  A regular semigroup is weakly generated by a set X if it has no proper
regular subsemigroups containing X. In this paper, we study the regular
semigroups weakly generated by idempotents. We show there exists a regular
semigroup FI(X) weakly generated by |X| idempotents such that all other regular
semigroups weakly generated by |X| idempotents are homomorphic images of FI(X).
The semigroup FI(X) is defined by a presentation $\langle
G(X),\rho_e\cup\rho_s\rangle$ and its structure is studied. Although each of
the sets $G(X)$, $\rho_e$, and $\rho_s$ is infinite for $|X|\geq 2$, we show
that the word problem is decidable as each congruence class has a canonical
form. If $FI_n$ denotes FI(X) for $|X|=n$, we prove also that $FI_2$ contains
copies of all $FI_n$ as subsemigroups. As a consequence, we conclude that (i)
all regular semigroups weakly generated by a finite set of idempotents, which
include all finitely idempotent generated regular semigroups, strongly divide
$FI_2$; and (ii) all finite semigroups divide $FI_2$.
