Involutive residuated lattices based on modular and distributive lattices. (Q466876)

Any residuated lattice, i.e. lattice ordered monoid with residuation, with a dualizing element also admits a canonical involution -- and, endowed with this additional operation, such is called an involutive residuated lattice, IRL. The paper under review deals with the question which involutive lattices \(L\) are reducts of IRLs. This happens if \(L\) is bounded and has an atom -- which is defined as the neutral element \(e\) of the monoid; such IRL are called \(e\)-minimal. An identity is provided which defines the variety generated by these. On the other hand, finite distributive \(L\) are constructed which are not reducts of any integral (\(e=\max L\)) nor odd (\(e\) dualizing) IRL. Finally, all IRL are described which have a finite height 2 lattice reduct.