My understanding of morphology is that a word is taken and many different words are glued to it.
Is not this true for both agglutinative and polysynthetic languages? Or what is the finer level of distinction?
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"A word is taken and many different words are glued to it" — that's wrong for both agglutinative and polysynthetic languages.
In agglutinative languages, a string of affixes is "glued" to a root, each affix with its own grammatical meaning, an affix doesn't combine several grammatical meanings, like in Latin 'pueris' (from boys) the affix '-is' means plural + ablative case at the same time. An example of a word of an agglutinative language:
Turkish evlerimden: ev-ler-im-den "from my houses" - HOUSE-plural-my-ablative.case
In polysynthetic languages there are also many morphemes in a word, so that noun, verb and adverb morphemes can combine in one word in such a way that a whole English sentence is needed to translate such a word:
Yupik tuntussuqatarniksaitengqiggtuq: tuntu-ssur-qatar-ni-ksaite-ngqiggte-uq "He had not yet said again that he was going to hunt reindeer." REINDEER-HUNT-future-SAY-negation-AGAIN-third.person.singular.indicative
Polysynthetic languages are often agglutinative.
Agglutination is a form of inflection. So is fusion (aka amalgamation).
The major difference is that agglutinative paradigms are one-dimensional,
while fusional paradigms are multi-dimensional. Consequently one fusional inflection can refer to
many categories (e.g, Latin -tis '2nd person plural subject of verb in present tense, active voice, indicative mood), whereas one agglutinative inflection refers to one category (e.g, Turkish '-im_ '1st person subject' and -iz 'plural subject' -- -imiz together means '1st person plural subject').
Languages with a lot of inflections are called Synthetic languages.
Their inflection may be either agglutinative or fusional.
Languages that have so much inflection that there is no simple way
to distinguish an inflected word from a clause are called Polysynthetic languages.
The inflection in a polysynthetic language may be agglutinative or fusional or compound-root.
Generally there's some of each, but there are an awful lot of complex paradigms.