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?
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$\begingroup$ $\Cup$ and $\Cap$ maybe? I doubt there's a great choice if you need to avoid all three types of symbols you mentioned. $\endgroup$– Mark S.Dec 7, 2020 at 22:40
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$\begingroup$ i just need somthing meet/join symbols other then $\lor,\land,\cup,\cap,\sqcup$ $\endgroup$– user3865391Dec 7, 2020 at 23:07
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2$\begingroup$ 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. $\endgroup$– amrsaDec 8, 2020 at 10:29
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1$\begingroup$ 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. $\endgroup$– William DeMeoDec 8, 2020 at 11:48
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1 Answer
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Supremum and infimum, or greatest lower bound and least upper bound.
$$a\vee b=\sup(a,b)=\text{lub}(a,b)$$
$$a\wedge b=\inf(a,b)=\text{glb}(a,b)$$
