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 "special" $TM$, while saying that every time the head moves left, it will stay put. Couldn't prove it, though.