The product property of the almost fixed point property for digital spaces

Abstract

Consider two digital spaces $ (X_i, k_i), i \in \{1, 2\} $, (in the sense of Rosenfeld model ) satisfying the almost fixed point property( AFPP for brevity). Then, the problem of whether the AFPP for the digital spaces is, or is not necessarily invariant under Cartesian products plays an important role in digital topology, which remains open. Given a Cartesian product $ (X_1 \times X_2, k) $ with a certain $ k $-adjacency, after proving that the AFPP for digital spaces is not necessarily invariant under Cartesian products, the present paper proposes a certain condition of which the AFPP for digital spaces holds under Cartesian products. Indeed, we find that the product property of the AFPP is strongly related to both the sets $ X_i $ and the $ k_i $-adjacency, $ i \in \{1, 2\} $. Eventually, assume two $ k_i $-connected digital spaces $ (X_i, k_i), i \in \{1, 2\} $, and a digital product $ X_1 \times X_2 $ with a normal $ k $-adjacency such that $ N_k^\star(p, 1) = N_k(p, 1) $ for each point $ p \in X_1 \times X_2 $ (see Remark 4.2(1)). Then we obtain that each of $ (X_i, k_i), i \in \{1, 2\} $, has the AFPP if and only if $ (X_1 \times X_2, k) $ has the AFPP .