The command ConstructibleSet(lrs, R) returns a constructible set defined by the list lrs of regular systems.

A point belongs to a constructible set if and only if it is a solution of one of its defining regular systems. That is, a constructible set is the union of the solution sets of its defining regular systems.

Since a regular system always defines a nonempty set, a constructible set is empty if and only if its list of defining regular systems is empty.

This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form ConstructibleSet(..) only after executing the command with(RegularChains[ConstructibleSetTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ConstructibleSetTools][ConstructibleSet](..).

See ConstructibleSetTools and RegularChains for the related mathematical concepts, in particular for the ideas of a constructible set, a regular system, and a regular chain.

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,a\,b\,c\,d\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{sys}≔\left[ax+by\,cx+dy\right]$

$\mathrm{sys}≔\left[ax+by\,cx+dy\right]$

$\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\,\mathrm{output}=\mathrm{lazard}\right)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

$\mathrm{lrs}≔\mathrm{map}\left(\mathrm{RegularSystem}\,\mathrm{dec}\,\left[1\right]\,R\right)$

$\mathrm{lrs}≔\left[\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\right]$

$\mathrm{cs}≔\mathrm{ConstructibleSet}\left(\mathrm{lrs}\,R\right)$

$\mathrm{cs}≔\mathrm{constructible\_set}$

$\mathrm{Info}\left(\mathrm{cs}\,R\right)$

$\left[\left[x\,y\right]\,\left[1\right]\right],\left[\left[cx+yd\,da-bc\right]\,\left[1\right]\right],\left[\left[y\,a\,c\right]\,\left[1\right]\right],\left[\left[x\,b\,d\right]\,\left[1\right]\right],\left[\left[ax+yb\,c\,d\right]\,\left[1\right]\right],\left[\left[a\,b\,c\,d\right]\,\left[1\right]\right]$