:: Basic Notions and Properties of Orthoposets
:: by Markus Moschner
::
:: Received February 11, 2003
:: Copyright (c) 2003-2012 Association of Mizar Users


begin

:: Some basic definitions and theorems for later examples;
theorem :: OPOSET_1:1
for L being non empty reflexive antisymmetric RelStr
for x, y being Element of L st x <= y holds
( sup {x,y} = y &amp; inf {x,y} = x )
proofend;

:: for various types of relations needed for Posets
:: Some existence conditions on non-empty relations
registration
let X be set ;
cluster Relation-like irreflexive asymmetric transitive for Element of K19(([#] (X,X)));
existence
 ex b1 being Relation of X st
( b1 is irreflexive &amp; b1 is asymmetric &amp; b1 is transitive )
proofend;
end;

registration
let X be non empty set ;
let R be Relation of X;
let C be UnOp of X;
cluster OrthoRelStr(# X,R,C #) ->  non empty ;
coherence
 not OrthoRelStr(# X,R,C #) is empty ;
end;

registration
cluster non empty strict for OrthoRelStr ;
existence
 ex b1 being OrthoRelStr st
( not b1 is empty &amp; b1 is strict )
proofend;
end;

:: Double negation property of the internal Complement
registration
let O be set ;
cluster Relation-like Function-like V14(O) V18(O,O) involutive for Element of K19(([#] (O,O)));
existence
 ex b1 being Function of O,O st b1 is involutive
proofend;
end;

:: Small example structures
definition
func TrivOrthoRelStr  ->  strict OrthoRelStr  equals :Def1: :: OPOSET_1:def 1
 OrthoRelStr(# 1,(id 1),op1 #);
coherence
 OrthoRelStr(# 1,(id 1),op1 #) is strict OrthoRelStr ;
end;

:: deftheorem Def1 defines TrivOrthoRelStr OPOSET_1:def_1_:_
 TrivOrthoRelStr = OrthoRelStr(# 1,(id 1),op1 #);

notation
synonym  TrivPoset for  TrivOrthoRelStr ;
end;

registration
cluster TrivOrthoRelStr  -> 1 -element strict ;
coherence
 TrivOrthoRelStr is 1 -element
proofend;
end;

definition
func TrivAsymOrthoRelStr  ->  strict OrthoRelStr  equals :: OPOSET_1:def 2
 OrthoRelStr(# 1,({} (1,1)),op1 #);
coherence
 OrthoRelStr(# 1,({} (1,1)),op1 #) is strict OrthoRelStr ;
end;

:: deftheorem  defines TrivAsymOrthoRelStr OPOSET_1:def_2_:_
 TrivAsymOrthoRelStr = OrthoRelStr(# 1,({} (1,1)),op1 #);

registration
cluster TrivAsymOrthoRelStr  ->  non empty strict ;
coherence
 not TrivAsymOrthoRelStr is empty ;
end;

definition
let O be non empty OrthoRelStr ;
attrO is Dneg  means :Def3: :: OPOSET_1:def 3
 ex f being Function of O,O st
( f = the Compl of O &amp; f is V252() );
end;

:: deftheorem Def3 defines Dneg OPOSET_1:def_3_:_
 for O being non empty OrthoRelStr holds
( O is Dneg iff ex f being Function of O,O st
( f = the Compl of O &amp; f is V252() ) );

registration
cluster TrivOrthoRelStr  ->  strict Dneg ;
coherence
 TrivOrthoRelStr is Dneg by Def3;
end;

registration
cluster non empty Dneg for OrthoRelStr ;
existence
 ex b1 being non empty OrthoRelStr st b1 is Dneg by Def1;
end;

:: InternalRel based properties
definition
let O be non empty RelStr ;
attrO is SubReFlexive  means :Def4: :: OPOSET_1:def 4
 the InternalRel of O is reflexive ;
end;

:: deftheorem Def4 defines SubReFlexive OPOSET_1:def_4_:_
 for O being non empty RelStr holds
( O is SubReFlexive iff the InternalRel of O is reflexive );

theorem :: OPOSET_1:2
for O being non empty RelStr st O is reflexive holds
O is SubReFlexive
proofend;

theorem Th3: :: OPOSET_1:3
TrivOrthoRelStr is reflexive
proofend;

registration
cluster TrivOrthoRelStr  ->  reflexive strict ;
coherence
 TrivOrthoRelStr is reflexive by Th3;
end;

registration
cluster non empty reflexive strict for OrthoRelStr ;
existence
 ex b1 being non empty OrthoRelStr st
( b1 is reflexive &amp; b1 is strict ) by Th3;
end;

definition
let O be non empty RelStr ;
redefine attr O is irreflexive  means :Def5: :: OPOSET_1:def 5
 the InternalRel of O is irreflexive ;
compatibility
 ( O is irreflexive iff the InternalRel of O is irreflexive )
proofend;
redefine attr O is irreflexive  means :Def6: :: OPOSET_1:def 6
 the InternalRel of O is_irreflexive_in the carrier of O;
compatibility
 ( O is irreflexive iff the InternalRel of O is_irreflexive_in the carrier of O )
proofend;
end;

:: deftheorem Def5 defines irreflexive OPOSET_1:def_5_:_
 for O being non empty RelStr holds
( O is irreflexive iff the InternalRel of O is irreflexive );

:: deftheorem Def6 defines irreflexive OPOSET_1:def_6_:_
 for O being non empty RelStr holds
( O is irreflexive iff the InternalRel of O is_irreflexive_in the carrier of O );

theorem Th4: :: OPOSET_1:4
TrivAsymOrthoRelStr is irreflexive
proofend;

registration
cluster TrivAsymOrthoRelStr  ->  irreflexive strict ;
coherence
 TrivAsymOrthoRelStr is irreflexive by Th4;
end;

registration
cluster non empty irreflexive strict for OrthoRelStr ;
existence
 ex b1 being non empty OrthoRelStr st
( b1 is irreflexive &amp; b1 is strict ) by Th4;
end;

:: Symmetry
definition
let O be non empty RelStr ;
redefine attr O is symmetric  means :Def7: :: OPOSET_1:def 7
 the InternalRel of O is symmetric Relation of the carrier of O;
compatibility
 ( O is symmetric iff the InternalRel of O is symmetric Relation of the carrier of O )
proofend;
end;

:: deftheorem Def7 defines symmetric OPOSET_1:def_7_:_
 for O being non empty RelStr holds
( O is symmetric iff the InternalRel of O is symmetric Relation of the carrier of O );

theorem Th5: :: OPOSET_1:5
TrivOrthoRelStr is symmetric by PARTIT_2:23, NECKLACE:def_3;

registration
cluster non empty symmetric strict for OrthoRelStr ;
existence
 ex b1 being non empty OrthoRelStr st
( b1 is symmetric &amp; b1 is strict ) by Th5;
end;

:: Antisymmetry
definition
let O be non empty RelStr ;
redefine attr O is antisymmetric  means :: OPOSET_1:def 8
 the InternalRel of O is antisymmetric Relation of the carrier of O;
compatibility
 ( O is antisymmetric iff the InternalRel of O is antisymmetric Relation of the carrier of O )
proofend;
end;

:: deftheorem  defines antisymmetric OPOSET_1:def_8_:_
 for O being non empty RelStr holds
( O is antisymmetric iff the InternalRel of O is antisymmetric Relation of the carrier of O );

Lm1:  TrivOrthoRelStr is antisymmetric
;

registration
cluster TrivOrthoRelStr  ->  symmetric strict ;
coherence
 TrivOrthoRelStr is symmetric by PARTIT_2:23, NECKLACE:def_3;
end;

registration
cluster non empty antisymmetric symmetric strict for OrthoRelStr ;
existence
 ex b1 being non empty OrthoRelStr st
( b1 is symmetric &amp; b1 is antisymmetric &amp; b1 is strict ) by Lm1;
end;

:: Asymmetry
definition
let O be non empty RelStr ;
redefine attr O is asymmetric  means :Def9: :: OPOSET_1:def 9
 the InternalRel of O is_asymmetric_in the carrier of O;
compatibility
 ( O is asymmetric iff the InternalRel of O is_asymmetric_in the carrier of O )
proofend;
end;

:: deftheorem Def9 defines asymmetric OPOSET_1:def_9_:_
 for O being non empty RelStr holds
( O is asymmetric iff the InternalRel of O is_asymmetric_in the carrier of O );

theorem Th6: :: OPOSET_1:6
TrivAsymOrthoRelStr is asymmetric by PARTIT_2:28, Def9;

registration
cluster TrivAsymOrthoRelStr  ->  asymmetric strict ;
coherence
 TrivAsymOrthoRelStr is asymmetric by PARTIT_2:28, Def9;
end;

registration
cluster non empty asymmetric strict for OrthoRelStr ;
existence
 ex b1 being non empty OrthoRelStr st
( b1 is asymmetric &amp; b1 is strict ) by Th6;
end;

:: Transitivity
definition
let O be non empty RelStr ;
redefine attr O is transitive  means :Def10: :: OPOSET_1:def 10
 the InternalRel of O is transitive Relation of the carrier of O;
compatibility
 ( O is transitive iff the InternalRel of O is transitive Relation of the carrier of O )
proofend;
end;

:: deftheorem Def10 defines transitive OPOSET_1:def_10_:_
 for O being non empty RelStr holds
( O is transitive iff the InternalRel of O is transitive Relation of the carrier of O );

registration
cluster non empty reflexive transitive antisymmetric symmetric strict for OrthoRelStr ;
existence
 ex b1 being non empty OrthoRelStr st
( b1 is reflexive &amp; b1 is symmetric &amp; b1 is antisymmetric &amp; b1 is transitive &amp; b1 is strict ) by Lm1;
end;

theorem :: OPOSET_1:7
TrivAsymOrthoRelStr is transitive by PARTIT_2:26, ORDERS_2:def_3;

registration
cluster TrivAsymOrthoRelStr  ->  transitive asymmetric irreflexive strict ;
coherence
 ( TrivAsymOrthoRelStr is irreflexive &amp; TrivAsymOrthoRelStr is asymmetric &amp; TrivAsymOrthoRelStr is transitive ) by PARTIT_2:26, ORDERS_2:def_3;
end;

registration
cluster non empty transitive asymmetric irreflexive strict for OrthoRelStr ;
existence
 ex b1 being non empty OrthoRelStr st
( b1 is irreflexive &amp; b1 is asymmetric &amp; b1 is transitive &amp; b1 is strict ) by Th4;
end;

theorem :: OPOSET_1:8
for O being non empty RelStr st O is symmetric &amp; O is transitive holds
O is SubReFlexive
proofend;

theorem :: OPOSET_1:9
for O being non empty RelStr st O is irreflexive &amp; O is transitive holds
O is asymmetric
proofend;

theorem Th10: :: OPOSET_1:10
for O being non empty RelStr st O is asymmetric holds
O is irreflexive
proofend;

begin

:: Quasiorder (Preorder)
definition
let O be non empty RelStr ;
attrO is SubQuasiOrdered  means :Def11: :: OPOSET_1:def 11
( O is SubReFlexive &amp; O is transitive );
end;

:: deftheorem Def11 defines SubQuasiOrdered OPOSET_1:def_11_:_
 for O being non empty RelStr holds
( O is SubQuasiOrdered iff ( O is SubReFlexive &amp; O is transitive ) );

notation
let O be non empty RelStr ;
synonym  SubPreOrdered O for  SubQuasiOrdered ;
end;

definition
let O be non empty RelStr ;
attrO is QuasiOrdered  means :Def12: :: OPOSET_1:def 12
( O is reflexive &amp; O is transitive );
end;

:: deftheorem Def12 defines QuasiOrdered OPOSET_1:def_12_:_
 for O being non empty RelStr holds
( O is QuasiOrdered iff ( O is reflexive &amp; O is transitive ) );

notation
let O be non empty RelStr ;
synonym  PreOrdered O for  QuasiOrdered ;
end;

theorem Th11: :: OPOSET_1:11
for O being non empty RelStr st O is QuasiOrdered holds
O is SubQuasiOrdered
proofend;

registration
cluster non empty QuasiOrdered  ->  non empty SubQuasiOrdered for OrthoRelStr ;
correctness
coherence
for b1 being non empty OrthoRelStr st b1 is QuasiOrdered holds
b1 is SubQuasiOrdered ;
by Th11;
end;

registration
cluster TrivOrthoRelStr  ->  strict QuasiOrdered ;
coherence
 TrivOrthoRelStr is QuasiOrdered by Def12;
end;

:: QuasiPure means complementation-order-like combination of properties
definition
let O be non empty OrthoRelStr ;
attrO is QuasiPure  means :Def13: :: OPOSET_1:def 13
( O is Dneg &amp; O is QuasiOrdered );
end;

:: deftheorem Def13 defines QuasiPure OPOSET_1:def_13_:_
 for O being non empty OrthoRelStr holds
( O is QuasiPure iff ( O is Dneg &amp; O is QuasiOrdered ) );

registration
cluster non empty strict Dneg QuasiOrdered QuasiPure for OrthoRelStr ;
existence
 ex b1 being non empty OrthoRelStr st
( b1 is QuasiPure &amp; b1 is Dneg &amp; b1 is QuasiOrdered &amp; b1 is strict )
proofend;
end;

registration
cluster TrivOrthoRelStr  ->  strict QuasiPure ;
coherence
 TrivOrthoRelStr is QuasiPure by Def13;
end;

definition
mode QuasiPureOrthoRelStr is non empty QuasiPure OrthoRelStr ;
end;

:: Partial Order ---> Poset
definition
let O be non empty OrthoRelStr ;
attrO is PartialOrdered  means :Def14: :: OPOSET_1:def 14
( O is reflexive &amp; O is antisymmetric &amp; O is transitive );
end;

:: deftheorem Def14 defines PartialOrdered OPOSET_1:def_14_:_
 for O being non empty OrthoRelStr holds
( O is PartialOrdered iff ( O is reflexive &amp; O is antisymmetric &amp; O is transitive ) );

registration
cluster non empty PartialOrdered  ->  non empty reflexive transitive antisymmetric for OrthoRelStr ;
coherence
 for b1 being non empty OrthoRelStr st b1 is PartialOrdered holds
( b1 is reflexive &amp; b1 is antisymmetric &amp; b1 is transitive ) by Def14;
cluster non empty reflexive transitive antisymmetric  ->  non empty PartialOrdered for OrthoRelStr ;
coherence
 for b1 being non empty OrthoRelStr st b1 is reflexive &amp; b1 is antisymmetric &amp; b1 is transitive holds
b1 is PartialOrdered by Def14;
end;

:: Pureness for partial orders
definition
let O be non empty OrthoRelStr ;
attrO is Pure  means :Def15: :: OPOSET_1:def 15
( O is Dneg &amp; O is PartialOrdered );
end;

:: deftheorem Def15 defines Pure OPOSET_1:def_15_:_
 for O being non empty OrthoRelStr holds
( O is Pure iff ( O is Dneg &amp; O is PartialOrdered ) );

registration
cluster non empty strict Dneg PartialOrdered Pure for OrthoRelStr ;
existence
 ex b1 being non empty OrthoRelStr st
( b1 is Pure &amp; b1 is Dneg &amp; b1 is PartialOrdered &amp; b1 is strict )
proofend;
end;

registration
cluster TrivOrthoRelStr  ->  strict Pure ;
coherence
 TrivOrthoRelStr is Pure by Def15;
end;

definition
mode PureOrthoRelStr is non empty Pure OrthoRelStr ;
end;

:: Strict Poset
definition
let O be non empty OrthoRelStr ;
attrO is StrictPartialOrdered  means :Def16: :: OPOSET_1:def 16
( O is asymmetric &amp; O is transitive );
end;

:: deftheorem Def16 defines StrictPartialOrdered OPOSET_1:def_16_:_
 for O being non empty OrthoRelStr holds
( O is StrictPartialOrdered iff ( O is asymmetric &amp; O is transitive ) );

notation
let O be non empty OrthoRelStr ;
synonym  StrictOrdered O for  StrictPartialOrdered ;
end;

theorem Th12: :: OPOSET_1:12
for O being non empty OrthoRelStr st O is StrictPartialOrdered holds
O is irreflexive
proofend;

registration
cluster non empty StrictPartialOrdered  ->  non empty irreflexive for OrthoRelStr ;
coherence
 for b1 being non empty OrthoRelStr st b1 is StrictPartialOrdered holds
b1 is irreflexive by Th12;
end;

registration
cluster non empty StrictPartialOrdered  ->  non empty irreflexive for OrthoRelStr ;
coherence
 for b1 being non empty OrthoRelStr st b1 is StrictPartialOrdered holds
b1 is irreflexive ;
end;

registration
cluster TrivAsymOrthoRelStr  ->  irreflexive strict StrictPartialOrdered ;
coherence
 ( TrivAsymOrthoRelStr is irreflexive &amp; TrivAsymOrthoRelStr is StrictPartialOrdered ) by Def16;
end;

registration
cluster non empty irreflexive strict StrictPartialOrdered for OrthoRelStr ;
existence
 ex b1 being non empty strict OrthoRelStr st
( b1 is irreflexive &amp; b1 is StrictPartialOrdered )
proofend;
end;

theorem :: OPOSET_1:13
for QO being non empty QuasiOrdered OrthoRelStr st QO is antisymmetric holds
QO is PartialOrdered
proofend;

registration
cluster non empty PartialOrdered  ->  non empty reflexive transitive antisymmetric for OrthoRelStr ;
correctness
coherence
for b1 being non empty OrthoRelStr st b1 is PartialOrdered holds
( b1 is reflexive &amp; b1 is transitive &amp; b1 is antisymmetric );
;
end;

definition
let PO be non empty RelStr ;
let f be UnOp of the carrier of PO;
attrf is Orderinvolutive  means :Def17: :: OPOSET_1:def 17
( f is V252() &amp; f is antitone );
end;

:: deftheorem Def17 defines Orderinvolutive OPOSET_1:def_17_:_
 for PO being non empty RelStr
for f being UnOp of the carrier of PO holds
( f is Orderinvolutive iff ( f is V252() &amp; f is antitone ) );

definition
let PO be non empty OrthoRelStr ;
attrPO is OrderInvolutive  means :Def18: :: OPOSET_1:def 18
 the Compl of PO is Orderinvolutive ;
end;

:: deftheorem Def18 defines OrderInvolutive OPOSET_1:def_18_:_
 for PO being non empty OrthoRelStr holds
( PO is OrderInvolutive iff the Compl of PO is Orderinvolutive );

theorem Th14: :: OPOSET_1:14
the Compl of TrivOrthoRelStr is Orderinvolutive
proofend;

registration
cluster TrivOrthoRelStr  ->  strict OrderInvolutive ;
coherence
 TrivOrthoRelStr is OrderInvolutive by Def18, Th14;
end;

registration
cluster non empty PartialOrdered Pure OrderInvolutive for OrthoRelStr ;
existence
 ex b1 being non empty OrthoRelStr st
( b1 is OrderInvolutive &amp; b1 is Pure &amp; b1 is PartialOrdered )
proofend;
end;

definition
mode PreOrthoPoset is non empty PartialOrdered Pure OrderInvolutive OrthoRelStr ;
end;

definition
let PO be non empty RelStr ;
let f be UnOp of the carrier of PO;
predf QuasiOrthoComplement_on PO means :Def19: :: OPOSET_1:def 19
( f is Orderinvolutive &amp; ( for y being Element of PO holds
( ex_sup_of {y,(f . y)},PO &amp; ex_inf_of {y,(f . y)},PO ) ) );
end;

:: deftheorem Def19 defines QuasiOrthoComplement_on OPOSET_1:def_19_:_
 for PO being non empty RelStr
for f being UnOp of the carrier of PO holds
( f QuasiOrthoComplement_on PO iff ( f is Orderinvolutive &amp; ( for y being Element of PO holds
( ex_sup_of {y,(f . y)},PO &amp; ex_inf_of {y,(f . y)},PO ) ) ) );

definition
let PO be non empty OrthoRelStr ;
attrPO is QuasiOrthocomplemented  means :Def20: :: OPOSET_1:def 20
 ex f being Function of PO,PO st
( f = the Compl of PO &amp; f QuasiOrthoComplement_on PO );
end;

:: deftheorem Def20 defines QuasiOrthocomplemented OPOSET_1:def_20_:_
 for PO being non empty OrthoRelStr holds
( PO is QuasiOrthocomplemented iff ex f being Function of PO,PO st
( f = the Compl of PO &amp; f QuasiOrthoComplement_on PO ) );

Lm2:  id {{}} = {[{},{}]}
by SYSREL:13;

theorem Th15: :: OPOSET_1:15
TrivOrthoRelStr is QuasiOrthocomplemented
proofend;

definition
let PO be non empty RelStr ;
let f be UnOp of the carrier of PO;
predf OrthoComplement_on PO means :Def21: :: OPOSET_1:def 21
( f is Orderinvolutive &amp; ( for y being Element of PO holds
( ex_sup_of {y,(f . y)},PO &amp; ex_inf_of {y,(f . y)},PO &amp; "\/" ({y,(f . y)},PO) is_maximum_of the carrier of PO &amp; "/\" ({y,(f . y)},PO) is_minimum_of the carrier of PO ) ) );
end;

:: deftheorem Def21 defines OrthoComplement_on OPOSET_1:def_21_:_
 for PO being non empty RelStr
for f being UnOp of the carrier of PO holds
( f OrthoComplement_on PO iff ( f is Orderinvolutive &amp; ( for y being Element of PO holds
( ex_sup_of {y,(f . y)},PO &amp; ex_inf_of {y,(f . y)},PO &amp; "\/" ({y,(f . y)},PO) is_maximum_of the carrier of PO &amp; "/\" ({y,(f . y)},PO) is_minimum_of the carrier of PO ) ) ) );

definition
let PO be non empty OrthoRelStr ;
attrPO is Orthocomplemented  means :Def22: :: OPOSET_1:def 22
 ex f being Function of PO,PO st
( f = the Compl of PO &amp; f OrthoComplement_on PO );
end;

:: deftheorem Def22 defines Orthocomplemented OPOSET_1:def_22_:_
 for PO being non empty OrthoRelStr holds
( PO is Orthocomplemented iff ex f being Function of PO,PO st
( f = the Compl of PO &amp; f OrthoComplement_on PO ) );

theorem :: OPOSET_1:16
for PO being non empty OrthoRelStr
for f being UnOp of the carrier of PO st f OrthoComplement_on PO holds
f QuasiOrthoComplement_on PO
proofend;

:: PartialOrdered (non empty OrthoRelStr)
theorem Th17: :: OPOSET_1:17
TrivOrthoRelStr is Orthocomplemented
proofend;

registration
cluster TrivOrthoRelStr  ->  strict QuasiOrthocomplemented Orthocomplemented ;
coherence
 ( TrivOrthoRelStr is QuasiOrthocomplemented &amp; TrivOrthoRelStr is Orthocomplemented ) by Th15, Th17;
end;

registration
cluster non empty PartialOrdered QuasiOrthocomplemented Orthocomplemented for OrthoRelStr ;
correctness
existence
ex b1 being non empty OrthoRelStr st
( b1 is Orthocomplemented &amp; b1 is QuasiOrthocomplemented &amp; b1 is PartialOrdered );
proofend;
end;

definition
mode QuasiOrthoPoset is non empty PartialOrdered QuasiOrthocomplemented OrthoRelStr ;
mode OrthoPoset is non empty PartialOrdered Orthocomplemented OrthoRelStr ;
end;