{************************************************************
 Computation of the Dedekind cut completion of a partial or-
 der: The vector describing the cuts, their columnwise rep-
 resentation, the order of the complete lattice of the cuts, 
 the injective embedding of the order into the cut lattice, 
 and their visualization as subrelation (see Report 9205, 
 UniBw Muenchen). The vector argument cv in CutList until 
 CutVisualize stands for the result of CutVect and serves 
 for efficient computations
8**********************************************************}

CutVect(R)
  DECL Id, c, eps
  BEG  eps = epsi(dom(R));
       Id  = I(eps^ * eps);  
       c   = dom(syq(eps,mi(R,ma(R,eps))) & Id)
       RETURN c
  END.

CutList(R,cv) = epsi(dom(R)) * inj(cv)^.

CutLattice(R,cv)
  DECL emb, eps, incl
  BEG  eps = epsi(dom(R));
       incl = eps \ eps;
       emb = inj(cv)
       RETURN emb * incl * emb^
  END.

FastCutLattice(R,cv)
  DECL CL
  BEG  CL = CutList(R,cv);
       RETURN CL \ CL
  END.

CutEmb(R,cv) = syq(R,epsi(dom(R)) * inj(cv)^).

CutVisualize(R,cv)
  DECL Emb
  BEG  Emb = CutEmb(R,cv)  
       RETURN Emb^ * R * Emb
  END.

{************************************************************
 Computation of the ideal completion of a lattice. The vec-
 tor describing the ideals, their column-wise representati-
 on, the order of the complete lattice of the ideals the in-
 jective embedding of the given lattice into the ideal lat-
 tice, and their visualization as subrelation
************************************************************}

ConeVect(E) 
  DECL Epsi
  BEG  Epsi = epsi(dom(E))
       RETURN -dom(Epsi^ & -Epsi^*E)
  END.

DirectVect(E)
  DECL Epsi, Empty
  BEG  Epsi  = epsi(dom(E));
       Empty = Epsi \ On1(E)
       RETURN -Empty & ((-Epsi^ | (Epsi \ E*E^)) / L1n(E))
  END.

IdealVect(E)
  DECL Epsi
  BEG  Epsi  = epsi(dom(E))
       RETURN ConeVect(E) & DirectVect(E)
  END.

ConeList(E) = epsi(dom(E)) * inj(ConeVect(E))^.

DirectList(E) = epsi(dom(E)) * inj(DirectVect(E))^.

IdealList(E) = epsi(dom(E)) * inj(IdealVect(E))^.

IdealLattice(E)
  DECL Emb, Epsi, Incl
  BEG  Epsi = epsi(dom(E));
       Incl = Epsi \ Epsi;
       Emb = inj(IdealVect(E))
       RETURN Emb * Incl * Emb^
  END.

IdealEmb(E) = syq(E,epsi(dom(E)) * inj(IdealVect(E))^).

IdealVisualize(E)
  DECL Emb
  BEG  Emb = IdealEmb(E)  
       RETURN Emb^ * E * Emb
  END.