Regular left-continuous t-norms

AbstractA left-continuous (l.-c.) t-norm ⊙ is called regular if there is an n<ω such that the map x↦x⊙a has, for any a∈[0,1], at most n discontinuity points, and if the function mapping every a∈[0,1] to the set $\{x\in[0,1]\hspace {-0.15em}:\hspace{0.15em}\lim_{y\searrow x}y\odot a=x\}$ behaves in a specifically simple way. The t-norm algebras based on regular l.-c. t-norms generate the variety of MTL-algebras.With each regular l.-c. t-norm, we associate certain characteristic data, which in particular specifies a finite number of constituents, each of which belongs to one out of six different types. The characteristic data determines the t-norm to a high extent; we focus on those t-norms which are actually completely determined by it. Most of the commonly known l.-c. t-norms are included in the discussion.Our main tool of analysis is the translation semigroup of the totally ordered monoid ([0,1];≤,⊙,0,1), which consists of commuting functions from the real unit interval to itself.

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