One can turn the set of retractions of a lattice $\langle L,\leq\rangle$ into a poset $R_f(\mathbf L)$ by letting $f\leq g$ iff $f(x)\leq g(x)$ for all $x\in L$. In 1982 H. Crapo raised the following two problems: (1) Is it true that $R_f(\mathbf L)$ is a lattice for any lattice $\mathbf L$? (2) Is it true that $R_f(\mathbf L)$ is a complete lattice if $\mathbf L$ is a complete lattice? In 1990 and 1991 B. Li published two papers dealing with the above two questions. He showed that $R_f(\mathbf L)$ is not necessarily a lattice and that $\mathbf L$ is a complete lattice if and only if $R_f(\mathbf L)$ is a complete lattice. Motivated by the idea of extending the structure from the base set to the set of all retractions, we introduce the notion of $\mathrm R$-algebra as follows. Let $R_f(\mathbf A)$ denote the set of all retractions of an algebra $\mathbf A$. We say that $\mathbf A$ is an $\mathrm R$-algebra if the set $R_f(\mathbf A)$ is closed with respect to operations of $\mathbf A$ applied pointwise. We give some necessary and some sufficient conditions for $\mathbf A$ to be an $\mathrm R$-algebra. We show that the property of being an $\mathrm R$-algebra carries over to retracts of the algebra. In a set of examples we show that almost no classical algebra is an $\mathrm R$-algebra. In particular, a lattice $\mathbf L$ is an $\mathrm R$-algebra iff $|L|\leq 2$.