Algebraic aspects

Abstract

We start with the introduction and investigation of some elementary algebraic properties of t-norms: idempotent and nilpotent elements and zero divisors, then strict monotonicity, the cancellation law and, most importantly, the Archimedean property as well as strict and nilpotent t-norms. The relationship between these properties is described in full detail, including many (counter-)examples. An important result is that each continuous Archimedean t-norm is either strict or nilpotent.

Algebra is essentially concerned with calculating,that is, performing, on elements of a set, “algebraic operations”, the most well-known example of which is provided by the “four rules” of arithmetic. [...] It is no doubt the possibility of these successive extensions, in which the form of the calculations remained the same, whereas the nature of the mathematical entities subjected to these calculations varied considerably, which was responsible for the gradual isolation of the guiding principle of modern mathematics, namely that mathematical entities in themselves are of little importance; what matters are their relations...

Nicolas Bourbaki