Authors: Jiří Adámek ; Stefan Milius ; Lawrence S Moss ; Lurdes Sousa

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.

• 18C05 - Equational categories
• 18C10 - Theories (e.g., algebraic theories), structure, and semantics
• 68Q65 - Abstract data types; algebraic specification
• 68Q70 - Algebraic theory of languages and automata