# Notation for joins/meets in lattice theory

Most often I see $$\lor$$ and $$\land$$ used for the join and meet operations respectively in a lattice, however what I am writing makes concurrent use of both symbolic logic and a particular lattice, thus I am already using these operations to express logical conjunction/disjunction and am unsure what to use for the joins/meets. For example I don't want to use $$\cup$$ and $$\cap$$ since I think this might cause confusion with sets or enable some unconscious error where I treat them like their set counter parts i.e. I might accidentally distribute them over each other despite not dealing with a distributive lattice, likewise for similar reasons I'd rather not use $$\sqcup$$ and $$\sqcap$$ as I often use the latter for 'disjoint' unions. So with all of that in mind what are some standard alternatives to these that I could use for joins/meets in a lattice?

• $\Cup$ and $\Cap$ maybe? I doubt there's a great choice if you need to avoid all three types of symbols you mentioned. Dec 7, 2020 at 22:40
• i just need somthing meet/join symbols other then $\lor,\land,\cup,\cap,\sqcup$ Dec 7, 2020 at 23:07
• In the Boolean algebra setting, it is also common to use additive and multiplicative notation for join and meet, respectively. I don't remember now a reference, but I think that in ancient Lattice Theory papers this was also common, for some authors. Perhaps the reason was just that those symbols were more easily accessed. Personally, I think it looks weird, specially when dealing with equalities in which both operations appear; for example, one distributivity law would be $$x(y+z)=xy+xz,$$ which looks OK, but the other equivalent one, $$x+yz=(x+y)(x+z),$$ not really. Dec 8, 2020 at 10:29
• As @amrsa suggests, old lattice theorists used + for join and * or $\cdot$ for meet. I recommend using those since they were standard well before $\vee$ and $\wedge$ came into fashion. Dec 8, 2020 at 11:48

$$a\vee b=\sup(a,b)=\text{lub}(a,b)$$
$$a\wedge b=\inf(a,b)=\text{glb}(a,b)$$