How Are The Paradigmatic And Syntagmatic Axes Related?

System is more abstract than structure:

Halliday & Matthiessen (2004: 22):

It will be clear that [system] is a more abstract representation than that of structure, since it does not depend on how the categories are expressed.

so syntagmatic organisation realises paradigmatic organisation:

Halliday & Matthiessen (2004: 24):

Structural operations — inserting elements, ordering elements, and so on — are explained as realising systemic choices. … When we speak of structural features as ‘realising’ systemic choices, this is one manifestation of a general relationship that pervades every quarter of language. Realisation derives from the fact that a language is a stratified system.

Halliday & Matthiessen (1999: 13):

… on the one hand, syntagmatic organisation realises paradigmatic organisation; on the other hand, types in a network of paradigmatic organisation correspond to fragments of syntagmatic specification …