What question did this study set out to answer?

The aim is to propose an enriched version of universal algebra based on a suitable enrichment base.

February 19, 2026Open Access

Towards enriched universal algebra

Key Points

The aim is to propose an enriched version of universal algebra based on a suitable enrichment base.
Defined a language of function symbols based on a base of enrichment.
Constructed $$\mathbb{L}$$-terms recursively using symbols and morphisms from the enrichment.
Incorporated the monoidal structure of the enrichment into the algebraic framework.
Characterized algebras for finitary monads as models of enriched equational theories.
Established a new framework for understanding enriched universal algebra.
Defined interpretations for $$\mathbb{L}$$-structures to facilitate enriched equational theories.
Showed how finitary monads on the enrichment represent models of defined theories.

Abstract

Abstract Following the classical approach of Birkhoff, we suggest an enriched version of universal algebra. Given a suitable base of enrichment V V, we define a language L L to be a collection of (X, Y) -ary function symbols whose arities are taken among the objects of V V. The class of L L -terms is constructed recursively from the symbols of L L, the morphisms in V V, and by incorporating the monoidal structure of V V. Then, L L -structures and interpretations of terms are defined, leading to enriched equational theories. In this framework we characterize algebras for finitary monads on V V as models of enriched equational theories.

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