Elliptic solutions to nonsymmetric Monge-Amp\`{e}re type equations I. The $d$-concavity and the comparison principle

We introduce the so-called $d$-concavity, $d \geq 0,$ and prove that the nonsymmetric Monge-Amp\`{e}re type function of matrix variable is concave in an appropriate unbounded and convex set. We prove also the comparison principle for nonsymmetric Monge-Amp\`{e}re type equations in the case when they are so-called $\delta$-elliptic with respect to compared functions with $0 \leq \delta <1$.