z; consider u being Element of L such that A46: p9 <= u and A47: u in Z by A45; consider z being Element of L such that A48: z = u and A49: z in x or z in y or ex a, b being Element of L st a in x & b in y & z = a "\/" b by A47; per cases by A49; suppose z in x; then p in x9 by A44,A46,A48,WAYBEL_0:def 19; hence thesis by A43; end; suppose z in y; then p in y9 by A44,A46,A48,WAYBEL_0:def 19; hence thesis by A42; end; suppose A50: ex a, b being Element of L st a in x & b in y & z = a "\/" b; reconsider d9= d as Ideal of L by Th41; u in d9 by A43,A42,A48,A50,WAYBEL_0:40; hence thesis by A44,A46,WAYBEL_0:def 19; end; end; hence thesis by YELLOW_1:3; end; y c= DZ proof let a be object; A51: Z c= DZ by WAYBEL_0:16; assume A52: a in y; then reconsider a9= a as Element of L by Th42; a9 in Z by A52; hence thesis by A51; end; then A53: y <= DZ by YELLOW_1:3; x c= DZ proof let a be object; A54: Z c= DZ by WAYBEL_0:16; assume A55: a in x; then reconsider a9= a as Element of L by Th42; a9 in Z by A55; hence thesis by A54; end; then x <= DZ by YELLOW_1:3; hence thesis by A53,A39,YELLOW_0:18; end; registration let L be with_suprema Poset; cluster InclPoset(Ids L) -> with_suprema; correctness proof set P = InclPoset(Ids L); for x, y being Element of P ex z being Element of P st x <= z & y <= z & for z9 being Element of P st x <= z9 & y <= z9 holds z <= z9 proof let x, y be Element of P; take x "\/" y; ex Z being Subset of L st Z = {z where z is Element of L: z in x or z in y or ex a, b being Element of L st a in x & b in y & z = a "\/" b} & ex_sup_of {x, y},InclPoset(Ids L) & x "\/" y = downarrow Z by Th44; hence thesis by YELLOW_0:18; end; hence thesis; end; end; theorem Th45: for L being lower-bounded sup-Semilattice for X being non empty Subset of Ids L holds meet X is Ideal of L proof let L be lower-bounded sup-Semilattice; let X be non empty Subset of Ids L; A1: for J being set st J in X holds J is Ideal of L proof let J be set; assume J in X; then J in Ids L; then ex Y being Ideal of L st Y = J; hence thesis; end; X c= bool the carrier of L proof let x be object; assume x in X; then x is Ideal of L by A1; hence thesis; end; then reconsider F = X as Subset-Family of L; for J being Subset of L st J in F holds J is lower by A1; then reconsider I = meet X as lower Subset of L by Th36; for J being set st J in X holds Bottom L in J proof let J be set; assume J in X; then reconsider J9= J as Ideal of L by A1; set j = the Element of J9; Bottom L <= j by YELLOW_0:44; hence thesis by WAYBEL_0:def 19; end; then reconsider I as non empty lower Subset of L by SETFAM_1:def 1; for J being Subset of L st J in F holds J is lower directed by A1; then I is Ideal of L by Th38; hence thesis; end; theorem Th46: for L being lower-bounded sup-Semilattice for A being non empty Subset of InclPoset(Ids L) holds ex_inf_of A,InclPoset(Ids L) & inf A = meet A proof let L be lower-bounded sup-Semilattice; let A be non empty Subset of InclPoset(Ids L); set P = InclPoset(Ids L); reconsider A9= A as non empty Subset of Ids L by YELLOW_1:1; meet A9 is Ideal of L by Th45; then reconsider I = meet A as Element of P by Th41; A1: for b being Element of P st b is_<=_than A holds I >= b proof let b be Element of P; assume A2: A is_>=_than b; A3: for J being set st J in A holds b c= J by A2,YELLOW_1:3; b c= I proof let c be object; assume A4: c in b; for J being set st J in A holds c in J proof let J be set; assume J in A; then b c= J by A3; hence thesis by A4; end; hence thesis by SETFAM_1:def 1; end; hence thesis by YELLOW_1:3; end; I is_<=_than A proof let y be Element of P; assume A5: y in A; I c= y by A5,SETFAM_1:def 1; hence I <= y by YELLOW_1:3; end; hence thesis by A1,YELLOW_0:31; end; theorem Th47: for L being with_suprema Poset holds ex_inf_of {},InclPoset(Ids L) & "/\"({}, InclPoset(Ids L)) = [#]L proof let L be with_suprema Poset; set P = InclPoset(Ids L); reconsider I = [#]L as Element of P by Th41; A1: for b being Element of P st b is_<=_than {} holds I >= b proof let b be Element of P; reconsider b9= b as Ideal of L by Th41; assume {} is_>=_than b; b9 c= [#]L; hence thesis by YELLOW_1:3; end; I is_<=_than {}; hence thesis by A1,YELLOW_0:31; end; theorem Th48: for L being lower-bounded sup-Semilattice holds InclPoset(Ids L) is complete proof let L be lower-bounded sup-Semilattice; set P = InclPoset(Ids L); for A being Subset of P holds ex_inf_of A,InclPoset(Ids L) proof let A be Subset of P; per cases; suppose A = {}; hence thesis by Th47; end; suppose A <> {}; hence thesis by Th46; end; end; hence thesis by Th28; end; registration let L be lower-bounded sup-Semilattice; cluster InclPoset(Ids L) -> complete; correctness by Th48; end; begin :: Special Maps definition let L be non empty Poset; func SupMap L -> Function of InclPoset(Ids L), L means :Def3: for I being Ideal of L holds it.I = sup I; existence proof deffunc F(set) = "\/"($1, L); set P = InclPoset(Ids L); A1: for I being set st I in the carrier of P holds F(I) in the carrier of L; ex f being Function of the carrier of P, the carrier of L st for I being set st I in the carrier of P holds f.I = F(I) from FUNCT_2:sch 11(A1); then consider f being Function of the carrier of P, the carrier of L such that A2: for I being set st I in the carrier of P holds f.I = "\/"(I, L); reconsider f as Function of P, L; take f; for I being Ideal of L holds f.I = sup I proof let I be Ideal of L; I in the carrier of RelStr(#Ids L, RelIncl Ids L#); then I in the carrier of P by YELLOW_1:def 1; hence thesis by A2; end; hence thesis; end; uniqueness proof set P = InclPoset(Ids L); let f, g be Function of InclPoset(Ids L), L; assume that A3: for I being Ideal of L holds f.I = sup I and A4: for I being Ideal of L holds g.I = sup I; A5: the carrier of P = the carrier of RelStr(#Ids L, RelIncl(Ids L)#) by YELLOW_1:def 1 .= Ids L; A6: now let x be object; assume x in the carrier of P; then ex I being Ideal of L st x = I by A5; then reconsider I = x as Ideal of L; f.I = sup I by A3; hence f.x = g.x by A4; end; dom f = the carrier of P & dom g = the carrier of P by FUNCT_2:def 1; hence f = g by A6,FUNCT_1:2; end; end; theorem Th49: for L being non empty Poset holds dom SupMap L = Ids L & rng SupMap L is Subset of L proof let L be non empty Poset; set P = InclPoset(Ids L); thus dom(SupMap L) = the carrier of P by FUNCT_2:def 1 .= the carrier of RelStr(#Ids L, RelIncl(Ids L)#) by YELLOW_1:def 1 .= Ids L; thus thesis; end; theorem for L being non empty Poset holds x in dom SupMap L iff x is Ideal of L proof let L be non empty Poset; hereby assume x in dom SupMap L; then x in Ids L by Th49; then ex I being Ideal of L st x = I; hence x is Ideal of L; end; assume x is Ideal of L; then x in the set of all X where X is Ideal of L; hence thesis by Th49; end; theorem Th51: for L being up-complete non empty Poset holds SupMap L is monotone proof let L be up-complete non empty Poset; set P = InclPoset Ids L; set f = SupMap L; for x, y being Element of P st x <= y for a, b being Element of L st a = f.x & b = f.y holds a <= b proof let x, y be Element of P such that A1: x <= y; reconsider I = x, J = y as Ideal of L by Th41; A2: I c= J by A1,YELLOW_1:3; A3: ex_sup_of I,L & ex_sup_of J,L by WAYBEL_0:75; A4: f.x = sup I & f.y = sup J by Def3; let a, b be Element of L; assume a = f.x & b = f.y; hence thesis by A3,A2,A4,YELLOW_0:34; end; hence thesis; end; registration let L be up-complete non empty Poset; cluster SupMap L -> monotone; correctness by Th51; end; definition let L be non empty Poset; func IdsMap L -> Function of L, InclPoset(Ids L) means :Def4: for x being Element of L holds it.x = downarrow x; existence proof deffunc F(Element of L) = downarrow $1; A1: for x being Element of L holds F(x) is Element of InclPoset(Ids L) by Th41; thus ex m being Function of L, InclPoset Ids L st for x being Element of L holds m.x = F(x) from FUNCT_2:sch 9(A1); end; uniqueness proof let f1,f2 be Function of L,InclPoset(Ids L) such that A2: for x being Element of L holds f1.x = downarrow x and A3: for x being Element of L holds f2.x = downarrow x; now let x be Element of L; thus f1.x = downarrow x by A2 .= f2.x by A3; end; hence thesis by FUNCT_2:63; end; end; theorem Th52: for L being non empty Poset holds IdsMap L is monotone proof let L be non empty Poset; let x1,x2 be Element of L; assume x1 <= x2; then downarrow x1 c= downarrow x2 by WAYBEL_0:21; then (IdsMap L).x1 c= downarrow x2 by Def4; then A1: (IdsMap L).x1 c= (IdsMap L).x2 by Def4; let a, b be Element of InclPoset(Ids L); assume a = (IdsMap L).x1 & b = (IdsMap L).x2; hence a <= b by A1,YELLOW_1:3; end; registration let L be non empty Poset; cluster IdsMap L -> monotone; correctness by Th52; end; begin :: A Family of Elements in a Lattice definition let L be non empty RelStr; let F be Relation; func \\/(F, L) -> Element of L equals "\/"(rng F, L); coherence; func //\(F, L) -> Element of L equals "/\"(rng F, L); coherence; end; notation let J be set, L be non empty RelStr; let F be Function of J, the carrier of L; synonym Sup F for \\/(F, L); synonym Inf F for //\(F, L); end; reserve J for non empty set, j for Element of J; theorem for F being Function of J, the carrier of L holds F.j <= Sup F & Inf F <= F.j proof let F be Function of J, the carrier of L; F.j in rng F by FUNCT_2:4; hence thesis by Th22; end; theorem for F being Function of J, the carrier of L holds (for j holds F.j <= a) implies Sup F <= a proof let F be Function of J, the carrier of L; assume A1: for j holds F.j <= a; now let c be Element of L; assume c in rng F; then consider j being object such that A2: j in dom F and A3: c = F.j by FUNCT_1:def 3; reconsider j as Element of J by A2; c = F.j by A3; hence c <= a by A1; end; then rng F is_<=_than a; hence thesis by YELLOW_0:32; end; theorem for F being Function of J, the carrier of L holds (for j holds a <= F. j) implies a <= Inf F proof let F be Function of J, the carrier of L; assume A1: for j holds a <= F.j; now let c be Element of L; assume c in rng F; then consider j being object such that A2: j in dom F and A3: c = F.j by FUNCT_1:def 3; reconsider j as Element of J by A2; c = F.j by A3; hence a <= c by A1; end; then a is_<=_than rng F; hence thesis by YELLOW_0:33; end;