Directional Kronecker algebra for ${\mathbb{Z}}^q$-actions

In this paper, directional sequence entropy and directional Kronecker algebra for ${\mathbb{Z}}^q$-systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, direcitonal discrete spectrum systems and directional null systems are defined. It is shown that a ${\mathbb{Z}}^q$-system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a ${\mathbb{Z}}^q$-system has directional discrete spectrum along $q$ linearly independent directions if and only if it has discrete spectrum.