Device for opening toilet flushing systems, which comprises a water cistern supplied by a valve connected to a water-distribution system, and which is capable of being opened in order to cause the cistern to be filled when the water level descends below a determined bottom limit, the said cistern comprising a bottom discharge orifice and a vertical movable discharge valve element associated with this orifice, which is capable of occupying a low closed position in order to keep the water inside the cistern or a high open position in order to discharge the water to a toilet pan, and the said device comprising an element, secured to the valve element and comprising a float, which is capable of moving the valve element to its open position under the effect of buoyancy when at least a determined fraction of the float is submerged, a top bearing face of said element being capable of being selectively blocked or unblocked by a free bottom end of a vertical blocking rod, substantially coaxial with the element, a free top end of which comprises a metal plate, and which can be moved vertically between: - a blocking position, in which a permanent magnet secured to the cistern, arranged facing the metal plate of the rod, retains the metal plate under the influence of its permanent magnetic field in order to keep the blocked rod in contact with the top bearing face of the element, in order to immobilize the element, - an unblocked position, in which a first solenoid, arranged close to the permanent magnet, is controlled electrically to oppose a temporary field to the magnetic field created by the permanent magnet and thus release the metal plate from the rod in order to release the element, so that the said element can be moved between: - a bottom position for closing the full cistern, associated with the blocking of the element by the rod against the buoyancy effect being applied to the float, in which the said element closes the valve element, - a top position for opening the cistern, associated with the unblocked position of the element by the rod, in which the float, emerging to the maximum, operates the element in order to open the valve element, the rod accompanying the element over a determined maximum travel, - a bottom position for closing the empty cistern, prior to a new blocking of the element by the rod, in which the said element closes the valve element, characterized in that it comprises a transverse retractable abutment finger, which is capable of being interposed selectively at right angles to the rod in order to limit the travel of the rod to a determined portion of its maximum travel, in order to limit the rise of the element to an intermediate open position of the cistern in which the float emerges only partly by moving the element in order to open the valve element, and from which the element returns to a bottom position of reclosing the at least partially emptied cistern, in which position the joint forces of the weight of the element and of the rod, moved to its blocking position by the permanent magnet, overcome the buoyancy effect being exerted on the float, so that the element closes the valve element, so as to discharge only a portion of the water from the cistern to the toilet pan. - a blocking position, in which a permanent magnet secured to the cistern, arranged facing the metal plate of the rod, retains the metal plate under the influence of its permanent magnetic field in order to keep the blocked rod in contact with the top bearing face of the element, in order to immobilize the element, - an unblocked position, in which a first solenoid, arranged close to the permanent magnet, is controlled electrically to oppose a temporary field to the magnetic field created by the permanent magnet and thus release the metal plate from the rod in order to release the element, - a bottom position for closing the full cistern, associated with the blocking of the element by the rod against the buoyancy effect being applied to the float, in which the said element closes the valve element, - a top position for opening the cistern, associated with the unblocked position of the element by the rod, in which the float, emerging to the maximum, operates the element in order to open the valve element, the rod accompanying the element over a determined maximum travel, - a bottom position for closing the empty cistern, prior to a new blocking of the element by the rod, in which the said element closes the valve element,