Unextendible maximally entangled bases and mutually unbiased bases

We study unextendible maximally entangled basis in arbitrary bipartite spaces. A systematic way of constructing a set of $d^2$ orthonormal maximally entangled states in ${\mathbb{C}}^d⨂{\mathbb{C}}^{d^{\mathrm{\prime }}}(\frac{d^{\mathrm{\prime }}}{2}<d<d^{\mathrm{\prime }})$ is provided. The complementary space of the set of these $d^2$ orthonormal maximally entangled states contains no maximally entangled states that are orthogonal to all of them. Furthermore, we investigate mutually unbiased bases in which all the bases are unextendible maximally entangled ones. We present two unextendible maximally entangled bases in ${\mathbb{C}}^2⨂{\mathbb{C}}^3$ which are mutually unbiased.