**Financial support through the following projects:**

UID/MAT/00297/2019

UID/MAT/00297/2013

PEst-OE/MAT/UI0297/2014

PEst-OE/MAT/UI0297/2011

This lecture deals with semigroups of transformations preserving a partial ordered set $(X;\leq )$, i.e. $a\leq b\Rightarrow a\alpha \leq b\alpha $ for all $a,b\in X$. We will give a survey about the current knowledge concerning semigroups of transformations preserving a poset, where the linear order will mainly be in our target.

In a second part, we introduce the concept of a fence. A fence is a particular partially ordered set. Each element in a fence is either maximal or minimal, and fences have several interesting applications in mathematics whose we will discuss. In this lecture, we will consider a particular fence $% (\mathbb{N};\leq )$, where $\ \mathbb{N}$ is the set of all natural numbers and obtain that the semigroup $OP_{\mathbb{N}}$ of all partial transformations on $\mathbb{N}$ preserving the fence $(\mathbb{N};\leq )$ is not regular. We determine several maximal regular subsemigroups of $OP_{% \mathbb{N}}$, i.e. subsemigroups of $OP_{\mathbb{N}}$ which are not proper subsemigroups of any other regular subsemigroup of $OP_{\mathbb{N}}$ It will also be proved that there are countable infinite many maximal regular subsemigroups of $OP_{\mathbb{N}}$.