Chajda Ivan, Länger Helmut
1Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech Republic.
2Faculty of Mathematics and Geoinformation, Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria.
Soft comput. 2019;23(14):5385-5392. doi: 10.1007/s00500-018-3493-2. Epub 2018 Aug 31.
We study conditions under which the lattice of ideals of a given a commutative semiring is complemented. At first we check when the annihilator of a given ideal of is a complement of . 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.
我们研究给定交换半环的理想格为可补格的条件。首先，我们考察何时给定理想的零化子是其补元。进一步地，我们研究零化子理想的补元。接下来，我们研究所谓的卢卡西维茨半环。它们与MV - 代数相对应，MV - 代数在量子结构中被使用，因为它们构成了多值逻辑以及量子力学逻辑的代数语义。我们描述了这些具有对合的半环的理想和同余核。最后，利用有限酉布尔环，给出了一种具有可补理想格的交换半环的构造。
