Quantifier-free descriptions for interval-quantifier linear systems

A system of relations of the form $Ax\,\sigma\,b$ is considered, where $\sigma$ is a relation vector with components $=$, $\geq$, and $\leq$ and the parameters (the elements of the matrix $A$ and of the right-hand side $b$) take values from given intervals. What is considered to be the set of solutions of this system depends on which quantifier is related to each interval-valued parameter and on the order of quantifier prefixes for individual parameters. For sets of solutions with a quantifier prefix of a rather general form, we obtain equivalent quantifier-free descriptions in the classical interval arithmetic, in the Kaucher interval arithmetic, and in the usual real arithmetic.