Talk:Triangular norms and conorms
There are problems with links to citations and with \L in the name of \L ukasiewicz. --Navara
PLEASE, HELP ME IF YOU CAN! MN
1. The expression
\left]a_j,b_j\right[\subseteq[0,1],\ j\in J,
has to appear in a somewhat different way.
PLEASE, SPECIFY! I DO NOT SEE ANYTHING WRONG ON IT. MN
2. I don't like the "definition" of Archimedean t-norms. It is true that this formulation of being Archimedean is equivalent to the real definition but one should follow the mathematical traditions in this respect. Therefore I would call a t-norm Archimedean in this paragraph, if for any $x,y\in]0,1[$ there exists $n\in N$ such that $T(\underbrace{x,x,\ldots,x}_n)<y$. This formulation makes a link to other fields of mathematics, where the notion of Archimedean is defined in a similar way.
OF COURSE, I MAY CHANGE IT; I TRIED TO FIND THE SHORTEST WAY, WITHOUT THE NEED TO INTRODUCE NEW NOTATION FOR AN n-ARY OPERATION. AS AN ALTERNATIVE, I CONSIDERED A DEFINITION USING THE SEQUENCE \(x_{n+1}=T(x,x_n)\) MN
3. The expression
\mu_A,\mu_B
has to appear in a somewhat different way.
PLEASE, SPECIFY! I DO NOT SEE ANYTHING WRONG ON IT. MN
4. Very important remark! A t-norm DOES NOT have residuum, in general. Residuation is an old algebraic notion, a kind of dual notion of Galois connections. One has to follow the much older tradition of algebra in the newer fuzzy theory to maintain a consistent notation! Therefore, ONLY left-continuous t-norms have residuum, and it is defined with max instead of sup in the corresponding expression.
HERE WE MAY ADD THE ASSUMPTION OF LEFT-CONTINUITY, ALTHOUGH IT IS NOT USED IN [KLEMENT, MESIAR, PAP]. MN
5. The name of P. Hájek must not appear as Hajek in the text.
THIS WAS AIMED ONLY AS A TEMPORARY LINK UNTIL I LEARN HOW TO LINK IT TO THE REFERENCES AT THE END. MN
6. I would suggest: "Then they were used as a natural interpretation of the conjunction in the semantics of mathematical fuzzy logics [Hájek] ..." ACCEPTED.
7. The word "familis" is not correct. ACCEPTED.
8. Finally a general remark. I miss the strong emphasism of the role of left-continuous t-norms. Left-continuity of the t-norm is a crucial property in mathematical fuzzy logic, see e.g. my remark concerning residuation. This property makes it possible to define the so-called residual implication, which plays the role of the semantic interpretation of implication in mathematical fuzzy logics. Implication is the most important connective in any logical system.
Left-continuity makes a link to the well-established theory of residuated lattices in algebra.
I remember that in early discussions about t-norms some scientists (e.g. C. Drossos) had thought that left-continuity had to be added as a fifth condition to the BASIC definition of t-norms.
Could you add a paragraph on left-continuous t-norms.
I THINK THAT FUZZY IMPLICATIONS ARE WORTH A SEPARATE ENTRY. I NOBODY ELSE, I MIGHT WRITE IT. THERE THE ROLE OF LEFT-CONTINUITY WOULD BE TREATED IN DETAIL. MN
Some comments
Hi. This page is written in a way that is easy to read. Great! However, I would suggest adding an introduction that includes some motivating information before the main text. Something like this: A triangular norm (abbreviation t-norm) is an operation which generalizes the logical conjunction in fuzzy logic. They are a natural interpretation of the conjunction in the semantics of mathematical fuzzy logics [Hájek (1998)] and are used to combine criteria in multi-criteria decision making.
(Or some other important applications.)
In the next section, you could continue with the formal definition: A triangular norm is a binary operation T on the interval [0,1] satisfying the following conditions:
I'm only suggesting this because, though the presentation is clear, my first question is "why is this important?" and I have to scroll down to find out.
User 4: Neutral Element
Section Triagular Norms > Definition > Neutral Element:
I think the result of applying the neutral element is \(T(x,1)=x\) instead of \(T(x,1)=1\).
Of course, yes. The same applies to t-conorms. Thank you. MN
