Exemplary embodiments of pumps, refill units and dispenser systems are disclosed herein. An exemplary refill unit for a soap, sanitizer or lotion includes a container for holding a fluid and a pump secured to the container. The pump includes a housing, a liquid piston, a liquid pump chamber, an air piston, an air pump chamber; and a piston holder. The liquid piston is connected to the air piston. The piston holder is connected to one of the liquid piston and the air piston. The connection comprises one or more projections and one or more grooves. The volume of the liquid pump chamber and the air pump chamber both change as a function of the engagement of the one or more projections with the one or more grooves.