First Girl/Alpha slave
When more than one slave stands in a relationship of slave to the same master/mistress, as when they serve in the same shop or house, or adorn the same pleasure gardens, it is common for their owner to appoint a "first girl" or alpha slave.
That slave's authority is then to the other slaves as that of the master or mistress, much like a majordomo.
It is generally her job to see to it that the other slaves in the household, business, camp, etc. mind their station, stay in line and do their duties in the owner's absence.
This slave usually has switch and/or whip rights over her sister or brother slaves, and will often not hesitate to use said rights, as the first girl can be punished for failing in HER duties as well as the errant slave if the slaves under her misbehave.
They should resist any urges to be cruel or petty, and strive to govern with intelligence and justice, as, at any time, another slave, at the owner's whim, may become first, and they themselves may then be under his or her almost absolute power.
The FG/AS is commonly addressed by other slaves in the household/establishment as Master or Mistress, just as though they were free.
The rest of the slaves are generally ranked according to length in service and/or level of training, in addition to their favor with their owner.
It should be noted that visiting slaves are almost always ranked below those slaves in residence. For instance, even the first girl belonging to a king or tribal chief visiting a humble shopkeeper would be placed under the authority of the host's first slave.
Kettle/Kitchen slave
This is a lower slave. Usually, the newest slaves are assigned to kitchen duty. They generally are expected to follow the directions of most of the other slaves in the household or establishment and are given tasks that require less skill or are more unpleasant. A slave may also be "demoted" to kettle slave status as a punishment.
Low slave
This ought to be self explanatory.
There is usually only one "low slave" at any given time and it is almost always assigned as a disciplinary measure.
The low slave must address ALL other slaves in the house or establishment as Master or Mistress and is required to serve and obey them as though they were free. The filthiest and least pleasant tasks generally fall to the low slave. Commonly, the low slave is also stripped of all clothing, personal hygiene items, bedding and food privileges, and must beg his or her chain sisters/brothers for anything that they may deign to give up for their use.
If the low slave is very much out of favor with their fellow slaves, it may be necessary to supplement their diet with slave gruel, a bland and often unpleasant tasting, but highly nutritious, thin, porridge-like substance to keep them from being malnourished.
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As you can see, It is not only important to remain pleasing to the free, but be pleasant to your chain siblings, lest you find yourself one day at their utter mercy.
There are other, more subtle "levels", but these are the primary distinctions.