As magistrates, they had their pet points (5)

I believe the answer is:

doges

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'as magistrates they had' is the definition.
The definition suggests an adverb but the answer is not.

'pet points' is the wordplay.
'pet' becomes 'dog'.
'points' becomes 'Es' (two compass points).
'dog'+'es'='DOGES'

'their' is the link.

Can you help me to learn more?

(Other definitions for doges that I've seen before include "Old Venetian magistrates" , "Chief magistrates of Venice and Genoa, formerly" , "Former rulers of Venice" , "Former lords of Venice" .)

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