Multilinear equations in free inverse monoids. (Q863473)

The free inverse monoid over a set \(A\) is denoted by \(\text{FIM}(A)\). Let \(X\cup X^{-1}=\{x_1,\dots,x_q\}\cup\{x_1^{-1},\dots,x_q^{-1}\}\) be a set disjoint from \(A\cup A^{-1}\). An equation in \(\text{FIM}(A)\) is a pair \((u,v)\) of elements from \(\text{FIM}(A\cup X)\). A map \(\varphi\colon X\to\text{FIM}(A)\) is called a solution of an equation \((u,v)\) if \(\widetilde\varphi(u)=\widetilde\varphi (v)\), where \(\widetilde\varphi\colon\text{FIM}(A\cup X)\to\text{FIM}(A)\) is the unique homomorphism determined by \(\varphi\). The consistency problem for systems of equations is considered. \textit{B. V. Rozenblat} [Sib. Mat. Zh. 26, No. 6(154), 101-107 (1985; Zbl 0584.03005)] has shown that when \(|A|\geq 2\) the consistency problem for finite systems of equations in \(\text{FIM}(A)\) is undecidable. The author considers the class of so called multilinear equations. It is proved that for singular multilinear equations the problem of finding solutions is decidable but the consistency problem for finite systems of multilinear equations is undecidable if \(|A|\geq 2\).