Join-semilattice morphism from a join-semilattice $\mathfrak{A}$ to a join-semilattice $\mathfrak{B}$ is a function $\alpha$ conforming to the formula $\alpha(X\sqcup Y) = \alpha X\sqcup\alpha Y$ (where $\sqcup$ denotes the semilattice operations).
Equivalently transforming this formula (under the supposition that $\alpha$ is a monotone function), we get: $\alpha (X \sqcup Y) \sqsubseteq \alpha X \sqcup \alpha Y$; $\forall Z \in \mathfrak{B}: ((Z \sqsupseteq \alpha X \wedge Z \sqsupseteq \alpha Y) \Rightarrow Z \sqsupseteq \alpha (X \sqcup Y))$; $\forall Z \in \mathfrak{B}: ((Z \sqsupseteq \alpha X \wedge Z \sqsupseteq \alpha Y) \Rightarrow \exists T \in \mathfrak{A}: (T \sqsupseteq X \wedge T \sqsupseteq Y \wedge Z \sqsupseteq \alpha T))$
(here $\sqsubseteq$ is our partial order).
The last formula does not use semi-lattice operation and is defined for every posets $\mathfrak{A}$ and $\mathfrak{B}$.
Is this definition of "generalized semilattice morphism" known? What is its name? What are its properties? Where to read about it?