Regular $\Gamma-$incline and field $\Gamma-$semiring

M. Murali Krishna Rao, B. Venkateswarlu

We introduce the notion of $\Gamma -$incline as a generalization of incline, which is an algebraic structure with an additional poset structure. We introduce the notions of regular $\Gamma -$incline, integral $\Gamma -$incline, field $\Gamma -$incline, field $\Gamma -$semiring, simple $\Gamma -$semiring and pre-integral $\Gamma -$semiring. We study their properties and relations between them. We prove that if $M$ is a linearly ordered regular $\Gamma -$incline, then $M$ is a commutative $\Gamma -$incline and $M$ is a field $\Gamma -$semiring with an additional property if and only if $M$ is an integral, simple and commutative $\Gamma -$semiring.