Morpheme

In Morpheme-based morphology, a morpheme is the smallest language unit that carries a semantic interpretation. Morphemes are, generally, a distinctive collocation of phonemes (as the free form pin or the bound form -s of pins) having no smaller meaningful members.

English example: The word "unbelievable" has three morphemes "un-", (negatory) a bound morpheme, "-believe-" a free morpheme, and "-able". "un-" is also a prefix, "-able" is a suffix. Both are affixes.

Types of morphemes

• Free morphemes like town, dog can appear with other lexemes (as in town-hall or dog-house) or they can stand alone, or "free". Allomorphs are variants of a morpheme, e.g. the plural marker in English is sometimes realized as /-z/, /-s/ or /-ɪz/.
• Bound morphemes like "un-" appear only together with other morphemes to form a lexeme. Bound morphemes in general tend to be prefixes and suffixes. Morphemes existing in only one bound form are known as "cranberry" morphemes, from the "cran" in that very word.
• Inflectional morphemes modify a word's tense, number, aspect, and so on. (as in the dog morpheme if written with the plural marker morpheme s becomes dogs).
• Derivational morphemes can be added to a word to create (derive) another word: the addition of "-ness" to "happy", for example, to give "happiness".

Reference

Spencer, Andrew (1992). Morphological Theory. Oxford:Blackwell. 

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Wikipedia:Morpheme