Mysql – How to show multiple names for one record but not duplicate the record

I might be missing something here, but maybe it only works for the first match because it sets t1.amount = 0, which means that record is excluded from a repeated invocation of the same command (whose WHERE clause requires t1.amount > 0)?

SELECT widget, MAX(`timestamp`) AS ts
FROM tableX AS t
WHERE state = 'down'
GROUP BY widget
HAVING NOT EXISTS
       ( SELECT *
         FROM tableX AS tt
         WHERE tt.widget = t.widget
           AND tt.state <> 'down'
           AND tt.`timestamp` > MAX(t.`timestamp`)
       ) ;

I think that you'll need two indices, one on (widget, state, timestamp) and one on (widget, timestamp, state) for efficiency.

This will work, too, and will be needing only one index, on (widget, timestamp, state):

SELECT t.widget, t.`timestamp`
FROM 
        tableX AS t
    JOIN
        ( SELECT widget, MAX(`timestamp`) AS ts
          FROM tableX
          GROUP BY widget
        ) AS tm
            ON  tm.widget = t.widget
            AND tm.ts = t.`timestamp`
WHERE t.state = 'down' ;

Tested both at SQL-Fiddle: test