Amalgamation for Reducts of Polyadic Equality Algebras, both a Negative Result and a Positive Result

Amalgamation for Reducts of Polyadic Equality Algebras, a Negative Result

Let G ⊆ ω be a semigroup. G polyadic algebras with equality, or simply G algebras, are reducts of polyadic algebras with equality obtained by restricting the similarity type and axiomatization of polyadic algebras to substitutions in G, and possibly weakening the axioms governing diagonal elements. Such algebras were introduced in the context of ’finitizing’ first order logic with equality. We ...

Reducts of Polyadic Equality Algebras without the Amalgamation Property

Let G ⊆ ω be a monoid such that {[i|j ] : i, j ∈ ω} ⊆ G. Let RPEAG be the reduct of representable polyadic equality algebras obtained by restricting the similarity type of RPEAω to substitutions in G and PEAG be the class of abstract polyadic algebras obtained from PEAω by restricting the similarity type and axiomatization of PEAω to substitutions in G and to finite quantifiers. Assume that G c...

On Amalgamation of Reducts of Polyadic Algebras

Neat embeddings as adjoint situations

Looking at the operation of forming neat α-reducts as a functor, with α an infinite ordinal, we investigate when such a functor obtained by truncating ω dimensions, has a right adjoint. We show that the neat reduct functor for representable cylindric algebras does not have a right adjoint, while that of polyadic algebras is an equivalence. We relate this categorical result to several amalgamati...

On the Equational Theory of Representable Polyadic Equality

Among others we will see that the equational theory of ω dimensional representable polyadic equality algebras (RPEAω’s) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable (and hence the equational theory of this class is finite schemaaxi...