let M be a magma, then then function cen defined by a function from a subset of M, X to cen(X) submagma of M are the largest set containing all elements commuting with elements of X

Their main property is that the cen(union of X_i) = intersection of cen(X_i) and the bicentralizer, cen^2(X) contains X. from this we deduce that cen^3(X)=cen(X) by choosing X_i to be X and the bicentralizer of X. we can also deduce that if X>Y then cen(Y)>cen(X). > being a non strict order. Then we know that the set X froms a cyclic semigroup starting from X then going into a 2 cycle.

these properties feel very general for many functions with similar definitions. I think this has to do with the functions being defined using the "for all" quantifier which is known for the property

for all (OR X_i) = AND for all(X_i).

are there results on functions that satisfy the union/intersection property?