The loop upper central series of an algebraic loop 
is an ascending sequence of normal subloops 
defined by 
and
 |
where 1 is the identity element (Semanišinová and Stanovský 2023). The center 
consists of the elements 
satisfying 
, 
, 
, and 
for every 
. Thus, central elements commute with every element
and associate with every pair in each position.
An algebraic loop is centrally nilpotent iff 
for some nonnegative integer 
. The least such 
is its class of central nilpotence. For a group,
this construction is the group upper central
series.
