[{"word":"ideal","phonetic":"/aɪˈdiː.əl/","phonetics":[{"text":"/aɪˈdiː.əl/","audio":"https://api.dictionaryapi.dev/media/pronunciations/en/ideal-us.mp3","sourceUrl":"https://commons.wikimedia.org/w/index.php?curid=1749869","license":{"name":"BY-SA 3.0","url":"https://creativecommons.org/licenses/by-sa/3.0"}}],"meanings":[{"partOfSpeech":"noun","definitions":[{"definition":"A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.","synonyms":[],"antonyms":[]},{"definition":"A subring closed under multiplication by its containing ring.","synonyms":[],"antonyms":[]},{"definition":"(lattice theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).","synonyms":[],"antonyms":[]},{"definition":"A collection of sets, considered small or negligible, such that every subset of each member and the union of any two members are also members of the collection.","synonyms":[],"antonyms":[]},{"definition":"(Lie theory) A Lie subalgebra (subspace that is closed under the Lie bracket) 𝖍 of a given Lie algebra 𝖌 such that the Lie bracket [𝖌,𝖍] is a subset of 𝖍.","synonyms":[],"antonyms":[]}],"synonyms":["Lie ideal"],"antonyms":["filter"]},{"partOfSpeech":"adjective","definitions":[{"definition":"Optimal; being the best possibility.","synonyms":[],"antonyms":[]},{"definition":"Perfect, flawless, having no defects.","synonyms":[],"antonyms":[]},{"definition":"Pertaining to ideas, or to a given idea.","synonyms":[],"antonyms":[]},{"definition":"Existing only in the mind; conceptual, imaginary.","synonyms":[],"antonyms":[]},{"definition":"Teaching or relating to the doctrine of idealism.","synonyms":[],"antonyms":[],"example":"the ideal theory or philosophy"},{"definition":"Not actually present, but considered as present when limits at infinity are included.","synonyms":[],"antonyms":[],"example":"An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle."}],"synonyms":[],"antonyms":[]}],"license":{"name":"CC BY-SA 3.0","url":"https://creativecommons.org/licenses/by-sa/3.0"},"sourceUrls":["https://en.wiktionary.org/wiki/ideal"]}]