By analogy with rings and rngs, we will assume all lattices are bounded, and refer to the unbounded case as a “lattce” (pronounced ‘lats’). Explicitly, a lattce is a pair of Abelian semigroups connected by the absorption axiom . It is clear that this absorption forces the component structures to be idempotent and hence semilattices, or rather semilattces.
Now consider the operation defined by . Obviously this is simply the left zero band, and this leads to a generalization of latt(i)ces.
A circumlatt(i)ce is a pair of semilatt(i)ces wherein the term-defined absorption magma is a band - and since idempotence is inherited from idempotence of join and meet, we actually only need to enforce associativity. That means we could also give it as a pair of semilatt(i)ces connected by the compatibilities and . Then a circumlatt(i)ce is a latt(i)ce precisely when the absorption magma is the left-zero band. Note that any semilatt(i)ce forms a circumlatt(i)ce when it is both join and meet, so let us exclude this case as trivial.
There are several interesting cases to narrow in on. A normal circumlatt(i)ce is a circumlatt(i)ce whose absorptive band is left normal (). A rectangular circumlatt(i)ce is a circumlatt(i)ce whose absorptive band is rectangular (). Every latt(i)ce is both a normal and rectangular circumlatt(i)ce and in fact latt(i)ces are precisely those circumlatt(i)ces which are both normal and rectangular.
You may be tempted to define a triple circumlatt(i)ce as a pair of (non-identical) semilatt(i)ces with a commutative absorption band (hence also a semilattce), but in fact all such structures necessarily have identical join and meet.