Weak Partial Complete Lattice and Homomorphisms 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"?
 A: There is a whole range in between posets and complete lattices. For instance, $\sigma$-complete lattices are well studied lattices that admit all countable meets and joins. Similarly, you can consider $\kappa$-complete lattices for any cardinal $\kappa$. Other possibilities include posets (or lattices) where every chain has a meet and join. All of these possibilities produce interesting categories that lie between $Pos$ and $CLat$. The article you cite considers more examples. 
As for a function between weak partial complete lattices that preserves joins, you can call it a join preserving function. I am not aware of a standard name. Notice however that such a function will not automatically be monotone, unless all binary joins are taken in.
Fianlly, it is more customary to use $\bigvee$ for the join operation, though $\Sigma$ is also used. You don't need to be too specific about it since it will be clear from the context. 
A: Since you posted a reference request for terminology/notation, you may want to look at George A. Grätzer, General Lattice Theory. (Google Books link). 
