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# 7 Graphic ideal lattices (XGAP only)

If you run SONATA under XGAP or Gap.app, it is possible to study ideal lattices of nearrings graphically.

• `GraphicIdealLattice( `nr`, `string` )`

The function `GraphicIdealLattice` computes one- and two-sided nearring ideals and uses XGAP's graphic capabilities to draw the ideal lattice of the nearring nr. The string string determines, which ideals are shown:

If string contains the letter `l`, left ideals are shown, if string contains the letter `r`, right ideals are shown, and if string contains the letter `i`, two-sided ideals are shown. Any combination of these letters is possible.

Right ideals of the nearring are represented by squares, left ideals by diamonds and two-sided ideals by circles. It is possible, that two-sided ideals are shown as right or left ideals, if the two-sided ideal property has not yet been tested.

Left clicking on an ideal allows one to select the ideal and determine more information about the ideal. Choosing `Ideal type` from the `Ideals`-menu determines, whether the selected ideal is two-sided (in which case its shape changes to a circle immediately). On a right click on an ideal a window opens showing more information about the ideal, such as its size and the isomorphism class of the factor nearring by the ideal (if it is two-sided). The information is computed and displayed as soon as you click on the corresponding entry in the window. Finally clicking on `Export ideal to GAP` makes the ideal the last output of XGAP's main window, where you may then go on working with it.

It is also possible to select several ideals at the same time pressing the Ctrl-key while left clicking on the ideals. When several ideals are selected it is possible to compute their sums and intersection from the corresponding `Ideals`-menu entries. The result of the computation is then indicated by green color.

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SONATA manual
December 2022