algebraic logic via arrows

23 Jun 1997

      I'd be grateful for comments on the following question motivated by a problem 
in categorizing algebraic logic a la Tarski.

Let C be a  complete category with a factorization
system (E,M). Given an object A\in C, let us call a pair (m,e) with 
m\in M, cod(m)=A and e\in E, dom(e)=A, {\it compatible}, m~e, if the
square

                                * --m--> A
                                |               |
                               e'              e
                               |                |
                               v              v 
                               *--m'--> A/e

is pull-back  (where (e',m') factorizes m;e ).            

In SET with the standard surjection-injection factorization, m~e iff 
for all a,b\in A, a\in A_m and e(a)=e(b)  entail b\in A_m where A_m 
is the subset of A corresponding to m.

Now let (e_i, i\in I) be a family of congruences compatible with
some m:*--->A,  e_i ~ m for all i\in I.  The question is what
properties of (C,E,M) are required to provide sup(e_i, i\in I) ~ m ?
(the collection CongA of e:A-->*, e\in E is a meet-complete semilattice due
to products, sup is join-via-meets in this lattice). 

If C is a category of finitary algebras over SET with the standard
epi-mono factorization, then sup(e_i) ~ m always holds due to
the finitary deduction property of taking sup in the congruence 
lattice: 

(a,b)\in sup(e_i) iff there exists a finite subfamily  e_1,...,e_k 
and c_0,...,c_k \in A s.t. a=c_0, b=c_k and e_j(c_{j-1}) =e_j(c_j) 
for  all j=1,...,k .

Zinovy Diskin