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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"?

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If some finite subsets of $L$ does not have join/meet, then $L$ is not a lattice. (By definition a lattice admits binary joins and meets; by induction it admits finite joins and meets.) By definition every lattice $L$ is "weak partial complete": just define $\wedge$ and $\vee$ to be the join and meet operators on finite elements in the power set, and undefined for everything else. –  Willie Wong Nov 6 '12 at 13:11
    
But is the "weak partial complete lattice" still a worthy object of study? It is discussed in the paper I linked to in the post, after all. The definition in that paper does not require finite subsets to have a join/meet. –  Herng Yi Nov 6 '12 at 13:19
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The point is that these objects are not lattices or semilattices, and you shouldn't think of them as such. On any poset you can define the trivial partial functions $\wedge,\vee$ (so that they are undefined for any subset) and you would have a so-called "weak partial complete lattice". So whether it is worth studying mostly boils down to "what can you say about these guys that you couldn't say about posets"? (And in your question, you probably should say that these are generalisations of "lattice" and not just "complete lattice".) –  Willie Wong Nov 6 '12 at 13:29

2 Answers 2

up vote 2 down vote accepted

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.

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But the function should be monotone if the binary join between comparable elements is defined as the maximum, right? $a \leq b \implies f(b) = f(b \vee a) = f(b) \vee (a) \implies f(a) \leq f(b)$ –  Herng Yi Nov 6 '12 at 13:42
    
yes, that would suffice. –  Ittay Weiss Nov 6 '12 at 13:43

Since you posted a reference request for terminology/notation, you may want to look at George A. Grätzer, General Lattice Theory. (Google Books link).

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