You can only split geometry along polygon boundaries. That is to say, where two or more polygons meet. Within that constraint, here is how to do it in Poser.

Select the thing that you want to split. From the Editing Tools palette, click on the Grouping Tool icon. Click the 'New Group' button, and give the group a descriptive name (eg "Half_Mellon"). Don't use any spaces in the name, but you can use an underscore or hyphen to separate words.

The new group should now be the active group, and you should see its name in the Group Editor, if not, click on the down arrow and select it from the list.

Now you need to select the polygons to be included in your new prop. Use the cursor to draw a rectangle around polygons that you want included in your prop, or click on a polygon to select it individually. Selected polygons will turn red. You will usually need to use the Track Ball in the Camera Controls to rotate the camera around the model so that you can select polygons on different sides.

If you select some polygons by mistake, holding down the keyboard Ctrl key whilst drawing a rectangle around them will deselect them.

Sometimes in the process of selecting polygons, a camera or other item may inadvertently become the selected item. If that happens, you will need to reselect the model and the group that you are working on.

Once you have the required polygons selected, click the 'Create Prop' button, and give the prop a name. This will create a new prop from the selected group. You may not be able to see the new prop, as it will overlay the original model, you can hide the original, or move it out of the way, so that you can see the new prop. Now you can either save the new prop to the Props library palette, or export the prop as an obj file.

To save it to the library: Open the Library (Shift+Ctrl+B), select a Props library, click the large "+" sign near the bottom of the palette, and type in a name for the prop.

To export it as an obj file: From the menu bar File menu select Export > Wavefront OBJ. Click 'OK' in the 'Export Range' window. Click in the 'UNIVERSE' box of the 'Select Object' window to clear all selections, then click in the box next to your prop to select it, then click 'OK'. In the 'Export Options' window, select only the 'Include existing groups...' option, leave the other options blank.