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.
