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Yes I understand the noun stem so it means discussing together.Consider that the construction is essentially:
{action 1} の上 {action 2}
Do you understand that 誘いあわせ is the verb-stem noun form of a single compound verb? (誘い合わせる)?
Sorry I meant ask each other.
So if it is transitive I am to ask/gather my friends to participate
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The dictionary has the explanation about that usage of 上.My guess would be the 上 would be the equivalent to above or beyond in English.
e.g. not just you, the invitation is extended beyond to your friends.
うえ【上】の意味 - goo国語辞書うえ【上】
1[名]
6 ある事柄と他の事柄とを関係させていう時に用いる。
㋑…したのち。…した結果。「相談した―で返事する」「知的探検家の努力の―に現代科学は築かれている」
上の英語・英訳 - goo辞書 英和和英うえ【上】
8 〔「…の上で」の形で,…を条件に入れて,…した結果〕
悪いことは承知の上で彼はやったらしい
It seems that he did it knowing it was wrong.
困難は承知の上でこの問題に取り組んでみる
I'll see what I can do about the problem, though I know it's difficult.
十分な論議の上で決定した
We decided after discussing it carefully.