Etymology
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-machy 

word-forming element meaning "battle, war, contest, fighting, warfare," from Latinized form of Greek -makhia, from makhē "a battle, fight," related to makhesthai "to fight." Beekes suspects it is from an isolated root, perhaps Pre-Greek: "In the domain of fighting and battle, old inherited expressions can hardly be expected."

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-happy 
word-forming element used in World War II armed forces slang and after, meaning "crazed or frazzled from stress due to" the thing specified (as in bomb-happy (1942), flak-happy (1943), trigger-happy (1942). The model might have been slap-happy in pugilism from 1936 as a slang variant of "punch-drunk."
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-ary 
adjective and noun word-forming element, in most cases from Latin -arius, -aria, -arium "connected with, pertaining to; the man engaged in," from PIE relational adjective suffix *-yo- "of or belonging to." The neuter of the adjectives in Latin also were often used as nouns (solarium "sundial," vivarium, honorarium, etc.). It appears in words borrowed from Latin in Middle English. In later borrowings from Latin to French, it became -aire and passed into Middle English as -arie, subsequently -ary.
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-gram 

noun word-forming element, "that which is written or marked," from Greek gramma "that which is drawn; a picture, a drawing; that which is written, a character, an alphabet letter, written letter, piece of writing;" in plural, "letters," also "papers, documents of any kind," also "learning," from stem of graphein "to draw or write" (see -graphy). Some words with it are from Greek compounds, others are modern formations. Alternative -gramme is a French form.

From telegram (1850s) the element was abstracted by 1959 in candygram, a proprietary name in U.S., and thereafter put to wide use as a second element in forming new commercial words, such as Gorillagram (1979), stripagram (1981), and, ultimately, Instagram (2010). The construction violates Greek grammar, as an adverb could not properly form part of a compound noun. An earlier instance was the World War II armed services slang latrinogram "latrine rumor, barracks gossip" (1944).

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-er (1)

English agent noun ending, corresponding to Latin -or. In native words it represents Old English -ere (Old Northumbrian also -are) "man who has to do with," from Proto-Germanic *-ari (cognates: German -er, Swedish -are, Danish -ere), from Proto-Germanic *-arjoz. Some believe this root is identical with, and perhaps a borrowing of, Latin -arius (see -ary).

Generally used with native Germanic words. In words of Latin origin, verbs derived from past participle stems of Latin ones (including most verbs in -ate) usually take the Latin ending -or, as do Latin verbs that passed through French (such as governor); but there are many exceptions (eraser, laborer, promoter, deserter; sailor, bachelor), some of which were conformed from Latin to English in late Middle English.

The use of -or and -ee in legal language (such as lessor/lessee) to distinguish actors and recipients of action has given the -or ending a tinge of professionalism, and this makes it useful in doubling words that have a professional and a non-professional sense (such as advisor/adviser, conductor/conducter, incubator/incubater, elevator/elevater).

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-able 

common termination and word-forming element of English adjectives (typically based on verbs) and generally adding a notion of "capable of; allowed; worthy of; requiring; to be ______ed," sometimes "full of, causing," from French -able and directly from Latin -abilis. It is properly -ble, from Latin -bilis (the vowel being generally from the stem ending of the verb being suffixed), and it represents PIE *-tro-, a suffix used to form nouns of instrument, cognate with the second syllables of English rudder and saddle (n.).

A living element in English, used in new formations from either Latin or native words (readable, bearable) and also with nouns (objectionable, peaceable). Sometimes with an active signification (suitable, capable), sometimes of neutral signification (durable, conformable). It has become very elastic in meaning, as in a reliable witness, a playable foul ball, perishable goods. A 17c. writer has cadaverable "mortal."

To take a single example in detail, no-one but a competent philologist can tell whether reasonable comes from the verb or the noun reason, nor whether its original sense was that can be reasoned out, or that can reason, or that can be reasoned with, or that has reason, or that listens to reason, or that is consistent with reason; the ordinary man knows only that it can now mean any of these, & justifiably bases on these & similar facts a generous view of the termination's capabilities; credible meaning for him worthy of credence, why should not reliable & dependable mean worthy of reliance & dependence? [Fowler]

In Latin, -abilis and -ibilis depended on the inflectional vowel of the verb. Hence the variant form -ible in Old French, Spanish, English. In English, -able tends to be used with native (and other non-Latin) words, -ible with words of obvious Latin origin (but there are exceptions). The Latin suffix is not etymologically connected with able, but it long has been popularly associated with it, and this probably has contributed to its vigor as a living suffix.

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