discrete dualities for monteiro's tetravalent modal algebras

Discrete duality refers to a form of duality in which a class of abstract relational systems serves as the dual counterpart to a class of algebras. these relational systems are called textit{frames}, following the terminology of non-classical logics. no topology is involved in the construction of these frames; hence, they can be considered as having a discrete topology. in 1978, a. monteiro introduced a class of algebras known as tetravalent modal algebras, which constitute a generalization of the three-valued {l}ukasiewicz algebras defined by moisil. the theory of these tetravalent modal algebras was initially developed by i. loureiro, followed by significant contributions from a. v. figallo, later enriched by the work of j. font and m. rius, and more recently by the works of m. coniglio and m. figallo. in this paper, we present two discrete dualities for monteiro's tetravalent modal algebras. each of these dualities involves a distinct class of frames and a unique definition of a complex algebra.