*This chapter considers the problem of solving a system of Boolean equations over a finite (atomic) Boolean algebra other than the two-valued one. A prominent “misnomer” in mathematical and engineering circles is the term ‘Boolean algebra’. This term is widely used to refer to switching algebra, which is just one particular case of a ‘Boolean algebra’ that has 0 generators, 1 atom and two elements belonging to B={0,1}.The chapter outlines classical and novel direct methods for deriving the general parametric solution of such a system and for listing all its particular solutions. A detailed example over Bis used to illustrate these two methods as well as a third method that starts by deriving the subsumptive solution first. The example demonstrates how the consistency condition forces a collapse of the underlying Boolean algebra to a subalgebra, and also how to list a huge number of particular solutions in a very compact space. Subsequently, the chapter proposes some potential applications for the techniques of Boolean-equation solving. These techniques are very promising as useful extensions of classical techniques based on two-valued Boolean algebra.*

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