What is the proper nomenclature for a generalization of a lattice $L$ such that not all subsets of $L$ may have a join/meet, sometimes not even for finite subsets? This paper calls it a "weak partial complete lattice", but could I check if this is standard terminology? What about semilattices, where only join is defined?
Given two such "weak partial complete lattices" $L$ and $M$, consider a function $f : L \to M$ such that whenever the $\bigvee L'$ is defined for some $L' \subseteq L$, $\bigvee f(L')$ is defined and $f(\bigvee L') = \bigvee f(L')$. Should I call $f$ a "weak partial complete semilattice homomorphism", or is there some snappier standard name?
Finally, if my join operation for a semilattice is $\sum$, may I abbrieviate "a join-semilattice with $\sum$ as join" to "a $\sum$-semilattice"?