Representation of ordered projection semigroups

This paper studies the following relations on a semigroup of functions $(\Phi,\circ)$: the relation $\subset$ of inclusion of functions (i.e. $f\subset\varphi$ if the function $\varphi$ is an extension of the function $f$), and the relations $\lceil$ and $\lfloor$ of inclusion of the first and second projections, respectively (i.e. inclusions of the domains of definition or the ranges of values of the functions).
Necessary and sufficient conditions are found for an algebraic system $G$ to be isomorphic to a semigroup of single-valued functions taken together with the relations $\subset$, $\lceil$ and $\lfloor$.
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