Introduction

(not even a draft)

Review¶

Given functors $T:\sf Set\to Set$ and $L:\sf BA\to BA$ and the contravariant adjunction “homming into 2”

the meaning of the logic $L$ is determined by a natural transformation

Moreover, $LP\to PT$ determines, and is determined by, its so-called mate

which maps a one-step behaviour to its theory.

One-Step Properties¶

It is possible to express properties of the logic in terms of the properties of these natural transformations. Below

• $n$ is a finite set,

• $\twoheadrightarrow$ is onto,

• $\Rightarrow$ is split epi (onto and has a half-inverse),

• $\rightarrowtail$ is injective,

• $\hookrightarrow$ is a section (injective and has a half-inverse).

References¶

The algebraic approach to coalgebraic logic was proposed in

• Kupke, Kurz, Pattinson: Algebraic Semantics for Coalgebraic Logics. 2004.

This paper proves that one-step completeness implies completeness. That one-step expressiveness implies expressiveness is due to

• Klin: (reference to be added) ...

...

• Kurz, Rosicky: (reference to be added) ...