descriptive grammar, funkcje zdania - help!

Witam, bardzo prosze o pomoc studentów, absolwentow albo w ogole osob orientujących się w tym temacie.

Proszę wyjaśnijcie mi kwestie:
Co to jest COMPLEMENT, a co to jest OBJECT
Do tej pory myślałam, że complement to przydawka, a object to dopełnienie (bliższe i dalsze) czyli wg mojej teorii, np:
I'm tired.
I - subject
am - verb
tired - complement

Aż tu nagle takie zdanie:
Jane laughted at her sister.
Jane - subject
laughted - verb
at her sister - również COMPLEMENT

O co chodzi?

≫ O co chodzi ?

In its most broadest term it’s about verb valency and its major complementation patterns.-:)
I can improve on your above examples :

I’m tired – standing alone this is the type of copular complementation where “I’ indeed is the subject ( as a personal pronoun), ‘am’ is the copular verb and ‘tired’ is the subject complement realized by a participial adjective. The simplest flat structure is : SVC which can further be expanded into S→Vcop→Cs or even further ( but now syntactically ) into [NP[[VP]][AdjP]] . But keep in mind that ‘tired’ functions as a subject related adjective in this complementation.
In this particular case with ‘tired” further complementation after ‘tired’ is not obligatory ( which in turn would be called ‘adjective complementation by a prepositional phrase if it was, say, “I’m tired of doing something” ) as opposed to , for example, “ *I’m averse to “
“Jane laughed at her sister” – “at her sister” is a purely intransitive complementation. “Laugh” doesn’t require complementation, hence “at her sister “ is clearly an adjunct realized by PP with NP ‘sister’ as the head of the former.

Many thanks for your explanation.

I'd be very grateful if you would give a few more examples of transitive and intransitive complementation or at least recommend any book.
BTW, what's the difference between Cs and Co?

'at my sister' - why an adjunct? It's is only manner, place and time that adjunct describes.

Terribly sorry for my curosity, but I'm all alone with this issue and the exam is coming.

>>>a few more examples of transitive and intransitive complementation

Here are three major types of complementation aside form that copular one ( SVA,SVC) you already know or intransitive, like, for example, ‘Jane (S) is lying (V) on her back ( C-loc)-locative.

1. Monotransitive - SVO type
“Jane(S) enjoys(V) jazz(O)”

2.Ditransitive ( with a verb that takes both Oi and Od ) -SVOO type
“Jane(S) offered(V) her sister(Oi) some help(Od) - what a crafty double-bagger, first she offered help to her sister but later she laughed at her. :)

3.Complex transitive – type SVOA, SVOC
“Jane (S) showed (V) her sister (O) into the sitting room(Ao-object related adverbial )
“ Jane (S) found (V) her sister (O) incredibly naive (Co)

>>>BTW, what's the difference between Cs and Co?

Please don’t do it to me. It’s like asking ‘what is the difference between a chinese cyclist and the former Barry’s Chief of Staff Rahm Emanuel :) What is the difference between subject and object?
Well, unless it’s ‘raising to object’ in generative syntax where such distinction might indeed be “somewhat blurred” as in “ Jane believes her to have lied “.
Here ‘her’ is syntactically the subject of ‘to have lied’ but by convention it’s the direct object of ‘Jane’ because of objective case-marking ( her )
But forget it.

Subject Complement
Jane (S) is (Vcop) someone who is laughing at her sister (Cs).
Here the relative clause describes, characterizes or defines, if you like, the subject and is linked to its complement by copular ‘is’.

Object complement
Jane (S) called (V) her sister(Od) an angel (Co)
Whom does the NP ‘angel’ refer to ? ‘Jane’ or ‘sister’ ?

>>>'at my sister' - why an adjunct? It's is only manner, place and time that adjunct describes.

Agree but let me elaborate my argument on that ‘adjunct’ thing then. Read it carefully please, first the quote and then my ‘dance and song’ :)

Quote,
“9.87 Adjunct and preposition:
There are several different types of combinations of verbs plus particles ( my note – in serious grammar books prepositions or adverbs are sometimes called ‘particles’ ). If the verb is intransitive, we can recognize the particle as a prepositional adverb functioning as adjunct.”, unquote.

Let me go on.
As I said “laugh” alone doesn’t require complementation , e.g. “[Jane was laughing]… [ at her sister ]” nor does it take any Oi or Od, for that matter, because it’s an intransitive verb and that we know, right ? ( though there’s at least a couple of ‘peculiar’ cases with “laugh” such as “laugh oneself sick/somebody out /one’s head off/ ” where, for example, a reflexive pronoun in the former is recognized as the object but then again the Co “sick” ( object Complement) would be obligatory.
“ * I laughed myself “ is a big no-no, “myself” neds to be complemented. But these are the fixed idiomatic expressions and they’re not of the greatest importance to my “soapbox oration” here -:)
Moreover , in transformational generative grammar or ‘syntactically’ the PP that is a sister to V and a daughter of an N is considered to be an adjunct but this is a completely different story.

There’s that one thing that I didn’t tell you. This ‘Jane’, you know, she must be the grumpiest kvetch that ever lived. Hope I didn’t bore you to death. -:)

haha.. apparenly, thanks to Jane, I've managed to get through it :)

None of my friends and colleagues at work were able to explain me that. You're the only person that didn't disregard my request.
Big thanks, you're awesome!

Sorry for my laic comments about complements. Now I'm ashamed of myself

Unfortunately, he haven't done such coding on classes, which I think is crutial, so let me make sure:
Oi - indirect object
Od - direct Am I right?

I hope I haven't become your trainer of patiency yet :)))

>Unfortunately, he haven't done such coding on classes, which I think is crutial, so >let me make sure:
>Oi - indirect object
>Od - direct Am I right?

Yes.

This ‘taxonomy’ has been a well-established way of abbreviations of such terms since Bullokar, I guess.

I had no idea, that my professor('s knowlegde) is THAT old...