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.
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 multidimensionally 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.
Abstract.A semigroup S is called regular semigroup if for every a ∈ S there exists x in S such that axa = a introduced by J. A. Green. In this paper, some preliminaries and basic concept of regular semigroups were presented. And proved that a cancellative semigroup S is left(right) regular semigroup if and only if it is a: (i) completely regular semigroup (ii) Clifford semigroup (iii) E-inversive semigroup (iv) -regular semigroup.
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