Freedom in polyadic algebras and two theorems of Beth and Craig.

Amalgamation, interpolation and epimorphisms in algebraic logic

“…In fact, we state in Proposition 3.1 below something stronger, namely that ES fails for such classes of algebras even if the righthand most algebra is in CA ω . This appears as (3,10) and (4,10) in Table 3. Also ES fails for any RCA ω ⊆ K ⊆ CA ω , which appears as (5,10) in Table 3.…”

“…Daigneault [10] was the first to explicitly make the connection between the Craig interpolation theorem and amalgamation in the context of polyadic algebras. The first systematic use of the link to obtain results about interpolation properties from results of amalgamation, or vice versa, can be found in Pigozzi's paper [31].…”

On amalgamation of reducts of polyadic algebras

“…The concept of super amalgamation also exists in Model Theory, under the name Heir-coheir amalgams [35] p. 136. Daigneault [25] was the first to explicitly make the connection between the Craig interpolation theorem and amalgamation in the context of polyadic algebras. The first systematic use of the link to obtain results about interpolation properties from results of amalgamation, or vice versa, can be found in [56].…”

“…Let A ∈ V be atomic, i.e below every non-zero element there is an atom. 25 Let At A denote the set of all atoms of A. Then the atom structure of A is the (first order) structure…”

“…When the present author solved four remaining problems and provided new proofs for some of the other problems as well, it was decided that Judit Madarász and the present author write a joint paper [43] expanding the above mentioned manuscript. 25 An atom is a minimal non-zero element. Cm and At, viewed as operators on atom structures and algebras, respectively, are therefore in some way dual to each other.…”