Abstract
Distributive quasi relation algebras (DqRAs) are distributive residuated
lattices equipped with three order-reversing unary operations. DqRAs are
a generalisation of relation algebras, which were first developed by Tarski
in the 1940s to study binary relations from an algebraic perspective. After
Tarski’s axiomatization of abstract relation algebras, the question arose
as to which are representable as algebras of binary relations. We examine
certain aspects of representable DqRAs using a recent definition
of representability for DqRAs developed by Craig and Robinson. This
definition involves the construction of algebras of binary relations using
posets. Our first new result is that when a DqRA is constructed from
a finite chain, the resulting algebra has no proper subalgebras. Building
on the work of Jipsen, we define specific elements in DqRAs called
symmetric central negative idempotent elements (SCNI-elements) which
we use to find homomorphic images of DqRAs. We prove that in the
finite case, SCNI-elements are in a one-to-one correspondence with congruences
on a DqRA, and hence can be used to study various universal
algebraic properties of DqRAs. In particular we use SCNI-elements to
identify subdirectly irreducible DqRAs in the finite case, and to prove
that the variety of DqRAs does not have the Congruence Extension Property.
Further, SCNI-elements are used to prove that the class of finitely
representable DqRAs is closed under homomorphic images.