Classes of Representable Algebras with the Amalgamation Property

We introduce an abstract class of algebras that is a proper reduct of both Pinter’s substitution algebras and cylindric algebras. We show that the class of representable algebras have the amalgamation property. Pinter’s substitution algebras were introduced by Pinter [13], [14]. Such algebras are obtained from cylindric algebras by dropping the diagonal elements and adding unary operations of substitutions. These substitutions, which are term definable in cylindric algebras, reflect the metalogical operation of substituting in formulas one variable for another, such that the substitution is free. Pinter proved that locally finite algebras are representable, and it seems that not much has been done for such algebras since. We start by defining the concrete versions of Pinter’s algebras. Let α be an ordinal. Let U be a set. Then we define for i, j < α and X ⊆ U : ciX = {s ∈ U : ∃t ∈ X, t(j) = t(i) for all i 6= j}, sjiX = {s ∈ U : s ◦ [i|j] ∈ X}, Here [i|j] is the replacement on α that takes i to j and leaves every other thing fixed. The extra non -boolean operations we deal with are as specified above. For set X, let B(X) = (℘(X),∪,∩, \, ∅, X) be the full boolean set algebra with universe ℘(X). Let S be the operation of forming subalgebras, and P be that of forming products. Then RSCα = SP{(B(U), ci, sji )i,j<α : U is a set }. 116 Tarek Sayed Ahmed From now on ordinals considered are infinite. It is not hard to show that RSCα is a variety and that is not finite schema axiomatizable. We shall need the notion of superamalgmation due to Maksimova [12].