Description
Let M(B) be the set of modal (possibility) operators on a nontrivial Boolean algebra B, and L be the bounded semilattice , where (f+g)(x) = f(x) + g(x), f0(x) = 0 for all x in B, and f1(x) = 0, if x = 0, and f1(x) = 1, otherwise. Clearly, f1 is the unary discriminator on B.
A discriminator decomposition algebra (DDA) is a bimodal algebra for which f+g = f1; DDAs have a close relationship to the weak mixed algebras of [1] and the logic K~ of [2]. A decomposition of f1 is a pair of possibility operators on B such that is a DDA. We investigate the question how the discriminator can be decomposed. Emphasis is given to the existence of pairs where f+g = f1 and both f and g are strictly below f1, and dual pseudocomplements in L.
[1] Düntsch, I., Orłowska, E., and Tinchev, T., Mixed algebras and their logics, Journal of Applied Non-Classical Logics, 2018
[2] Gargov, G., Passy, S., and Tinchev, T., Modal environment for Boolean speculations. In Skordev, D., editor, Mathematical Logic and Applications, 1987