Etymology
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Words related to lax

*sleg- 
*slēg-, also *lēg-, Proto-Indo-European root meaning "be slack, be languid."

It forms all or part of: algolagnia; catalectic; laches; languid; languish; lax; lease; lessor; lush; relax; release; relish; slack (adj.); sleep.

It is the hypothetical source of/evidence for its existence is provided by: Greek legein "to leave off, stop," lagnein "to lust;" Latin languere "to be faint, weary," laxus "wide, spacious, roomy;" Old Church Slavonic slabu "lax, weak;" Lithuanian silpnas "weak."
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lox (n.)

1934, American English, from Yiddish laks, from Middle High German lahs "salmon," from Proto-Germanic *lakhs-, from the common IE root for the fish, *laks- (source also of Lithuanian lašiša, Russian losos, Polish łosoś "salmon").

lacrosse (n.)
1850, American English, from Canadian French jeu de la crosse (18c.), literally "game of the hooked sticks," from crosse "hooked stick," such as that used in the game to throw the ball. This French word is, perhaps via a Gallo-Romance *croccia, from Proto-Germanic *kruk- (see crook (n.)). Originally a North American Indian game; the native name is represented by the Ojibwa (Algonquian) verb baaga'adowe "to play lacrosse." Modern form and rules of the game were laid down 1860 in Canada.
delay (v.)

c. 1300, delaien, "to put off, postpone;" late 14c., "to put off or hinder for a time," from Old French delaiier, from de- "away, from" (see de-) + laier "leave, let." This is perhaps a variant of Old French laissier, from Latin laxare "slacken, undo" (see lax). But Watkins has it from Frankish *laibjan, from a Proto-Germanic causative form of PIE root *leip- "to stick, adhere." Intransitive sense of "linger, move slowly" is from c. 1500. Related: Delayed; delaying.

lackadaisical (adj.)
"sentimentally woebegone" [Century Dictionary], 1768, lack-adaysical (Sterne), from interjection lackadaisy "alas, alack" (1748), a ludicrous alteration of lack-a-day (1690s), an exclamation of sorrow or regret, from alack the day (1590s). Hence, "given to crying 'lack-a-day,' vapidly sentimental." Sense probably altered by influence of lax. Related: Lackadaisically.
laissez-faire 
also laissez faire, 1822, French, literally "let (people) do (as they think best)," from laissez, second person plural imperative of laisser "to let, to leave" (10c., from Latin laxare, from laxus "loose;" see lax) + faire "to do" (from Latin facere "to make, to do" (from PIE root *dhe- "to set, put"). From the phrase laissez faire et laissez passer, motto of certain 18c. French economists, chosen to express the ideal of government non-interference in business and industry. Compare laisser-faire "a letting alone," taken to mean "non-interference with individual freedom of action" as a policy in government and political economy.
languor (n.)
c. 1300, "disease, sickness; distress, mental suffering," from Old French langor "sickness; weakness" (12c., Modern French langueur), from Latin languorem (nominative languor) "faintness, feebleness, lassitude," from languere "be weak or faint" (see lax). Sense in English shifted to "faintness, weariness" (1650s) and "habitual want of energy" (1825).
laxative (adj.)
late 14c., "causing relaxation or looseness," from Old French laxatif (13c.), from Medieval Latin laxativus "loosening," from Latin laxat-, past participle stem of laxare "loosen," from laxus "loose, lax" (see lax). The noun meaning "a laxative medicine, a medicine that relieves constipation by relaxing the intestines" is from late 14c.
laxity (n.)

1520s, from French laxité, from Latin laxitatem (nominative laxitas) "width, spaciousness," from laxus "loose, lax" (see lax). An earlier noun was laxation (late 14c.). Laxness is from 1630s.

leash (n.)
c. 1300, "thong for holding a dog or hound," from Old French lesse, laisse "hound's leash," ultimately from Latin laxus "loose" (see lax), perhaps via noun use of fem. form laxa. The notion seems to be of a string loosely held. Figurative sense attested from early 15c. The meaning "a set of three, three creatures of a kind" is from early 14c., originally in sporting language and especially of greyhounds, foxes, bucks, or hares.