  In this article, we study monomial curves, toric ideals and monomial algebras
associated to $4$-generated pseudo symmetric numerical semigroups. Namely, we
determine indispensable binomials of these toric ideals, give a
characterization for these monomial algebras to have strongly indispensable
minimal graded free resolutions. We also characterize when the tangent cones of
these monomial curves at the origin are Cohen-Macaulay.
