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.
Keywords: variety, ordered algebras, effective congruences, subcongruence, varietal generator
2020 MSC: 18C05, 08C05
Theory and Applications of Categories, Vol. 43, 2025, No. 3, pp 39-67.
Published 2025-03-02.
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