Halliday & Matthiessen (1999: 605-6):
From a semantic point of view, such systems constitute contexts for language; they can thus be modelled as part of a general linguistic theory, being interpreted as a higher stratum of language itself. That is to say, we can extend the series:the system of phonology realises that of lexicogrammar;the system of [lexicogrammar realised in phonology] realises that of semantics;the system of [semantics realised in [lexicogrammar realised in phonology]] — which is what we call "language" — realises the system of context (i.e. the "culture", considered as a semiotic potential).Such higher-level systems (theories, institutions, genres), since they are realised in language, are realised as subsystems within the semantics and the grammar. These subsystems are what we have referred to as registers;
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Halliday also uses the term 'institution' as the general term for variant subsystems of context:
To be clear, this is 'genre' in Hasan's sense of rhetorical mode, not in Martin's sense of a stratum above a register stratum, which confuses text type, rhetorical mode and the different semantic structures that realise different rhetorical modes.