Some explanation here which may help you.

I wonder about The "Let"

Ben: -eyim -ayým

Sen: -esin -asýn

O: -e -a

Biz: -elim -alým

Siz: -esiniz - asýnýz

Onlar: -eler -alar

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When we say eg. Kalsýn, then it means let it stay, why not saying Kala, doesn´t it means let it stay..

and when we say Kalasýn, does it means let you stay or.? explain it to me please..

Thank you so much

The root verb "kal" has a few meanings such as stay, remain and in special contexts it may have different meanings. The suffix -sýn  gives it a meaning like in in "Let (it) ....

It is also used as an imperative for third parties. There is another use which gives us a sentence that functions the same as for example  "may it be old or young". Here are the examples:

Dokunma öyle kalsýn. (Don´t touch. Let it stay the way it is.)

Yaz ki aklýnda kalsýn. Write it so that you (can) remember it.

Üstü kalsýn. (Keep the change).

O otel pahalý, daha ucuz bir otelde kalsýn. (That hotel is expensive. (Tell him to) stay in a cheaper one.

Bu öðrenci tembel, sýnýfta kalsýn. (This student is lazy, let him fail (this year).

"Kalmak" can denote:

Yolda kalmak.   (To be stuck somewhere.)

iki arada bir derede kalmak (to remain undecided)

üç gün kalmak (to have three more days to go)

parasýz kalmak  (to be penniless)

az parasý kalmak  (to be short of money)

sýnýfta kalmak (to fail and lose the entire year in school)

tedavülde kalmak (to remain in circulation)

Kala is possible but it is archaic. In Ottoman Turkish, Sultan would issue firmans using this form.

Paþa Edirne´de kala.    Let Pasha stay in Edirne. (This form sounds exagerated and is no longer used.)

Furthermore, "kala"  is frequently used in another context today:

Bire üç kala.  At three to one.

In Azeri Turkish kala means castle or Turkish kale.

I wonder about The "Let"

Ben: -eyim -ayým

Sen: -esin -asýn

O: -e -a

Biz: -elim -alým

Siz: -esiniz - asýnýz

Onlar: -eler -alar

_____________________________________________________________

When we say eg. Kalsýn, then it means let it stay, why not saying Kala, doesn´t it means let it stay..

and when we say Kalasýn, does it means let you stay or.? explain it to me please..

Thank you so much

usage of only first singular and plural are common, others are, as also said, not used in daily language any more and used only in poetry.

for 2nd person, e.g."kalasýn", we can say it is used as a wish but usually a bad one, i.e. cursing

e.g. a song:

ela gözlü, benli dilber, sen de olasýn benim gibi

beautiful woman with the hazel eyes and moles, i wish you also become like me