Simplicity, and stability in there

Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T. canonical base of an amalgamation class is the union of names of ψ-definitions of , ψ ranging over stationary L-formulas in . Also, we prove that the same is true with stable formulas for an 1-based theory having elimination of hyperimaginaries. For such a theory, the stable forking property holds, too.

Filtral powers of structures

Among the results of this paper are the following:1. Every Boolean (ultra)power is the union of an updirected elementary family of direct ultrapowers.2. Under certain conditions, a finitely iterated Boolean ultrapower is isomorphic to a single Boolean ultrapower.3. A ω-bounded filtral power is an elementary substructure of a filtral power.4. Let be an elementary class closed under updirected unions (e.g., if is an amalgamation class); then is closed under finite products if and only if is closed under reduced products if and only if is a Horn class.

Absolute Retracts and Amalgamation in Certain Congruence Distributive Varieties

It is shown that if V is a congruence distributive variety whose members have one element subalgebras, then the class of absolute retracts of V is closed under direct products. If V is residually small, then a characterisation of the amalgamation class of V is given.