Digital products with $ PN_k $-adjacencies and the almost fixed point property in $ DTC_k^\blacktriangle $

<abstract><p>Given two digital images $ (X_i, k_i), i \in \{1, 2\} $, first of all we establish a new $ PN_k $-adjacency relation in a digital product $ X_1 \times X_2 $ to obtain a relation set $ (X_1 \times X_2, PN_k) $, where the term $ ''$$ PN $" means $ ''$pseudo-normal". Indeed, a $ PN $-$ k $-adjacency is softer or broader than a normal $ k $-adjacency. Next, the present paper initially develops both the notion of $ PN $-$ k $-continuity and a $ PN $-$ k $-isomorphism. Furthermore, it proves that these new concepts, the $ PN $-$ k $-continuity and a $ PN $-$ k $-isomorphism, need not be equal to the typical $ k $-continuity and a $ k $-isomorphism, respectively. Precisely, we prove that none of the typical $ k $-continuity (<italic>resp.</italic> typical $ k $-isomorphism) and the $ PN $-$ k $-continuity (<italic>resp.</italic> $ PN $-$ k $-isomorphism) implies the other. Then we prove that for each $ i \in \{1, 2\} $, the typical projection map $ P_i: X_1 \times X_2 \to X_i $ preserves a $ PN_k $-adjacency relation in $ X_1 \times X_2 $ to the $ k_i $-adjacency relation in $ (X_i, k_i) $. In particular, using a $ PN $-$ k $-isomorphism, we can classify digital products with $ PN_k $-adjacencies. Furthermore, in the category of digital products with $ PN_k $-adjacencies and $ PN $-$ k $-continuous maps between two digital products with $ PN_k $-adjacencies, denoted by $ DTC_k^\blacktriangle $, we finally study the (almost) fixed point property of $ (X_1 \times X_2, PN_k) $.</p></abstract>