Well-orders in the transfinite Japaridze algebra

This paper studies the transfinite propositional provability logics ${\text{glp}}_{\mathrm{\Lambda }}$ and their corresponding algebras. These logics have for each ordinal $\xi <\mathrm{\Lambda }$ a modality $\text{la}\alpha \text{ra}$. We will focus on the closed fragment of ${\text{glp}}_{\mathrm{\Lambda }}$ (i.e., where no propositional variables occur) and \emph{worms} therein. Worms are iterated consistency expressions of the form $\text{la}{\xi }_n\text{ra}\dots \text{la}{\xi }_1\text{ra}\mathrm{\top }$. Beklemishev has defined well-orderings ${<}_{\xi }$ on worms whose modalities are all at least $\xi$ and presented a calculus to compute the respective order-types. In the current paper we present a generalization of the original ${<}_{\xi }$ orderings and provide a calculus for the corresponding generalized order-types $o_{\xi }$. Our calculus is based on so-called {\em hyperations} which are transfinite iterations of normal functions. Finally, we give two different characterizations of those sequences of ordinals which are of the form $\text{la}{\text{formerOmega}}_{\xi }(A){\text{ra}}_{\xi \in \text{ord}}$ for some worm $A$. One of these characterizations is in terms of a second kind of transfinite iteration called {\em cohyperation.}