  We investigate a new algebraic structure which always gives rise to a
set-theoretic solution of the Yang-Baxter equation. Specifically, a weak (left)
brace is a non-empty set $S$ endowed with two binary operations $+$ and $\circ$
such that both $(S,+)$ and $(S, \circ)$ are inverse semigroups and they hold
  \begin{align*}
  a \circ \left(b+c\right) = a\circ b - a +a\circ c \qquad \text{and} \qquad
  a\circ a^- = - a + a,
  \end{align*}
  for all $a,b,c \in S$, where $-a$ and $a^-$ are the inverses of $a$ with
respect to $+$ and $\circ$, respectively. In particular, such structures
include that of skew braces and form a subclass of inverse semi-braces. Any
solution $r$ associated to an arbitrary weak brace $S$ has a behavior close to
bijectivity, namely $r$ is a completely regular element in the full
transformation semigroup on $S\times S$. In addition, we provide some methods
to construct weak braces.
