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identity has 6 distinct synonyms spread across 3 of its 4 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
the distinct personality of an individual regarded as a persisting entity
“you can lose your identity when you join the army”
the individual characteristics by which a thing or person is recognized or known
“geneticists only recently discovered the identity of the gene that causes it”
an operator that leaves unchanged the element on which it operates
“the identity under numerical multiplication is 1”
exact sameness
“they shared an identity of interests”
Synonyms, antonyms and the broader and narrower relations for identity come from Princeton University’s WordNet. Full sources and licences.