A process receives a SIGPIPE when it attempts to write to a pipe (named or not) or socket of type SOCK_STREAM that has no reader left.

It's generally wanted behaviour. A typical example is:

find . | head -n 1

You don't want find to keep on running once head has terminated (and then closed the only file descriptor open for reading on that pipe).

The yes command typically relies on that signal to terminate.

yes | some-command

Will write "y" until some-command terminates.

Note that it's not only when commands exit, it's when all the reader have closed their reading fd to the pipe. In:

yes | ( sleep 1; exec <&-; ps -fC yes)
      1 2       1        0

Their will be 1 (the subshell), then 2 (subshell + sleep), then 1 (subshell) then 0 fd reading from the pipe after the subshell explicitely closes its stdin, and that's when yes will receive a SIGPIPE.

Above, most shells use a pipe(2) while ksh93 uses a socketpair(2), but the behaviour is the same in that regard.

When a process ignores the SIGPIPE, the writing system call (generally write, but could be pwrite, send, splice...) returns with a EPIPE error. So processes wanting to handle the broken pipe manually would typically ignore SIGPIPE and take action upon a EPIPE error.

(f)

Writing failed because there are no processes which could read from the pipe.

Although unless you duplicate descriptors and fork, there can only be one process to start with: generally a pipe has one reader and one writer, and when one of them closes the connection, the pipe is defunct. If you are using a named pipe, you can can make multiple connections (in serial) with it, but each one represents a new pipe in this sense. So a "pipe" to a thread or process is synonymous with a file descriptor.

From man 7 pipe:

If all file descriptors referring to the read end of a pipe have been closed, then a write(2) will cause a SIGPIPE signal to be generated for the calling process. If the calling process is ignoring this signal, then write(2) fails with the error EPIPE.

So a "broken pipe" is to the writer what EOF is to the reader.

A broken pipe happens when the reading process exits before the writing process. So I would go with (f)