Two notes on \(n\)-lattices (Q2759832)

\(L= (L;+_1,\dots,+_n)\) is called an \(n\)-fold semilattice if every \((L;+_i)\) is a semilattice, \(i\leq n\). An \(n\)-fold semilattice \(L\) is said to be an \(n\)-lattice if \(L\) satisfies the absorption identity NEWLINE\[NEWLINE(\cdots(x+_{i_1}y)+_{i_2}y)\cdots) +_{i_n} y= y,NEWLINE\]NEWLINE where \(i_j= j\tau\) for every permutation \(\tau\in S_n\). An \(n\)-lattice is fully absorptive if it satisfies the absorption identity for all sequences \(i_1,\dots, i_k\) such that \(\{1,\dots,n\}= \{i_1,\dots, i_k\}\).NEWLINENEWLINENEWLINEClearly, an \(n\)-lattice need not be fully absorptive. The author presents two non-trivial examples of an \(n\)-lattice that is fully absorptive.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00014].