We consider the problem of nding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions deening representable relation algebras (for the nite case) and a… (More)
A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n ≥ 3, the class of all strongly representable n-dimensional… (More)
A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any xed dimension are elementary.
Given a representation of a relation algebra we construct relation algebras of pairs and of intervals. If the representation happens to be complete, homogeneous and fully universal then the pair and interval algebras can be constructed direct from the relation algebra. If, further, the original relation algebra is !-categorical we show that the interval… (More)
A boolean algebra is shown to be completely representable if and only if it is atomic whereas it is shown that the class of completely representable relation algebras is not elementary.
There is a version of this in JSL but that version contains an error and is followed by an erratum. The erratum is encorporated into the text here. We show, for any ordinal γ ≥ 3, that the class RaCAγ is pseudo-elementary and has a recursively enumerable elementary theory. ScK denotes the class of strong subalgebras of members of the class K. atomic… (More)
Hirsch and Hodkinson proved, for 3 ≤ m < ω and any k < ω, that the class SNrmCA m+k+1 is strictly contained in SNrmCA m+k and if k ≥ 1 then the former class cannot be defined by any finite set of first order formulas, within the latter class. We generalise this result to the following algebras of m-ary relations for which the neat reduct operator Nrm is… (More)
Two complexity problems in algebraic logic are surveyed: the satisfaction problem and the network satisfaction problem. Various complexity results are collected here and some new ones are derived. Many examples are given. The network satisfaction problem for most cylindric algebras of dimension four or more is shown to be intractable. Complexity is tied-in… (More)
We study relation algebras with n-dimensional relational bases in the sense of Maddux. Fix n with 3 ≤ n ≤ ω. Write B n for the class of non-associative algebras with an n-dimensional relational basis, and RA n for the variety generated by B n. We define a notion of relativised representation for algebras in RA n , and use it to give an explicit (hence… (More)