To say that monos are regular (in a variety or, more generally, in a general category satisfying some very minor hypotheses) amounts to the so-called Intersection Property of Amalgamations: for any two algebras A, B with a common subalgebra C, if there are monomorphisms f : A --> D, g: B --> D that coincide on C, then one can choose f, g with the additional property that C is their pullback. References: C.M. Ringel: JPAA 2 (1972) 341-42 W. Tholen: Algebra Univ. 14 (1982) 391-397 E.W. Kiss, L. Marki, P. Prohle, W. Tholen: Studia Sci. Math. Hungaricum 18 (1983) 79-141. The last paper contains a large table of specific categories, including lattices (the affirmative answer is attributed to Gratzer in this case), plus an extensive list to the literature. For some categories, the question whether monos are regular can get quite involved, for example in the category of compact (Hausdorff) groups, for which an affirmative answer was provided by Poguntke (Math. Z. 130 (1973) 107-117). The property in question obviously implies "epimorphisms are surjective", but examples witnessing failure of the converse statement are harder to find: see again the four-author paper. Hope this helps. Walter Tholen. On Mar 13, 4:01pm, Andrej Bauer wrote:
Subject: categories: When are all monos regular? This might be an embarrassingly easy question, but I always get confused about it. When are all monos in an algebraic category regular (or more generally, when are all monos in a regular category regular)? What are some sensible sufficient or necessary conditions?
For example, all monos in the category of groups are regular. How about the category of lattices, or lattices with a top element?
Andrej Bauer