Decision problem for orthomodular lattices

Abstract

This paper answers a question of H. P. Sankappanavar who asked whether the theory of orthomodular lattices is recursively (finitely) inseparable (question 9 in [10]). A very similar question was raised by Stanley Burris at the Oberwolfach meeting on Universal Algebra, July 15–21, 1979, and was later included in G. Kalmbach’s monograph [6] as the problem 42. Actually Burris asked which varieties of orthomodular lattices are finitely decidable. Although we are not able to give a full answer to Burris’ question we have a contribution to the problem. Note here that each finitely generated variety of orthomodular lattices is semisimple arithmetical and therefore directly representable. Consequently each such a variety is finitely decidable. (For a generalization of this, i.e. a characterization of finitely generated congruence modular varieties that are finitely decidable see [5].) In section 3, we give an example of finitely decidable variety of orthomodular lattices that is not finitely generated.