{************************************************************ 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.