Orthomodular lattices that are horizontal sums of Boolean algebras
Ivan Chajda, Länger Helmut · Commentationes Mathematicae Universitatis Carolinae · 2020
The paper deals with orthomodular lattices which are so-called horizontal sums of Boolean algebras. It is elementary that every such orthomodular lattice is simple and its blocks are just these Boolean algebras. Hence, the commutativity relation plays a key role and enables us to classify these orthomodular lattices. Moreover, this relation is closely related to the binary commutator which is a term function. Using the class $\mathcal H$ of horizontal sums of Boolean algebras, we establish an identity which is satisfied in the variety generated by $\mathcal H$ but not in the variety of all orthomodular lattices. The concept of ternary discriminator can be generalized for the class $\mathcal H$ in a modified version. Finally, we present several results on varieties generated by finite subsets of finite members of $\mathcal H$.