Polynomial-Time Algorithms for Black-Box Distributive Expanded Groups

Let $\Omega$ be a finite set of finitary operation symbols. An $\Omega$-expanded group is a group (written additively and called the additive group of the $\Omega$-expanded group) with an $\Omega$-algebra structure. We use the black-box model of computation in $\Omega$-expanded groups. In this model, elements of a finite $\Omega$-expanded group $H$ are represented (not necessarily uniquely) by bit strings of the same length, say, $n$. Given representations of elements of $H$, equality testing and the fundamental operations of $H$ are performed by an oracle. Assume that $H$ is distributive, i.e., all its fundamental operations associated with nonnullary operation symbols in $\Omega$ are distributive over addition. Suppose $s=(s_1,\dots,s_m)$ is a generating system of $H$. In this paper, we present probabilistic polynomial-time black-box $\Omega$-expanded group algorithms for the following problems: (i) given $(1^n,s)$, construct a generating system of the additive group of $H$, (ii) given $(1^n,s,(t_1,\dots,t_k))$ with $t_1,\dots,t_k\in H$, find a generating system of the additive group of the ideal in $H$ generated by $\{t_1,\dots,t_k\}$, and (iii) given $(1^n,s)$, decide whether $H\in\mathfrak V$, where $\mathfrak V$ is an arbitrary finitely based variety of distributive $\Omega$-expanded groups with nilpotent additive groups. The error probability of these algorithms is exponentially small in $n$. In particular, this can be applied to groups, rings, $R$-modules, and $R$-algebras, where $R$ is a fixed finitely generated commutative associative ring with $1$. Rings and $R$-algebras may be here with or without $1$, where $1$ is considered as a nullary fundamental operation.