  We investigate the structure of $\omega$-limit (resp. $\alpha$-limit) sets
for a monotone map $f$ on a regular curve $X$. %Let $X$ be a regular curve and
let $f: X\longrightarrowX$ be a monotone map. We show that for any $x\in X$
(resp. for any negative orbit $(x_{n})_{n\geq 0}$ of $x$), the $\omega$-limit
set
  $\omega_{f}(x)$ (resp. $\alpha$-limit set $\alpha_{f}((x_{n})_{n\geq 0})$) is
a minimal set. This also hold for $\alpha$-limit set $\alpha_{f}(x)$ whenever
$x$ is not a periodic point.
  These results extend those of Naghmouchi \cite{n}
  %[J. Difference Equ. Appl., 23 (2017), 1485--1490]
  established whenever $f$ is a homeomorphism on a regular curve and those of
Abdelli \cite{a}
  %[Chaos, Solitons Fractals, 71 (2015), 66--72]
  , whenever $f$ is a monotone map on a local dendrite. Further results related
to the basin of attraction of an infinite minimal set are also obtained.
