Since the dual of a modular lattice is again modular, ?? is also the identity on [ " a ", " a " ( " " b " ], and therefore the two maps ? and ? are isomorphisms between these two intervals.
22.
In mathematics, the "'butterfly lemma "'or "'Zassenhaus lemma "', named after Hans Zassenhaus, is a technical result on the lattice of subgroups of a group or the lattice of submodules of a module, or more generally for any modular lattice.
23.
Maltsev's characterization explains a large number of similar results in groups ( take ), rings, quasigroups ( take, complemented lattices, Heyting algebras etc . Furthermore, every congruence-permutable algebra is congruence-modular, i . e . its lattice of congruences is modular lattice as well; the converse is not true however.
