Abstract
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.
Highlights
Semirings play an important role in both algebra and applications. They share several important properties of rings, but, on the other hand, every distributive lattice with the least element can be recognized as an idempotent semiring
In the second part we study certain semirings with involution, so-called Łukasiewicz semirings and their ideals
We show that by forming the direct product of such semiring with a finite unitary Boolean ring we obtain a commutative semiring whose lattice of ideals has similar properties as the lattice of ideals of the original semiring
Summary
Semirings play an important role in both algebra and applications They share several important properties of rings, but, on the other hand, every distributive lattice with the least element can be recognized as an idempotent semiring. In the second part we study certain semirings with involution, so-called Łukasiewicz semirings and their ideals. These semirings originated in the study of quantum structures, see, for example, Bonzio et al (2016) and Chajda et al (2018) for the concepts and motivation. The ideals of Łukasiewicz semirings have interesting properties, and the congruence kernels of these semirings can be described . We use the fact that in such a case we have no skew ideals in the direct product and the lattice of ideals of the new semiring is the direct product of the lattices of ideals of its factors
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