Re: what happens when a basic sentence is added ?

Dear Philippe, Your question about limit theories T:

Add a basic sentence to this theory to obtain a theory T'. So Mod(T') is accessible, Mod(T) is locally presentable. Is Mod(T') accessibly- embedded

has an affirmative answer, essentailly due to Coste's 1979 paper. Take a cardinal k larger than the arities of the symbols of your signature S, then both Mod(T) and Mod(T') are closed under k-filtered colimits in Str S. (See the ananlogous argument in part II of the proof of Theroem 5.9 of Rosicky's and mine book.) Consequently, the embedding Mod(T') -> Mod(T) preserves k-filtered colimits. Best regards Jiri [For admin and other information see: http://www.mta.ca/~cat-dist/ ]