Finite-field networks (FFNs) are a class of multi-agent systems over finite fields with sensing, computing, and communication capabilities. FFNs have been investigated extensively to save computing and communication resources. This paper summarizes the current research results to provide a direction for future research. First, different models of FFNs are reviewed, including FFNs with time-delays, switching topology, and leader-following structures. Then, the consensus and synchronization problems of multi-agent systems over finite fields are analyzed, and the necessary and sufficient conditions for consensus and synchronization of some autonomous systems have been derived in recent research. Finally, the distributed control of multi-agent systems over finite fields has been developed by many scholars based on various approaches.

Design of distributed pinning controllers for set stabilization of $k$-valued logical control networks is investigated in this paper. Firstly, by analyzing the coupling correlations among nodes, pinned node set is determined. Secondly, based on the solvability of a class of matrix equations, sufficient conditions which resort to local information are put forward for the design of pinning controllers. Furthermore, an algorithm for designing pinning feedback controllers is presented, where the designed controllers are related to part of state variables instead of all variables. Finally, two illustrative examples are presented to verify the effectiveness of the main results.

The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebras (PHAs) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product (STP) of matrices are reviewed. The zero sets are defined for non-invertible hypercomplex numbers in a given PHA, and characteristic functions are proposed for calculating zero sets. Then PHA of various dimensions are considered. First, classification of $2$-dimensional PHAs are investigated. Second, all the $3$-dimensional PHAs are obtained and the corresponding zero sets are calculated. Finally, $4$- and higher dimensional PHAs are also considered.

A kind of algebra, called numerical algebra, is proposed and investigated. As its opponent, non-numerical algebra is also defined. The numeralization and dis-numeralization, which convert non-numerical algebra to numerical algebra and vise versa, are considered. Product structure matrix (PSM) of a finite dimensional algebra is constructed. Using PSM, some fundamental properties of finite dimensional algebras are obtained. Then a necessary and sufficient condition for a numerical algebra to be a field is presented. Finally, the invertibility of Segre (commutative) quaternion and some related properties of matrices over Segre quaternion are investigated.