Ideals and their complements in commutative semirings

We study conditions under which the lattice [Formula: see text] of ideals of a given a commutative semiring [Formula: see text] is complemented. At first we check when the annihilator [Formula: see text] of a given ideal I of [Formula: see text] is a complement of I. Further, we study complements of annihilator ideals. Next we investigate so-called Łukasiewicz semirings. These form a counterpart to MV-algebras which are used in quantum structures as they form an algebraic semantic of many-valued logics as well as of the logic of quantum mechanics. We describe ideals and congruence kernels of these semirings with involution. Finally, using finite unitary Boolean rings, a construction of commutative semirings with complemented lattice of ideals is presented.