  We introduce two new algebras that we call \emph{tied--boxed Hecke algebra}
and \emph{tied--boxed Temperley--Lieb algebra}. The first one is a subalgebra
of the algebra of braids and ties introduced by Aicardi and Juyumaya, and the
second one is a tied--version of the well known Temperley--Lieb algebra. We
study their representation theory and give cellular bases for them.
Furthermore, we explore a strong connection between the tied--boxed
Temperley--Lieb algebra and the so--called partition Temperley--Lieb algebra
given by Juyumaya. Also, we show that both structures inherit diagrammatic
interpretations from a new class of monoids that we call \emph{boxed ramified
monoids}. Additionally, we give presentations for the singular part of the
ramified symmetric monoid and for the boxed ramified monoid associated to the
Brauer monoid.
