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"agreement," 1590s, from Latin compactum "agreement," noun use of neuter past participle of compacisci "come to agreement," from com- "together" (see com-) + pacisci "to covenant, contract" (see pact).
"make-up case," 1921, from compact (adj.), based on its containing compacted face powder.
early 15c., from Latin compactus, past participle of compingere "to fasten together" (see compact (adj.)). Related: Compacted; compacting.
d <= lub S => there exists s in S such that d <= s.
I.e. you always reach d (or better) after a finite number of steps up the chain.
("<=" is written in LaTeX as \sqsubseteq).