Should I use article "a" in this sentence?
Has anyone sent documents by post
Has anyone sent documents by a post?
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"Post" in this sense is an uncountable or mass noun (as noted by Longman), so you'd always say "by post" (or "in the post", "via post" or "by mail"), never "by a post".
No, by here shows the method in use, how the action of sending documents is done - by post. It's an uncountable noun which refers to the public system for collecting and delivering of letters, so a post is never the case.
Similarly, you can travel by train/car, you can pay by cheque, you can carry/ship goods by sea/air, you can read by candlelight.
We use by + zero article to talk about means of transport and communication, including
- go/travel by car/taxi/bus/plane/train/air/sea
- communicate/contact by phone/post/mail/email
- I generally go by bus to work.
- I generally take the bus to work.
Source: Advanced Grammar in Use Third Edition by M Hewings
It is normal to say "by post" in British English (or "by mail" in American English).
You would never use an indefinite article with "post" in this context, since there is only one post ("a post" would therefore be a fence-post or similar). However, it can be correct to use a definite article, as in "through the post".