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If some phonological process (like gemination from here) happens only at morpheme boundary (say, only in the coda of a syllable when a certain suffix is added to the stem whose final syllable is that syllable), is there a way to express this in the SPE notation? Or do just people say "this rule applies only at morpheme boundary" (or something like that)? Or if not, is there any standard way to express such things?

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As far as I'm aware, morpheme boundaries were indicated in SPE by a large + symbol. The standard way to address the type of phenomenon you describe was a combination of boundary erasure and the cycle. The basic idea is that all rules apply on one cycle, then (some) boundaries are erased, and then all rules apply on the next cycle, etc. This allows for distinguishing different kinds of morpheme, some of which trigger phonological rules and some of which do not. Cyclicity has survived in general in Rule-Based Phonology (post-SPE phonology), but I'm not up on the latest formalisms used.

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