  Simultaneously transmitting and reflecting reconfigurable intelligent surface
(STAR-RIS) provides a promising way to expand coverage in wireless
communications. However, limitation of single STAR-RIS inspire us to integrate
the concept of multi-hop transmissions, as focused on RIS in existing research.
Therefore, we propose the novel architecture of multi-hop STAR-RISs to achieve
a wider range of full-plane service coverage. In this paper, we intend to solve
active beamforming of the base station and passive beamforming of STAR-RISs,
aiming for maximizing the energy efficiency constrained by hardware limitation
of STAR-RISs. Furthermore, we investigate the impact of the on-off state of
STAR-RIS elements on energy efficiency. To tackle the complex problem, a
Multi-Agent Global and locAl deep Reinforcement learning (MAGAR) algorithm is
designed. The global agent elevates the collaboration among local agents, which
focus on individual learning. In numerical results, we observe the significant
improvement of MAGAR compared to the other benchmarks, including Q-learning,
multi-agent deep Q network (DQN) with golbal reward, and multi-agent DQN with
local rewards. Moreover, the proposed architecture of multi-hop STAR-RISs
achieves the highest energy efficiency compared to mode switching based
STAR-RISs, conventional RISs and deployment without RISs or STAR-RISs.
