Confusion in Pumping Lemma (PL) as presented in book by: Arthur Fleck.
 

Regarding the concept of pumping length, in Pumping Lemma (PL), have some confusion in the presentation in the book titled: Formal Models Of Computation, by Arthur Fleck.
It states on page #91 (attached herewith, & accessed via subscription to Perlego) that the pumping length (N) is given, for a regular language L ⊆ Σ*, as:

for each string z ∈ L, with len(z) ∈ N, there exist (sub-strings of the string z) u, v & w ∈ Σ*, so that:
(i) string z is a concatenation of u, v & w; i.e. z = uvw,
(ii) len(uv) ≤ N and len(uv) ≥ 1,
(iii) uv^kw L; for all non-negative integer values for k.

.........................................

Doubt #1: The length of string (z) is stated to be in the range from 0...N; rather than having length at-least equal to the pumping length (N).
Though, the point (ii) states that len(uv) ≤ N, still it does no help.

In the site here, it is clearly stated that the length of the string (s) is at least equal to the pumping length (p); i.e. states |s| ≤ p.

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The doubt intensifies when you do not get any clue in the book, in further down pages; as shown attached herewith (till page #95; and few more pages attached in the next message).

Further pages in the book, after page #95, till the end of the presentation of the topic of PL.
 

The book presents the topic of PL for regular languages, till page #97.
But, the book still has mention of PL, in the context of operations overcoming the limitations of PL, on page #98.
So, added till page #98.

I know that page #98, is unnecessary; but added as consider it as the best book on the topic (& in general, on formal languages, from the UG viewpoint); so might need help later on topic started on page #98.

Regarding the first red hilighted part - what would that even mean? You have substituted the symbol for element of, '∈', for what is clearly a typo in the image of page 91 which looks like the epsilon character, 'ε', and also makes no sense.

The typo looks like an OCR error, so I would be very cautious about reading any of those pages, especially of non-alphabetic characters which have a higher rate of OCR errors in my experience.

Item (ii) contains a transcription error.

Regarding the Doubt #1 hilight, I do not see that statement on that page, but have not looked at others. But in view of the probability of other OCR errors I think it is moot.

I would suggest that you find a better quality version of this text, or another source entirely, and start over.

That is self contradictory, so this question also makes no sense.

I hope your last line means that should have used '≥', instead of '≤', i.e.:
'... it is clearly stated that the length of the string (s) is at least equal to the pumping length (p); i.e. states |s| ≥ p.'

If so, kindly ignore it as a typo.

The Pumping Lemma, as presented in Arthur Fleck's book, implies that for a regular language \(L\), the length of \(z\) should be at least equal to the pumping length \(N\). Clarification may be sought in subsequent pages or additional resources. I found this https://stateofwriting.com/ website for writing service called State of Writing has completely transformed my academic experience. When I was overwhelmed with a stack of assignments, I sought out their custom writing and editing services. The effect was remarkable - my papers now exhibit a level of professionalism and precision that I had not previously achieved. The team at the State of Writing not only met deadlines but went beyond my expectations, clearly demonstrating their dedication to student success.