Abstract
We propose an abstract framework for modelling state-based systems with internal behaviour as e.g. given by silent or epsilon-transitions. Our approach employs monads with a parametrized fixpoint operator dagger to give a semantics to those systems and implement a sound procedure of abstraction of the internal transitions, whose labels are seen as the unit of a free monoid. More broadly, our approach extends the standard coalgebraic framework for state-based systems by taking into account the algebraic structure of the labels of their transitions. This allows to consider a wide range of other examples, including Mazurkiewicz traces for concurrent systems and non-deterministic transducers.

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Abstract
This paper takes a fresh look at the topic of trace semantics in the theory of coalgebras. In the last few years, two approaches, somewhat incomparable at first sight, captured successfully in a coalgebraic setting trace semantics for various types of transition systems. The first development of coalgebraic trace semantics used final coalgebras in Kleisli categories and required some non-trivial assumptions, which do not always hold, even in cases where one can reasonably speak of traces (like for weighted automata). The second development stemmed from the observation that trace semantics can also arise by performing a determinization construction and used final coalgebras in Eilenberg-Moore categories. In this paper, we develop a systematic study in which the two approaches can be studied and compared. Notably, we show that the two different views on trace semantics are equivalent, in the examples where both approaches are applicable.