  Pseudo-BCI-algebras generalize both BCI-algebras and pseudo-BCK-algebras,
which are a non-commutative generalization of BCK-algebras. In this paper,
following [J.G. Raftery and C.J. van Alten, Residuation in commutative ordered
monoids with minimal zero, Rep. Math. Log. 34 (2000) 23-57], we show that
pseudo-BCI-algebras are the residuation subreducts of semi-integral residuated
po-monoids and characterize those pseudo-BCI-algebras which are direct products
of pseudo-BCK-algebras and groups (regarded as pseudo-BCI-algebras). We also
show that the quasivariety of pseudo-BCI-algebras is relatively congruence
modular; in fact, we prove that this holds true for all relatively point
regular quasivarieties which are relatively ideal determined, in the sense that
the kernels of relative congruences can be described by means of ideal terms.
