Categories which are varieties of classical or ordered algebras
Categories which are varieties of classical or ordered algebras · Theory and applications of categories · 2025
Following ideas of Lawvere and Linton we prove that classical varieties are precisely the exact categories with a varietal generator.This means a strong generator which is abstractly finite and regularly projective.An analogous characterization of varieties of ordered algebras is also presented.We work with order-enriched categories, and introduce the concept of subcongruence (corresponding to congruence in ordinary categories): it is a relation which is order-reflexive and transitive.Varieties of ordered algebras are precisely the categories with effective subcongruences and a subvarietal generator.This means a strong generator which is abstractly finite and subregularly projective.