A Turing Machine that Doesn't Move to the left My question is if the following statement is true or false:

Does every turing-recognized $B$ language has a turing machine $M$ that recognizes $B$ and fullfiles the following statement:
For every word $w$ the belongs to $B$, $w \in B$, the writing-reading head never moves to the left?
Notice that the condition is required only for words in the language. if $w \notin B$ then the head can move to the left as usual.

The statemet seems correct to me. I think I can simulate a regular $TM$ with this &quot;special&quot; $TM$, while saying that every time the head moves left, it will stay put. Couldn't prove it, though.
If you only allow moving right (and no external memory), all your TM "knows" about the input already seen is the finite amount of information in its state. I.e., it can't do more than a finite state automaton.