Quasi-Polyadic Algebras and Their Dual Position

Abstract

Infinite-dimensional quasi-polyadic algebras and cylindric algebras are the best-known algebraizations of first-order logic. Quasi-polyadic equality algebras (QPEAs), due to Paul R. Halmos, are special polyadic algebras; nevertheless, for a long time, these algebras have been considered only as a version of cylindric algebras (CAs), because their behavior seemed to be very similar to that of CAs. However, in researching QPEAs, over time it turned out that many properties of QPEA are different from those of cylindric algebras. The reason is the presence of the polyadic operator, the transposition $p_{ij}$ (missing in CAs). The author states and documents that QPEAs have a dual position in algebraic logic. In many respects, this class is like CAs, following the cylindric paradigm, but, in other respects, as an effect of the polyadic paradigm (of the transposition), QPEAs are different from CAs. The paper is a survey on quasi-polyadic algebras with particular regard to the foregoing documentation, and it fills an existing gap in the literature.