The definition for ideal memory system is so strict that some physical elements cannot exist in the real world. In this paper, an ideal memory system can be extended to generate 15 different kinds of quasi-ideal memory systems, which are included in memory systems as its special cases and are different from ideal memory system. For a system to be a quasi-ideal memory system, it should show three unique fingerprints: 1) the pinched hysteretic loop of a quasi-ideal memory system must be odd symmetrical in the plane; 2) the pinched hysteretic loop of a quasi-ideal memory system must be "self-crossing"; and 3) the slope of tangent line for the pinched hysteresis loop must be strictly monotone in a given period.