Halliday & Matthiessen (1999: 376):
Such contextual regions evolve together with special functional varieties of the language or "registers". These are, in the first instance, registerial subsystems of the semantic system. A register is a semantic region within the overall semantic space. It is made up of contributions from all three metafunctions: a domain within the ideation base and similar regions within the interpersonal and textual parts of the overall semantic space. Our specifications of the meteorological and culinary domains constitute the ideational aspects of the registers of weather forecasting and recipes.
A given contextual specification of field, tenor and mode is thus realised by "preselecting" a register within the semantic system … . This contextual preselection within the semantics narrows down the overall potential to a registerial subpotential;
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Be aware that Martin and his students mistake register for context. That is, they mistake a functional variety of language — a subpotential of the semantic system — for the semiotic environment of language.