Representable Cylindric Algebras and Many-Dimensional Modal Logics

Abstract

The equationally expressible properties of the cylindrifications and the diagonals in finite-dimensional representable cylindric algebras can be divided into two groups:

• (i) ‘One-dimensional’ properties describing individual cylindrifications. These can be fully characterised by finitely many equations saying that each c _i , for i < n, is a normal (c _i 0 = 0), additive (c _i (x+y) = c _i x+c _i y) and complemented closure operator:

  $$ x \leqslant C_i x\quad \quad C_i c_i x \leqslant C_i x\quad \quad C_i \left( { - C_i x} \right) \leqslant - C_i x. $$

  (2.0.1)

• (ii) ‘Dimension-connecting’ properties, that is, equations describing the diagonals and interaction between different cylindrifications and/or diagonals. These properties are much harder to describe completely, and there are many results in the literature on their complexity.

I am grateful to Ian Hodkinson for discussions and for his many comments on the preliminary version. Thanks are also due to Rob Goldblatt, Stanislav Kikot, András Simon and Misha Zakharyaschev for discussions.