Article WAYBEL16, MML version 4.99.1005

:: WAYBEL16:th 1
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2, b3 being Element of the carrier of b1 holds
"/\"((uparrow b2) /\ uparrow b3,b1) = b2 "\/" b3;

:: WAYBEL16:th 2
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2, b3 being Element of the carrier of b1 holds
"\/"((downarrow b2) /\ downarrow b3,b1) = b2 "/\" b3;

:: WAYBEL16:th 3
theorem
for b1 being non empty RelStr
for b2, b3 being Element of the carrier of b1
      st b2 is_maximal_in (the carrier of b1) \ uparrow b3
   holds (uparrow b2) \ {b2} = (uparrow b2) /\ uparrow b3;

:: WAYBEL16:th 4
theorem
for b1 being non empty RelStr
for b2, b3 being Element of the carrier of b1
      st b2 is_minimal_in (the carrier of b1) \ downarrow b3
   holds (downarrow b2) \ {b2} = (downarrow b2) /\ downarrow b3;

:: WAYBEL16:th 5
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of bool the carrier of b1 ~
      st b2 = b4 & b3 = b5
   holds b2 "\/" b3 = b4 "/\" b5;

:: WAYBEL16:th 6
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of bool the carrier of b1 ~
      st b2 = b4 & b3 = b5
   holds b2 "/\" b3 = b4 "\/" b5;

:: WAYBEL16:th 7
theorem
for b1 being non empty reflexive transitive RelStr holds
   Filt b1 = Ids (b1 ~);

:: WAYBEL16:th 8
theorem
for b1 being non empty reflexive transitive RelStr holds
   Ids b1 = Filt (b1 ~);

:: WAYBEL16:modenot 1 => WAYBEL16:mode 1
definition
  let a1, a2 be non empty reflexive transitive antisymmetric complete RelStr;
  mode CLHomomorphism of A1,A2 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a2 means
    it is directed-sups-preserving(a1, a2) & it is infs-preserving(a1, a2);
end;

:: WAYBEL16:dfs 1
definiens
  let a1, a2 be non empty reflexive transitive antisymmetric complete RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is CLHomomorphism of a1,a2
it is sufficient to prove
  thus a3 is directed-sups-preserving(a1, a2) & a3 is infs-preserving(a1, a2);

:: WAYBEL16:def 1
theorem
for b1, b2 being non empty reflexive transitive antisymmetric complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is CLHomomorphism of b1,b2
   iff
      b3 is directed-sups-preserving(b1, b2) & b3 is infs-preserving(b1, b2);

:: WAYBEL16:prednot 1 => WAYBEL16:pred 1
definition
  let a1 be non empty reflexive transitive antisymmetric complete continuous RelStr;
  let a2 be Element of bool the carrier of a1;
  pred A2 is_FG_set means
    for b1 being non empty reflexive transitive antisymmetric complete continuous RelStr
    for b2 being Function-like quasi_total Relation of a2,the carrier of b1 holds
       ex b3 being CLHomomorphism of a1,b1 st
          b3 | a2 = b2 &
           (for b4 being CLHomomorphism of a1,b1
                 st b4 | a2 = b2
              holds b4 = b3);
end;

:: WAYBEL16:dfs 2
definiens
  let a1 be non empty reflexive transitive antisymmetric complete continuous RelStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is_FG_set
it is sufficient to prove
  thus for b1 being non empty reflexive transitive antisymmetric complete continuous RelStr
    for b2 being Function-like quasi_total Relation of a2,the carrier of b1 holds
       ex b3 being CLHomomorphism of a1,b1 st
          b3 | a2 = b2 &
           (for b4 being CLHomomorphism of a1,b1
                 st b4 | a2 = b2
              holds b4 = b3);

:: WAYBEL16:def 2
theorem
for b1 being non empty reflexive transitive antisymmetric complete continuous RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is_FG_set
   iff
      for b3 being non empty reflexive transitive antisymmetric complete continuous RelStr
      for b4 being Function-like quasi_total Relation of b2,the carrier of b3 holds
         ex b5 being CLHomomorphism of b1,b3 st
            b5 | b2 = b4 &
             (for b6 being CLHomomorphism of b1,b3
                   st b6 | b2 = b4
                holds b6 = b5);

:: WAYBEL16:funcreg 1
registration
  let a1 be non empty reflexive transitive antisymmetric upper-bounded RelStr;
  cluster Filt a1 -> non empty;
end;

:: WAYBEL16:th 9
theorem
for b1 being set
for b2 being non empty Element of bool the carrier of InclPoset Filt BoolePoset b1 holds
   meet b2 is non empty filtered upper Element of bool the carrier of BoolePoset b1;

:: WAYBEL16:th 10
theorem
for b1 being set
for b2 being non empty Element of bool the carrier of InclPoset Filt BoolePoset b1 holds
   ex_inf_of b2,InclPoset Filt BoolePoset b1 &
    "/\"(b2,InclPoset Filt BoolePoset b1) = meet b2;

:: WAYBEL16:th 11
theorem
for b1 being set holds
   bool b1 is non empty filtered upper Element of bool the carrier of BoolePoset b1;

:: WAYBEL16:th 12
theorem
for b1 being set holds
   {b1} is non empty filtered upper Element of bool the carrier of BoolePoset b1;

:: WAYBEL16:th 13
theorem
for b1 being set holds
   InclPoset Filt BoolePoset b1 is upper-bounded;

:: WAYBEL16:th 14
theorem
for b1 being set holds
   InclPoset Filt BoolePoset b1 is lower-bounded;

:: WAYBEL16:th 15
theorem
for b1 being set holds
   Top InclPoset Filt BoolePoset b1 = bool b1;

:: WAYBEL16:th 16
theorem
for b1 being set holds
   Bottom InclPoset Filt BoolePoset b1 = {b1};

:: WAYBEL16:th 17
theorem
for b1 being non empty set
for b2 being non empty Element of bool the carrier of InclPoset b1
      st ex_sup_of b2,InclPoset b1
   holds union b2 c= "\/"(b2,InclPoset b1);

:: WAYBEL16:th 18
theorem
for b1 being reflexive transitive antisymmetric upper-bounded with_infima RelStr holds
   InclPoset Filt b1 is complete;

:: WAYBEL16:funcreg 2
registration
  let a1 be reflexive transitive antisymmetric upper-bounded with_infima RelStr;
  cluster InclPoset Filt a1 -> strict complete;
end;

:: WAYBEL16:attrnot 1 => WAYBEL16:attr 1
definition
  let a1 be non empty RelStr;
  let a2 be Element of the carrier of a1;
  attr a2 is completely-irreducible means
    ex_min_of (uparrow a2) \ {a2},a1;
end;

:: WAYBEL16:dfs 3
definiens
  let a1 be non empty RelStr;
  let a2 be Element of the carrier of a1;
To prove
     a2 is completely-irreducible
it is sufficient to prove
  thus ex_min_of (uparrow a2) \ {a2},a1;

:: WAYBEL16:def 3
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1 holds
      b2 is completely-irreducible(b1)
   iff
      ex_min_of (uparrow b2) \ {b2},b1;

:: WAYBEL16:th 19
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
      st b2 is completely-irreducible(b1)
   holds "/\"((uparrow b2) \ {b2},b1) <> b2;

:: WAYBEL16:funcnot 1 => WAYBEL16:func 1
definition
  let a1 be non empty RelStr;
  func Irr A1 -> Element of bool the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
          b1 in it
       iff
          b1 is completely-irreducible(a1);
end;

:: WAYBEL16:def 4
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 = Irr b1
   iff
      for b3 being Element of the carrier of b1 holds
            b3 in b2
         iff
            b3 is completely-irreducible(b1);

:: WAYBEL16:th 20
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being Element of the carrier of b1 holds
      b2 is completely-irreducible(b1)
   iff
      ex b3 being Element of the carrier of b1 st
         b2 < b3 &
          (for b4 being Element of the carrier of b1
                st b2 < b4
             holds b3 <= b4) &
          uparrow b2 = {b2} \/ uparrow b3;

:: WAYBEL16:th 21
theorem
for b1 being non empty reflexive transitive antisymmetric upper-bounded RelStr holds
   not Top b1 in Irr b1;

:: WAYBEL16:th 22
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr holds
   Irr b1 c= IRR b1;

:: WAYBEL16:th 23
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1
      st b2 is completely-irreducible(b1)
   holds b2 is meet-irreducible(b1);

:: WAYBEL16:th 24
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being Element of the carrier of b1
   st b2 is completely-irreducible(b1)
for b3 being Element of bool the carrier of b1
      st ex_inf_of b3,b1 & b2 = "/\"(b3,b1)
   holds b2 in b3;

:: WAYBEL16:th 25
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being Element of bool the carrier of b1
      st b2 is order-generating(b1)
   holds Irr b1 c= b2;

:: WAYBEL16:th 26
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of the carrier of b1
      st ex b3 being Element of the carrier of b1 st
           b2 is_maximal_in (the carrier of b1) \ uparrow b3
   holds b2 is completely-irreducible(b1);

:: WAYBEL16:th 27
theorem
for b1 being transitive antisymmetric with_suprema RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 < b3 &
         (for b5 being Element of the carrier of b1
               st b2 < b5
            holds b3 <= b5) &
         not b4 <= b2
   holds b2 "\/" b4 = b3 "\/" b4;

:: WAYBEL16:th 28
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 < b3 &
         (for b5 being Element of the carrier of b1
               st b2 < b5
            holds b3 <= b5) &
         not b4 <= b2
   holds not b4 "/\" b3 <= b2;

:: WAYBEL16:th 29
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima complete RelStr
   st b1 ~ is meet-continuous
for b2 being Element of the carrier of b1
      st b2 is completely-irreducible(b1)
   holds (the carrier of b1) \ downarrow b2 is non empty filtered upper Open Element of bool the carrier of b1;

:: WAYBEL16:th 30
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima complete RelStr
   st b1 ~ is meet-continuous
for b2 being Element of the carrier of b1
      st b2 is completely-irreducible(b1)
   holds ex b3 being Element of the carrier of b1 st
      b3 in the carrier of CompactSublatt b1 & b2 is_maximal_in (the carrier of b1) \ uparrow b3;

:: WAYBEL16:th 31
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima algebraic RelStr
for b2, b3 being Element of the carrier of b1
      st not b3 <= b2
   holds ex b4 being Element of the carrier of b1 st
      b4 is completely-irreducible(b1) & b2 <= b4 & not b3 <= b4;

:: WAYBEL16:th 32
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima algebraic RelStr holds
   Irr b1 is order-generating(b1) &
    (for b2 being Element of bool the carrier of b1
          st b2 is order-generating(b1)
       holds Irr b1 c= b2);

:: WAYBEL16:th 33
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima algebraic RelStr
for b2 being Element of the carrier of b1 holds
   b2 = "/\"((uparrow b2) /\ Irr b1,b1);

:: WAYBEL16:th 34
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st b3 is completely-irreducible(b1) & b3 = "/\"(b2,b1)
   holds b3 in b2;

:: WAYBEL16:th 35
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete algebraic RelStr
for b2 being Element of the carrier of b1
      st b2 is completely-irreducible(b1)
   holds b2 = "/\"({b3 where b3 is Element of the carrier of b1: b3 in uparrow b2 &
    (ex b4 being Element of the carrier of b1 st
       b4 in the carrier of CompactSublatt b1 & b3 is_maximal_in (the carrier of b1) \ uparrow b4)},b1);

:: WAYBEL16:th 36
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete algebraic RelStr
for b2 being Element of the carrier of b1 holds
      ex b3 being Element of the carrier of b1 st
         b3 in the carrier of CompactSublatt b1 & b2 is_maximal_in (the carrier of b1) \ uparrow b3
   iff
      b2 is completely-irreducible(b1);