  Pseudo equality algebras were initially introduced by Jenei and $\rm
K\acute{o}r\acute{o}di$ as a possible algebraic semantic for fuzzy type theory,
and they have been revised by Dvure\v{c}enskij and Zahiri under the name of
JK-algebras. In this paper we define and study the commutative pseudo equality
algebras. We give a characterization of commutative pseudo equality algebras
and we prove that an invariant pseudo equality algebra is commutative if and
only if its corresponding pseudo BCK(pC)-meet-semilattice is commutative. Other
results consist of proving that every commutative pseudo equality algebra is a
distributive lattice and every finite invariant commutative pseudo equality
algebra is a symmetric pseudo equality algebra. We also introduce and
investigate the commutative deductive systems of pseudo equality algebras. As
applications of these notions and results we define and study the measures and
measure-morphisms on pseudo equality algebras, we prove new properties of state
pseudo equality algebras, and we introduce and investigate the
pseudo-valuations on pseudo equality algebras.
