# Varieties where any subalgebra is a regular sub-object

Following [LPAC] Chap.3, p.132 let $$S$$ a set of sorts and $$\Sigma$$ a $$S$$-sorted signature. From the [LPAC] treatment we have the category $$Alg(\Sigma)$$ with a $$(\mathit{regular.Epi,Mono})$$-factorization, and where all epimorphism are all regular, and are just the surjective morphisms. Consider a equational class $$\operatorname{Alg}(\Sigma, E)$$ defined by a set of equation $$E$$ as in [LPAC] p.138, from point $$(1),\ (5)$$ of remark 3.6 of [LPAC] p.140, follow that the full inclusion $$\operatorname{Alg}(\Sigma, E)\subset \operatorname{Alg}(\Sigma)$$ create the $$(\mathit{regular.Epi,Mono})$$-factorization. By [LPAC] corollary 3.7 $$\operatorname{Alg}(\Sigma, E)$$ is locally presented, then by [LPAC] 1.61 pag. 50 has a $$(\mathit{regular.Epi,Mono})$$-factorization and a $$(\mathit{Epi, regular.Mono})$$-factorization, then we have that:

*) all epimorphisms are regular iff all monomorphisms are regular:

let $$f: X\to Y$$ a epimorphism, and $$f=m\circ e: X\to I\to Y$$ with $$m$$ regular monomorphism, for $$f=m\circ e$$ and $$f$$ epimorphism, follow that $$m$$ is a epimorphism, then it is an isomorphism. Dually for the rest.

For $$E=\emptyset$$ the above result $$(*)$$ is true for $$\operatorname{Alg}(\Sigma)$$, but in $$\operatorname{Alg}(\Sigma)$$ we have that "Epi=regular Epi" i.e. all epimorphism are regular (and then all monomorphism are regular). Also for the (algebraic) theory of groups "Epi=regular Epi" is valid (Barry Mitchell, "Theory of Categories" pag.38, ex. 13). Of course "Epi=regular Epi" isn't even valid, consider "rings with unity" theory, and the inclusion $$\mathbb{Z}\subset \mathbb{Q}$$ is monomorphism, and is a epimorphism but no a regular epimorphism (bacause isnt a isomorphism and: (reg.Epi +Mono) $$\to$$ Iso).

My question is:

"for what kind of (finitary) algebraic theory $$\Sigma$$ and equation class $$E$$, is valid that "Epi=regular Epi" for the category $$Alg(\Sigma, E)$$" ?

[LPAC] Locally Presentable and Accessible Categories – J. Adamek, J.Rosicky.

• I believe an alternate formulation of the same question would be: Which (possibly multi-sorted) varieties are balanced (i.e. have the property that every monic epic is an isomorphism). – Tim Campion Sep 18 at 20:02