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 ci, for i < n, is a normal (ci0 = 0), additive (ci(x+y) = cix+ciy) and complemented closure operator: x ≤ cix cicix ≤ cix ci(−cix) ≤ −cix. (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. The main aim of this chapter is to study generalisations of (i) while keeping (ii) as unchanged as possible. In other words, we would like to analyse how much of the complexity of RCAn is due to its ‘many-dimensional ’ character and how much of it

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