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Abstract
In this dissertation, we study the solvability of equations in the free inverse monoid generated by a set A. This monoid is denoted by FIM(A). Our study is accomplished by using the following fact. A necessary condition for an equation to be consistent (i.e. have a solution) in FIM(A) is that the equation is consistent in the setting of the corresponding free group generated by A. This group is denoted by FG( A). If the equation is consistent in FG( A), we examine conditions on that equation to determine whether it is consistent in the setting of FIM(A ).
It has been shown that the problem of determining consistency of systems of equations in FIM(A) is undecidable. We consider specific classes of equations and show that consistency is decidable. In particular, we examine multilinear, single variable and quadratic equations in FIM(A).
We show that determining the consistency for systems of multilinear equations in FIM(A) is undecidable. In addition, we associate with each system of equations in FIM(A ), a single equation in a free inverse monoid generated by a set which contains A; this equation has the property that its solutions in FIM(A) are the same as the solutions to the original system of equations in FIM( A).
