Description of Research Interests

My primary and current research work is in the general spirit of algebras of logic, lattice theory, hyper structures and applications. A partially ordered set, or poset for short is a pair (P, ≤) where P is a set and ≤ a partial order on P. A poset (P, ≤) is called a lattice if every pair x, y ∈ P has a least upper bound x ∨ y and a greatest lowest bound x ∧ y in P. A lattice is bounded if it has a smallest element 0 and a greatest element 1. We denote such a lattice by (L, ∧, ∨, 0, 1). Bounded lattices arise in every corner of Mathematics. The simplest example of a bounded lattice is the power set (P(X), ∩, ∪, ∅, X) of a fixed set X. In ring theory, the set of ideals of a fixed ring R is a bounded lattice, where the infimum is the intersection and the supremum is the ideal generated by the union. Yet another natural example is found in topology, if X is a set and τ is a topology on X with the set of opens denoted by O(X); then (O(X), ∩, ∪, ∅, X) is a bounded lattice. A residuated lattice is a bounded lattice (L, ∧, ∨, 0, 1) together with two binary operations ⊗ and → satisfying: (RL1) (A, ⊗, 1) is a commutative monoid; (RL2) x ⊗ y ≤ z if and only if x ≤ y → z for all x, y, z ∈ L. In any residuated lattice, we have a natural complement ¯ x := x → 0. The terminology complement is motivated by the fact that the power set of a fixed set is a residuated lattice in a natural way, and this operation coincides with the usual set complementation. The are several subclasses of residuated lattices that have been studied: Boolean algebras, Heyting algebras, MV-algebras, BL-algebras. These classes are studied both for their rich algebraic properties, and also for their connections to various logical systems. A Heyting algebra is a residuated lattice with ∧ = ⊗. A BL-algebra is a residuated lattice satisfying: (BL1) x ∧ y = x ⊗ (x → y) (Divisibility); (BL2) (x → y) ∨ (y → x) = 1 (Prelinearity). An MV-algebra is a BL-algebra whose complement is an involution: x = x for all x ∈ L.