Spectral primal spaces

Let f: X → X be a mapping. Consider (f) = {O ⊆ X: F-1(O) ⊆ O}. Then, according to Echi, (f) is an Alexandroff topology. A topological space X is called a primal space if its topology coincides with an (f) for some mapping f: X → X. We denote by Fix(f):= {x X|f(x) = x} the set of all fixed points of f, and Per(f):= {x X|fn(x) = x,for some positive integer n} the set of all periodic points of f. The topology (f) induces a preorder ≤f defined on X by: X ≤fy if and only if y = fk(x), for some integer k ≥ 0. The main purpose of this paper is to provide necessary and sufficient algebraic conditions on the function f in order to get (f) (respectively, the one-point compactification of (f)) a spectral topology. More precisely, we show the following results. (1) (f) is spectral if and only if Per(f) is a finite set and every chain in the ordered set (X,≤f) is finite. (2) The one-point(Alexandroff) compactification of (f) is a spectral topology if and only if Per(f) = Fix(f) and every nonempty chain of (X,≤f) has a least element. (3) The poset (X,≤f) is spectral if and only if every chain is finite. As an application the main theorem [12, Theorem 3. 5] of Echi-Naimi may be derived immediately from the general setting of the above results.