Confusing definition about a free monoid and comparison operator

Question

In one of the works about tree languages I met the following definition:

Let $\mathcal{N}$ be the set of nonnegative integers and let $U$ be the free monoid generated by $\mathcal{N}$ with operation $\cdot$ and identify $0$. Let $a\leq b$ for $a,b\in U$ iff $\exists x\in U\ni a\cdot x=b$, let $a$ and $b$ incomparable iff $a\nleq b$ and $b\nleq a$, and let $a<b$ if $a<b$ and $a\neq b$.

This is exact citation from the paper (from ethical reasons I do not know should I give exact link to it or not).

It seems to me that there are 3 inaccurates in this definition. I just want to be sure that it is not my misunderstanding, but possible misprint in the paper. Could you please confirm my assumption or explain why it is wrong?

It should be not "indetify" but "identity"
The definition of $a\leq b$ implies that $a\neq b$, because if $x$ exists, then $a\cdot x\neq a$?
The definition of the strict inequality $a<b$ seems to me totally confusing. It probably should be: $a<b$ iff $a\leq b$ and $a\neq b$, but according to p.2 it seems to be useless.

There is a further inconsistency in that they have $0$ be one of the generators of the free monoid but also have it as the identity. — Tobias Kildetoft, Jul 27 '17 at 18:53
@TobiasKildetoft would you please be so kind and expand this point, may be as answer (I will definitely +1 it), because it is something that I do not quite understand. — Andremoniy, Jul 27 '17 at 18:55
There is not that much to expand on. The free monoid on a set is the set of all finite strings of symbols from that set. The identity is the empty string, which is in particular not an element of the original set. — Tobias Kildetoft, Jul 27 '17 at 18:56
May be they meant that it is a free monoid with* identity $0$ and generated with operation $\cdot$*? — Andremoniy, Jul 27 '17 at 18:58
@TobiasKildetoft Could you please give a clue, what would be correct form of this definition regarding identity? — Andremoniy, Jul 27 '17 at 19:05

score 1 · Accepted Answer · answered Jul 27 '17 at 18:50

1

(1) Yes, it should say "identity".

(2) No, because $x$ could be the identity element $0$. In that case, $a\cdot x=a$.

(3) Yes, it should say "let $a<b$ if $a\leq b$ and $a\neq b$".

answered Jul 27 '17 at 18:50

Eric Wofsey

330,363

Thank you. Just to clarify, shouldn't it be also "if and only if" in p.3? – Andremoniy Jul 27 '17 at 18:51
@Andremoniy In definitions, one need not use the extra f. – Tobias Kildetoft Jul 27 '17 at 18:51
When making a definition, "if" means "if and only if". See https://math.stackexchange.com/questions/566565/are-if-and-iff-interchangeable-in-definitions – Eric Wofsey Jul 27 '17 at 18:52
Yes, but in this definition appear to be different forms of "if", see above, it contains two iff's and one "if". – Andremoniy Jul 27 '17 at 18:53
That's just sloppiness on the part of the author--there is no difference in meaning intended here. – Eric Wofsey Jul 27 '17 at 18:54
Just to clarify, what could you tell about TobiasKildetoft's notice about that $0$ is identity and generator at the same time? – Andremoniy Jul 27 '17 at 19:06

Confusing definition about a free monoid and comparison operator

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