Well-Pointed Coalgebras
Ji\v{r}\'i Ad\'amek (Institut f\"ur Theoretische Informatik,, Technische Universit\"at Braunschweig, Germany), Stefan Milius (Institut, f\"ur Theoretische Informatik, Technische Universit\"ut Braunschweig,, Germany), Lawrence S Moss (Department of Mathematics, Indiana University
TL;DR
This paper characterizes the final coalgebra and initial algebra for certain endofunctors using well-pointed coalgebras, linking algebraic and coalgebraic structures with applications to automata and transition systems.
Contribution
It provides a new description of final coalgebras and initial algebras via well-pointed coalgebras, connecting well-foundedness with algebraic structures.
Findings
Final coalgebra consists of all well-pointed coalgebras.
Initial algebra consists of well-founded, well-pointed coalgebras.
Initial iterative algebra includes finite well-pointed coalgebras.
Abstract
For endofunctors of varieties preserving intersections, a new description of the final coalgebra and the initial algebra is presented: the former consists of all well-pointed coalgebras. These are the pointed coalgebras having no proper subobject and no proper quotient. The initial algebra consists of all well-pointed coalgebras that are well-founded in the sense of Osius and Taylor. And initial algebras are precisely the final well-founded coalgebras. Finally, the initial iterative algebra consists of all finite well-pointed coalgebras. Numerous examples are discussed e.g. automata, graphs, and labeled transition systems.
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