Using multi-valued logic can lead to reducing the interconnections in the chip. Reducing the interconnection, in turn, leads to decreasing the chip area and interconnections power dissipation. The design of the multi-valued logic circuits should be performed with the minimum complexity to fulfill the multi-valued logic aim. In the recent years, much research has been focused on the design of multi-valued logics in nanoelectronics due to the high capability of nanoelectronics to design them. In this paper, first, a novel single-supply ternary successor and predecessor are designed based on the multi-threshold voltage in CNFET, which is more energy efficient than those in the previous works. Then, these are used to design the ternary full adder. To reduce the number of transistors in the proposed full adder, the structure of this full adder is designed so that only one successor and predecessor are used and some common portions can be used in the sum and carry generator, and this is shown by equations. The number of transistors in the proposed single-supply full adder is reduced from 132 in the best previous single-supply full adder to 54. Also, to enhance the PDP, the successor and predecessor are used in the quad-state mode (‘0’, ‘1’, ‘2’ and ‘z’: high impedance), where in the ‘z’ mode, the direct current path is cut off. The circuits are simulated by the HSPICE software, using the Stanford 32 nm CNTFET library. The simulation results confirm the correct operation of the proposed circuit and PDP improvement in the proposed ternary full adder, which is about 81.12%, as compared to the best single supply reported in the previous works.