:: Algebraic Lattices
:: by Robert Milewski
::
:: Received December 1, 1996
:: Copyright (c) 1996-2021 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, RELAT_2, ORDERS_2, STRUCT_0, CAT_1, YELLOW_0, SUBSET_1,
RCOMP_1, WELLORD1, TARSKI, LATTICES, REWRITE1, WAYBEL_0, XXREAL_0,
WAYBEL_3, WAYBEL_6, CARD_FIL, LATTICE3, EQREL_1, ORDINAL2, FINSET_1,
ZFMISC_1, WAYBEL_4, MSSUBFAM, WAYBEL_7, INT_2, WAYBEL_1, FUNCT_1,
GROUP_6, RELAT_1, BINOP_1, SEQM_3, YELLOW_1, FILTER_1, SETFAM_1,
WAYBEL_8, CARD_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, CARD_1, FINSET_1,
RELAT_1, TOLER_1, FUNCT_1, RELSET_1, FUNCT_2, DOMAIN_1, RELAT_2,
STRUCT_0, ORDERS_2, LATTICE3, QUANTAL1, YELLOW_0, YELLOW_1, YELLOW_2,
WAYBEL_0, WAYBEL_1, WAYBEL_3, WAYBEL_4, WAYBEL_6, WAYBEL_7;
constructors DOMAIN_1, TOLER_1, QUANTAL1, ORDERS_3, WAYBEL_1, YELLOW_3,
WAYBEL_3, WAYBEL_4, WAYBEL_6, WAYBEL_7, RELSET_1, PRALG_1;
registrations XBOOLE_0, SUBSET_1, RELSET_1, FINSET_1, STRUCT_0, LATTICE3,
YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_2, WAYBEL_1, WAYBEL_2, WAYBEL_3,
WAYBEL_4, WAYBEL_6;
requirements SUBSET, BOOLE;
begin :: The Subset of All Compact Elements
definition :: DEFINITION 4.1
let L be non empty reflexive RelStr;
func CompactSublatt L -> strict full SubRelStr of L means
:: WAYBEL_8:def 1
for x be Element of L holds x in the carrier of it iff x is compact;
end;
registration
let L be lower-bounded non empty reflexive antisymmetric RelStr;
cluster CompactSublatt L -> non empty;
end;
theorem :: WAYBEL_8:1
for L be complete LATTICE for x,y,k be Element of L holds x <= k & k
<= y & k in the carrier of CompactSublatt L implies x << y;
theorem :: WAYBEL_8:2 :: REMARK 4.2
for L be complete LATTICE for x be Element of L holds uparrow x is
Open Filter of L iff x is compact;
theorem :: WAYBEL_8:3 :: REMARK 4.3
for L be lower-bounded with_suprema non empty Poset holds
CompactSublatt L is join-inheriting & Bottom L in the carrier of CompactSublatt
L;
definition
let L be non empty reflexive RelStr;
let x be Element of L;
func compactbelow x -> Subset of L equals
:: WAYBEL_8:def 2
{y where y is Element of L: x >= y
& y is compact};
end;
theorem :: WAYBEL_8:4
for L be non empty reflexive RelStr for x,y be Element of L holds
y in compactbelow x iff x >= y & y is compact;
theorem :: WAYBEL_8:5
for L be non empty reflexive RelStr for x be Element of L holds
compactbelow x = downarrow x /\ the carrier of CompactSublatt L;
theorem :: WAYBEL_8:6
for L be non empty reflexive transitive RelStr for x be Element
of L holds compactbelow x c= waybelow x;
registration
let L be non empty lower-bounded reflexive antisymmetric RelStr;
let x be Element of L;
cluster compactbelow x -> non empty;
end;
begin :: Algebraic Lattices
definition
let L be non empty reflexive RelStr;
attr L is satisfying_axiom_K means
:: WAYBEL_8:def 3
for x be Element of L holds x = sup compactbelow x;
end;
definition :: DEFINITION 4.4
let L be non empty reflexive RelStr;
attr L is algebraic means
:: WAYBEL_8:def 4
(for x being Element of L holds
compactbelow x is non empty directed) & L is up-complete satisfying_axiom_K;
end;
theorem :: WAYBEL_8:7 :: PROPOSITION 4.5
for L be LATTICE holds L is algebraic iff ( L is continuous & for
x,y be Element of L st x << y ex k be Element of L st k in the carrier of
CompactSublatt L & x <= k & k <= y );
registration
cluster algebraic -> continuous for LATTICE;
end;
registration
cluster algebraic -> up-complete satisfying_axiom_K for non empty reflexive
RelStr;
end;
registration
let L be non empty with_suprema Poset;
cluster CompactSublatt L -> join-inheriting;
end;
definition
let L be non empty reflexive RelStr;
attr L is arithmetic means
:: WAYBEL_8:def 5
L is algebraic & CompactSublatt L is meet-inheriting;
end;
begin :: Arithmetic Lattices
registration
cluster arithmetic -> algebraic for LATTICE;
end;
registration
cluster trivial -> arithmetic for LATTICE;
end;
registration
cluster 1-element strict for LATTICE;
end;
theorem :: WAYBEL_8:8
for L1,L2 be non empty reflexive antisymmetric RelStr st the
RelStr of L1 = the RelStr of L2 & L1 is up-complete for x1,y1 be Element of L1
for x2,y2 be Element of L2 st x1 = x2 & y1 = y2 & x1 << y1 holds x2 << y2;
theorem :: WAYBEL_8:9
for L1,L2 be non empty reflexive antisymmetric RelStr st the
RelStr of L1 = the RelStr of L2 & L1 is up-complete for x be Element of L1 for
y be Element of L2 st x = y & x is compact holds y is compact;
theorem :: WAYBEL_8:10
for L1,L2 be up-complete non empty reflexive antisymmetric
RelStr st the RelStr of L1 = the RelStr of L2 for x be Element of L1 for y be
Element of L2 st x = y holds compactbelow x = compactbelow y;
theorem :: WAYBEL_8:11
for L1,L2 be RelStr st the RelStr of L1 = the RelStr of L2 & L1 is non
empty holds L2 is non empty;
theorem :: WAYBEL_8:12
for L1,L2 be RelStr st the RelStr of L1 = the RelStr of L2 & L1
is reflexive holds L2 is reflexive;
theorem :: WAYBEL_8:13
for L1,L2 be RelStr st the RelStr of L1 = the RelStr of L2 & L1
is transitive holds L2 is transitive;
theorem :: WAYBEL_8:14
for L1,L2 be RelStr st the RelStr of L1 = the RelStr of L2 & L1
is antisymmetric holds L2 is antisymmetric;
theorem :: WAYBEL_8:15
for L1,L2 be non empty reflexive RelStr st the RelStr of L1 =
the RelStr of L2 & L1 is up-complete holds L2 is up-complete;
theorem :: WAYBEL_8:16
for L1,L2 be up-complete non empty reflexive antisymmetric
RelStr st the RelStr of L1 = the RelStr of L2 & L1 is satisfying_axiom_K & for
x be Element of L1 holds compactbelow x is non empty directed holds L2 is
satisfying_axiom_K;
theorem :: WAYBEL_8:17
for L1,L2 be non empty reflexive antisymmetric RelStr st the
RelStr of L1 = the RelStr of L2 & L1 is algebraic holds L2 is algebraic;
theorem :: WAYBEL_8:18
for L1,L2 be LATTICE st the RelStr of L1 = the RelStr of L2 & L1
is arithmetic holds L2 is arithmetic;
registration
let L be non empty RelStr;
cluster the RelStr of L -> non empty;
end;
registration
let L be non empty reflexive RelStr;
cluster the RelStr of L -> reflexive;
end;
registration
let L be transitive RelStr;
cluster the RelStr of L -> transitive;
end;
registration
let L be antisymmetric RelStr;
cluster the RelStr of L -> antisymmetric;
end;
registration
let L be with_infima RelStr;
cluster the RelStr of L -> with_infima;
end;
registration
let L be with_suprema RelStr;
cluster the RelStr of L -> with_suprema;
end;
registration
let L be up-complete non empty reflexive RelStr;
cluster the RelStr of L -> up-complete;
end;
registration
let L be algebraic non empty reflexive antisymmetric RelStr;
cluster the RelStr of L -> algebraic;
end;
registration
let L be arithmetic LATTICE;
cluster the RelStr of L -> arithmetic;
end;
theorem :: WAYBEL_8:19 :: PROPOSITION 4.7 a)
for L be algebraic LATTICE holds L is arithmetic iff CompactSublatt L
is LATTICE;
theorem :: WAYBEL_8:20 :: PROPOSITION 4.7 b)
for L be algebraic lower-bounded LATTICE holds L is arithmetic
iff L-waybelow is multiplicative;
theorem :: WAYBEL_8:21 :: COROLLARY 4.8.a)
for L be arithmetic lower-bounded LATTICE, p be Element of L holds p
is pseudoprime implies p is prime;
theorem :: WAYBEL_8:22 :: COROLLARY 4.8.b)
for L be algebraic distributive lower-bounded LATTICE st for p being
Element of L st p is pseudoprime holds p is prime holds L is arithmetic;
registration
let L be algebraic LATTICE;
let c be closure Function of L,L;
cluster non empty directed for Subset of Image c;
end;
theorem :: WAYBEL_8:23
for L be algebraic LATTICE for c be closure Function of L,L st c
is directed-sups-preserving holds c.:([#]CompactSublatt L) c= [#]CompactSublatt
Image c;
theorem :: WAYBEL_8:24 :: PROPOSITION 4.9.(i)
for L be algebraic lower-bounded LATTICE for c be closure Function of
L,L st c is directed-sups-preserving holds Image c is algebraic LATTICE;
theorem :: WAYBEL_8:25 :: PROPOSITION 4.9.(ii)
for L be algebraic lower-bounded LATTICE, c be closure Function of L,L
st c is directed-sups-preserving holds c.:([#]CompactSublatt L) = [#]
CompactSublatt Image c;
begin :: Boolean Posets as Algebraic Lattices
theorem :: WAYBEL_8:26
for X,x be set holds x is Element of BoolePoset X iff x c= X;
theorem :: WAYBEL_8:27
for X be set for x,y be Element of BoolePoset X holds x << y iff
for Y be Subset-Family of X st y c= union Y ex Z be finite Subset of Y st x c=
union Z;
theorem :: WAYBEL_8:28
for X be set for x be Element of BoolePoset X holds x is finite
iff x is compact;
theorem :: WAYBEL_8:29
for X be set for x be Element of BoolePoset X holds compactbelow
x = the set of all y where y is finite Subset of x;
theorem :: WAYBEL_8:30
for X be set for F be Subset of X holds F in the carrier of
CompactSublatt BoolePoset X iff F is finite;
registration
let X be set;
let x be Element of BoolePoset X;
cluster compactbelow x -> lower directed;
end;
theorem :: WAYBEL_8:31 :: EXAMPLES 4.11.(1b)
for X be set holds BoolePoset X is algebraic;
registration
let X be set;
cluster BoolePoset X -> algebraic;
end;