… In SFL terms, a semiotic is a system of features realised by structures.
Hjelmslev defined connotative semiotics as having another semiotic as its expression plane, including those 'involving interpretation and cultural norms’ (in your terms), whose expression plane is language. …
Connotative semiotics should be describable as systems of field, tenor, mode and genre, as language is describable as metafunctional systems. The unit of analysis for these strata is minimally a text, up to indefinitely large series of texts (of which our threads are microcosms). But such descriptions depend on how we analyse texts. …
 This is misleading, because it is untrue. In SFL terms, a 'semiotic' need not involve structures, as exemplified by the semogenesis of protolanguage and traffic lights. More fundamental to the architecture of a 'semiotic' are the global* dimensions of stratification and instantiation.
* See, for example, Halliday & Matthiessen (2014: 32) for 'the global and local semiotic dimensions of language in context'.
 To be clear, in SFL theory, the more abstract semiotic system that has language as its expression plane, context, is modelled metafunctionally in terms of field, tenor and mode. However, what Rose strategically omits, here, is that he follows Martin (1992) in mistaking field, tenor and mode as systems of register. As functional varieties of language, registers are coherently modelled in SFL as subsystems of language, not systems of the context that is realised by language. For some of the inconsistencies in modelling registerial varieties of language as context, instead of language, see the reasoning here (register) or here (context).
 The notion of genre as a more abstract semiotic system than language is invalidated for the same reasons as register, which is not surprising, given that, in SFL theory, register and genre (text type) are two different angles on the very same point of variation on the cline of instantiation. Register is the view from the system pole, whereas text type (genre) is the view from the instance pole. As varieties of language, genres are language, and as such, not more abstract than language. The notion of genre as also more abstract than register thus involves further inconsistencies. For some of the inconsistencies in modelling generic varieties of language as context, instead of language, see the reasoning here.
To be clear, the identification of genre with text type in the model of genre as context is acknowledged in Working With Discourse (Martin & Rose 2007: 8):
We use the term genre in this book to refer to different types of texts that enact various types of social contexts.
A further complicating inconsistency in this regard is that, despite arguing that register and genre are levels of a connotative semiotic whose expression plane is language, Martin & Rose (2007: 4) neverthesss include the higher level semiotic within the strata of language:
[relevant] levels of language: as grammar, as discourse, and as social context (known as the strata of language)
 To be clear, contrary to Rose's implication, the model of genre as context (Martin 1992: 546-73) is not described either in terms of metafunction, or in terms of system. See, for example, Martin's Reasons For Not Devising Genre Systems.
 Here Rose, having argued that register and genre are not language, but connotative semiotics whose expression plane is language, nevertheless identifies a unit of language, text, as a unit of the connotative semiotic. In terms of SFL theory, the inconsistency is thus a confusion of different levels of symbolic abstraction.