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Do outputs from Mealy and Moore machines get printed only when a transition is made or are lambda transitions allowed? Putting in a different way, what will be the output if empty string is given to the machines? I have read various answers from various sources. It will be helpful if the answer has some source, any answer is welcome.

basically Mealy and Moore are deterministic transducer which converts one input to another output. but as per my understanding only moore machine can accept null string because in mealy machine we have output on transition and there is no $\epsilon$ transition in both machines.
Any basis for your understanding?