Etymology
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up- 

prefix with various senses, from Old English up (adv.), corresponding to similar prefixes in other Germanic languages.

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ana- 

before vowels an-, word-forming element meaning: 1. "upward, up in place or time," 2. "back, backward, against," 3. "again, anew," from Greek ana (prep.) "up, on, upon; up to, toward; throughout; back, backwards; again, anew," from an extended form of PIE root *an- (1) "on, upon, above" (see on, which is the English cognate). In old medical prescriptions, ana by itself meant "an equal quantity of each."

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retro- 

word-forming element of Latin origin meaning "backwards; behind," from Latin retro (prep.) "backward, back, behind," usually in reference to place or position, rarely of time, "formerly, in the past," probably originally the ablative form of *reteros, based on re- "back" (see re-).

L. retro stands to re- as intro, "in, within"; to in, "in," and as citro, "hither," stands to cis, "on this side." [Klein]

Common in combinations in post-classical Latin (the classical equivalent was post-). Active in English as a word-forming element from mid-20c.

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rachio- 

also rhachio-, before vowels rachi-, word-forming element meaning "spinal, pertaining to the vertebrae," from Latinized form of Greek rhakhis "spine, back," metaphorically "ridge (of a mountain), rib of a leaf," a word of uncertain origin. Compare Greek rhakhos "thorn hedge."

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hypo- 

word-forming element meaning "under, beneath; less, less than" (in chemistry, indicating a lesser oxidation), from Greek hypo (prep. and adverb) "under, beneath; up from under; toward and under (i.e. into)," from PIE root *upo "under."

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paleo- 

before vowels pale- word-forming element used in scientific combinations (mostly since c. 1870) meaning "ancient, early, prehistoric, primitive, fossil," from Latinized form of Greek palaios "old, ancient," from palai "long ago, far back" (from PIE root *kwel- (2) "far" in space or time).

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hypso- 

word-forming element meaning "height," from Greek hypsos "height, top," from PIE *upso-, from root *upo "under," also "up from under," hence also "over" The Greek word is cognate with Sanskrit os "above, over," Old Church Slavonic vysoku "high."

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re- 

word-forming element meaning "back, back from, back to the original place;" also "again, anew, once more," also conveying the notion of "undoing" or "backward," etc. (see sense evolution below), c. 1200, from Old French re- and directly from Latin re- an inseparable prefix meaning "again; back; anew, against."

Watkins (2000) describes this as a "Latin combining form conceivably from Indo-European *wret-, metathetical variant of *wert- "to turn." De Vaan says the "only acceptable etymology" for it is a 2004 explanation which reconstructs a root in PIE *ure "back."

In earliest Latin the prefix became red- before vowels and h-, a form preserved in redact, redeem, redolent, redundant, redintegrate, and, in disguise, render (v.). In some English words from French and Italian re- appears as ra- and the  following consonant is often doubled (see rally (v.1)).

The many meanings in the notion of "back" give re- its broad sense-range: "a turning back; opposition; restoration to a former state; "transition to an opposite state." From the extended senses in "again," re- becomes "repetition of an action," and in this sense it is extremely common as a formative element in English, applicable to any verb. OED writes that it is "impossible to attempt a complete record of all the forms resulting from its use," and adds that "The number of these is practically infinite ...."   

Often merely intensive, and in many of the older borrowings from French and Latin the precise sense of re- is forgotten, lost in secondary senses, or weakened beyond recognition, so that it has no apparent semantic content (receive, recommend, recover, reduce, recreate, refer, religion, remain, request, require). There seem to have been more such words in Middle English than after, e.g. recomfort (v.) "to comfort, console; encourage;" recourse (n.) "a process, way, course." Recover in Middle English also could mean "obtain, win" (happiness, a kingdom, etc.) with no notion of getting something back, also "gain the upper hand, overcome; arrive at;" also consider the legal sense of recovery as "obtain (property) by judgment or legal proceedings." 

And, due to sound changes and accent shifts, re- sometimes entirely loses its identity as a prefix (rebel, relic, remnant, restive, rest (n.2) "remainder," rally (v.1) "bring together"). In a few words it is reduced to r-, as in ransom (a doublet of redemption), rampart, etc.

It was used from Middle English in forming words from Germanic as well as Latin elements (rebuild, refill, reset, rewrite), and was used so even in Old French (regret, regard, reward, etc.).

Prefixed to a word beginning with e, re- is separated by a hyphen, as re-establish, re-estate, re-edify, etc. ; or else the second e has a dieresis over it: as, reëstablish, reëmbark, etc. The hyphen is also sometimes used to bring out emphatically the sense of repetition or iteration : as, sung and re-sung. The dieresis is not used over other vowels than e when re is prefixed : thus, reinforce, reunite, reabolish. [Century Dictionary, 1895]
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mixo- 

word-forming element of Greek origin meaning "mixed," from Greek mixo-, from mixis "a mixing, mingling, intercourse," from root of mignynai "to mix, mix up, mingle" (from PIE root *meik- "to mix"). As in mixolydian in reference to a half-Lydian mode in ancient Greek music.

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permutate (v.)

1898 in the modern sense of "change the order of" (earlier "to change, alter, 16c. but obsolete thereafter), from Latin permutatus, past participle of permutare "change thoroughly, exchange" (see permutation). "Probably regarded by those who use it as a back-formation from permutation" [OED]. Compare permute. Related: Permutated; permutating.

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