The identity problem of finitely generated bi-ideals

Abstract

Finitely generated bi-ideals with letters from a selected alphabet A are considered. We solve the equivalence problem for generating systems of bi-ideals, i.e., look for an effective procedure which provides the means of determining if two generating systems ${\langle u_0, . . . , u_{m-1} \rangle}$ and ${\langle v_0, . . . , v_{n-1} \rangle}$ represent equal or different bi-ideals. We offer a method of constructing, for every generating system ${\langle u_0, . . . , u_{m-1} \rangle}$ , an equivalent generating system ${\langle u^{\prime}_{0}, . . . , u^{\prime}_{m-1} \rangle}$ with differing members. We also describe an algorithm for deciding if two generating systems ${\langle u_0, u_1 \rangle}$ and ${\langle v_0, v_1 \rangle}$ are equivalent or not. For a general case, the problem of existence of such an algorithm remains open.