You haven't given a whole lot of information! You said [itex]\beta[/itex] represents a blank (I guess we might call that a "hard" blank) so really is treated just as another symbol. But what is [itex]\lambda[/itex]? The only thing you tell us about that is "but x[itex]\lambda[/itex]y= xy". So [itex]\lambda[/itex] is a "soft" blank- like nothing? Is [itex]\beta^3[/itex] the same as [itex]\beta\beta\beta[/itex]? And what, exactly is the definition of [itex]\parallel \parallel[/itex]? It would guess it is the length of the string but it would be a good idea to say that explicitely.

You haven't given a whole lot of information! You said [itex]\beta[/itex] represents a blank (I guess we might call that a "hard" blank) so really is treated just as another symbol. But what is [itex]\lambda[/itex]? The only thing you tell us about that is "but x[itex]\lambda[/itex]y= xy". So [itex]\lambda[/itex] is a "soft" blank- like nothing? Is [itex]\beta^3[/itex] the same as [itex]\beta\beta\beta[/itex]? And what, exactly is the definition of [itex]\parallel \parallel[/itex]? It would guess it is the length of the string but it would be a good idea to say that explicitely.

[tex] \lambda [/tex] is according to definition a empty string - that is, the string consisting of no symbols taken from [tex]\Sigma[/tex].