Concerning the power of one-dimensional cellular automata recognizers, Ibarra and Jiang have proved that real time cellular automata (CA) and linear time CA are equivalent if and only if real time CA is closed under reverse. In this paper we investigate the question of equality of real time CA and linear time CA with respect to the operations of concatenation and cycle. In particular, we prove that if real time CA is closed under concatenation then real time CA is as powerful as linear time CA on the unary languages. We also prove that the question of knowing whether real time CA is as powerful than linear time CA is equivalent to the question of whether real time CA is closed under cycle. Moreover, in the case of two-dimensional CA recognizers, we investigate how restricted communication reduces the computational power. In particular, we show that real time CA and linear time CA with restricted variants of Moore and Von Neumann neighborhoods are not closed under rotation. Furthermore, they are not equivalent to real time CA with Moore or Von Neumann neighborhoods.