On the relationship between tetravalent modal algebras, symmetric Boolean algebras and modal algebras for S5
In this paper we prove, on the one hand, that A. Monteiro's tetravalent modal algebras plus a Boolean complement are the same as De Morgan algebras plus a Boolean complement or, equivalently, Boolean algebras plus a De Morgan negation. The latter are called symmetric (or involutive) Boolean algebras (IBAs), introduced by Gr. Moisil and mainly studied by A. Monteiro. On the other hand, we prove that IBAs are modal algebras for modal logic S5 satisfying additional equations such that the variety of IBAs is generated by the Henle algebra H2. Thus, the logic that can be naturally associated to IBAs is a proper
normal extension of S5.