Reducts of RAs and CAs
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Bases, relation algebra-cylindric algebra
connections
Classical results of Monk (1964, 1969) show that
the representable relation and n-dimensional cylindric algebras (n>2) are
not finitely axiomatisable in first-order logic. The question arises as
to which algebraic operations `cause' this non-finite axiomatisability.
Various results, both positive and negative, have been obtained by (e.g.)
Andréka, Bredikhin and Schein, and Hansen. Here, we use `rainbow'
constructions to obtain further results, including on the connection
of weakly representable relation algebras to Maddux's n-dimensional relation algebras, and the lack of canonicity of the former.
Ian Hodkinson and Szabolcs Mikulás
Algebra Universalis 43 (2000) 127-156
In this paper, we prove that any subreduct of the
class of representable relation algebras whose similarity type includes
intersection, relation composition and converse is a non-finitely axiomatizable
quasivariety and that its equational theory is not finitely based.
This was proved earlier and in a different way by Haiman. We show the same
result for subreducts of the class of representable cylindric algebras
of dimension at least three whose similarity types include intersection
and cylindrifications. A similar result is proved for subreducts of the
class of representable sequential algebras.
Ian Hodkinson and Szabolcs Mikulás
ILLC prepublication series, ML-1997-04, Sep. 1997,
c. 16 pages (earlier version of the preceding paper).
In this paper, we show that certain subreducts of
the class of representable relation algebras are not finitely axiomatizable.
We do the same for sequential and cylindric algebras.
Robin Hirsch, Ian Hodkinson, Roger Maddux
J. Symbolic Logic 76 (2011) 870-882.
It is known that for all finite n≥5, there are relation algebras with
n-dimensional relational bases but no weak representations. We prove that conversely, there
are finite weakly representable relation algebras with no n-dimensional relational bases.
In symbols: neither of the classes RAn and wRRA contains the other.
Ian Hodkinson and Szabolcs Mikulás
J. Symbolic Logic, to appear.
We show that the variety
of weakly representable relation
algebras is not canonical, nor closed under Monk completions.