  The fourth-order lattice Gel'fand-Dikii equations in quadrilateral form are
investigated. Utilizing the direct linearization approach, we present some
equations of the extended lattice Gel'fand-Dikii type. These equations are
related to a quartic discrete dispersion relation and can be viewed as
higher-order members of the extended lattice Boussinesq type equations. The
resulting lattice equations given here are in five-component form, and some of
them are multi-dimensionally consistent by introducing extra equations. Lax
integrability is discussed both by direct linearization scheme and also through
multi-dimensional consistent property. Some reductions of the five-component
lattice equations to the four-component forms are considered.
