Representation of a Boolean algebra by its triangular norms
Given a complete and atomic Boolean algebra $B$, there exists a family $\tau_{\gamma}$ of triangular norms on $B$ such that, under the partial ordering of triangular norms, $\tau_{\gamma}$ is a Boolean algebra isomorphic to $B$, where $\gamma$ is the set of all atoms in $B$. In other words, as we ha...
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 1997 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2099/3482 |
| Acceso en línea: | https://hdl.handle.net/2099/3482 |
| Access Level: | acceso abierto |
| Palabra clave: | Triangular norms Complete and atomic Boolean algebra Fuzzy groups Anells booleans Classificació AMS::06 Order, lattices, ordered algebraic structures::06E Boolean algebras (Boolean rings) Classificació AMS::46 Associative rings and algebras::46H Topological algebras, normed rings and algebras, Banach algebras |
| Sumario: | Given a complete and atomic Boolean algebra $B$, there exists a family $\tau_{\gamma}$ of triangular norms on $B$ such that, under the partial ordering of triangular norms, $\tau_{\gamma}$ is a Boolean algebra isomorphic to $B$, where $\gamma$ is the set of all atoms in $B$. In other words, as we have shown in this note, every complete and atomic Boolean algebra can be represented by its own triangular norms. What we have not shown in this paper is our belief that $\tau_{\gamma}$ is not unique for $B$ and that, for such a representation, $B$ needs neither to be complete, nor to be atomic. |
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