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Grammatical sense of "complement"

Hi,
I learned this noun here, and came to think if it as describing a word or expression which "completed" the "action" of a being verb, or verb of sense, etc.

Now the term "subject complement" seems to be in vogue.
I'm comfortable in saying that a linking verb can link a discriptor to a noun subject, but I don't feel that eg. an adjective complement "completes" anything about the subject, in the transitive sense.

When I say "adjective complement," I'm naming the part of speech of the complement (I think).
If I say "verb complement," I'm describing what the complement does to the verb.

When I say "subject complement," I presume I'm saying that the complement links to the subject (of the clause?). But I'm not comfortable with it at all. It seems like a misnomer.
I don't know if "subject" modifies "complement," or the other way around. In what sense does the complement complete the subject?

When a trained grammarian says "subject complement," how does he view the sense of "complement"?

Thanks. - A.
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This is all I think about, really:

Traditionally, a complement is a constituent of a clause, such as a noun phrase or adjective phrase, that is used to predicate a description of the subject or object of the clause.
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That's good enough for me! Thanks.
Hmmm, looking at resha's recent post, to which you and CJ replied, I'm beginning to understand that my early impression that a complement completes was totally out of whack.
I had taken the position that in the sentence "Jack should have been [fired]," (taking the view that "fired" is an adjective complement) if "fired" is omitted, the clause is incomplete.

When you look at resha's thread, "They selected Ram the leader (CJ)," the object complement is not "necessary."
"They selected Ram" is a perfectly fine sentence.

It seems like "the leader" would be better described as an object "enhancer."

I suppose you could say it completes the phrase semantically, in terms of the author's intention; but "complement" doesn't seem to describe the grammatical function.

http://www.EnglishForward.com/English/ObjectComplement/xklck/post.htm

Edit. I realize that MrM's definition of "complement" doesn't involve the idea of completion.
I'm just hanging on to my feeling that "complement" is a strange word to use.

Edit. Edit. Well, my AmHtg. offers 6. A word or words used after a verb to complete a predicate construction.

Okay.

I suppose "subject complement"; "object complement" are just additional qualifications.

Can we say "subject adjective complement"? "subject verb adjective complement"?

Jane is stubborn.

"Stubborn" is an adjective. It links to the subject. It completes the action of the verb.

Considering my AmHtg. reference, would it be fair to say that all complements are verb complements?

I think perhaps only the author decides when a clause is complete.