We study conditions under which the lattice Id R of ideals of a given a commutative semiring R is complemented. At first we check when the annihilator I ∗ of a given ideal I of 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.

中文翻译：

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交换半环中的理想及其补语。

我们研究给定可交换半环R的理想点阵Id R的互补条件。首先，我们检查R的给定理想I的an灭者I ∗是I的补数。此外，我们研究an灭者理想的补数。接下来，我们研究所谓的Łukasiewicz半环。它们与MV代数相对应，后者在量子结构中使用，因为它们形成了多值逻辑以及量子力学逻辑的代数语义。我们用对合来描述这些半环的理想和全同核。最后，使用有限unit布尔环，给出了具有理想晶格互补的交换半环的构造。