Ideals and their complements in commutative semirings
Ivan Chajda, Helmut Länger · Soft Computing · 2018
We study conditions under which the lattice $${{\mathrm{\mathbf {Id}}}}\mathbf R$$ of ideals of a given a commutative semiring $${\mathbf {R}}$$ is complemented. At first we check when the annihilator $$I^*$$ of a given ideal I of $${\mathbf {R}}$$ 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.