Let $C$ be the category of $\tau$-algebras for some type $\tau$. Consider the statements:


1. Every monomorphism is regular.
2. Every epimorphism in $C$ is surjective.


It is easy to see that 1. implies 2. what about the converse?



When are epimorphisms of algebraic objects surjective?
Update: the following exchange appeared on the categories mailing list several years ago: <http://article.gmane.org/gmane.science.mathematics.categories/3094>. Walter Tholen's response strongly suggests that the answer to Martin's question is that the converse does not hold, although I don't have access to the four-author article he cites as reference. It's probably worth a look though, and if I learn anything more I'll post another update.


**Second update**: My surmise was correct. Walter Tholen kindly emailed to me the relevant two pages (pp. 88-89) of the four-author paper


* E.W. Kiss, L. Marki, P. Prohle, W. Tholen: Studia Sci. Math. Hungaricum 18
(1983) 79-141.


where the following example is given on page 89: in the category of semigroups with zero such that all 4-fold products are zero, all epimorphisms are surjective but not all monos are regular [a specific nonregular mono is described]. (I can forward this email if you write to me at topological dot musings at gmail dot com.)


Assuming amalgamated products exist (as they do in categories of algebras of a Lawvere theory), a mono $i: A \rightarrowtail B$ is regular if it is the equalizer of the pair of canonical maps from $B$ to the amalgamated product $B \*\_A B$ (i.e., the coprojections of the pushout of $i$ with itself, aka the *cokernel pair* of $i$). The equalizer of the cokernel pair defines a closure operator on the lattice of subalgebras $\mathsf{Sub}(B)$, called the *dominion operator* $\mathsf{Dom}\_B$. So to prove a subalgebra is not regular is to show that it is not $\mathsf{Dom}$-closed. The key technical result needed to prove the claim above is Isbell's Zig-Zag theorem (given in his paper Epimorphisms and Dominions in the 1965 La Jolla conference proceedings on categorical algebra), as recalled in [Peter M. Higgins. "A short proof of Isbell's Zigzag Theorem." Pacific J. Math. 144(1):47–50 (1990)](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-144/issue-1/A-short-proof-of-Isbells-zigzag-theorem/pjm/1102645825.full), which gives a precise and useful criterion for an element to belong to the dominion (i.e., the $\mathsf{Dom}$-closure) of a subalgebra.


Hope this helps. I am voting up your question, Martin, since it's rather nontrivial!


