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 all 
. Thus, central elements commute with every element
and associate with every pair in each position.
Here, a normal subloop is the inverse image of the identity element under a map 
between algebraic loops satisfying 
and 
for all 
(Semanišinová and Stanovský 2023, §3.1).
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.
