Totally Ordered Monoids Based on Triangular Norms

A totally ordered monoid, or tomonoid for short, is a monoid together with a translation-invariant (i.e., compatible) total order. We consider in this paper tomonoids fulfilling the following conditions: the multiplication is commutative; the monoidal identity is the top element; all nonempty suprema exist; and the multiplication distributes over arbitrary suprema. The real unit interval endowed with its natural order and a left-continuous t-norm is our motivating example. A t-norm is a binary operation used in fuzzy logic for the interpretation of the conjunction.

Given a tomonoid of the indicated type, we consider the chain of its quotients induced by filters. The intention is to understand the tomonoid as the result of a linear construction process, leading from the coarsest quotient, which is the one-element tomonoid, up to the finest quotient, which is the tomonoid itself. Consecutive elements of this chain correspond to extensions by Archimedean tomonoids. If in this case the congruence classes are order-isomorphic to real intervals, a systematic specification turns out to be possible.

In order to deal with tomonoids and their quotients in an effective and transparent way, we follow an approach with a geometrical flavor: we work with transformation monoids, using the Cayley representation theorem. Our main results are formulated in this framework. Finally, a number of examples from the area of t-norms are included for illustration.