Ideals and their complements in commutative semirings
| Autor: | Ivan Chajda, Helmut Länger |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
0209 industrial biotechnology
Pure mathematics Complement 02 engineering and technology Unitary state Theoretical Computer Science 020901 industrial engineering & automation Lattice (order) 0202 electrical engineering electronic engineering information engineering Algebraic number Lattice of ideals Łukasiewicz semiring Commutative property Quantum Annihilator Mathematics Idempotent semiring Commutative semiring Mathematics::Commutative Algebra Boolean ring 16. Peace & justice Ideal Complemented lattice 020201 artificial intelligence & image processing Geometry and Topology Software Foundations |
| Zdroj: | Soft Computing |
| ISSN: | 1433-7479 1432-7643 |
| Popis: | We study conditions under which the lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{\mathbf {Id}}}}\mathbf R$$\end{document}IdR of ideals of a given a commutative semiring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {R}}$$\end{document}R is complemented. At first we check when the annihilator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^*$$\end{document}I∗ of a given ideal I of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {R}}$$\end{document}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. |
| Databáze: | OpenAIRE |
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