Some new class of ideals in semirings and their applications

Abstract

In the present paper, we focus on semirings, which are positive cones of a class of lattice-ordered rings. We establish a lattice isomorphism between semiring $l$-ideals and ring $l$-ideals of cancellative $l$-semirings and its difference $l$-ring, and from this, we obtain a structure of semiring $l$-ideals via ring $l$-ideals in cancellative $l$-semirings. Smith in \cite{Smith} defined $f$-semirings as a suitable class of $l$-semirings in which one can establish a structure theorem.
By introducing convexity properties on $f$-semirings, we give a necessary and sufficient condition on a cancellative positive $l$-semiring to be a Bezout semiring. A new class of $f$-semirings, namely $\mathcal{P}$-semirings, is defined to focus solely on positive cones of abundant function rings, e.g., $C(X)$. We bring notions of $z$-ideals and $z^\circ$-ideals into commutative semirings. It is shown that these ideals are equally important in investigating $\mathcal{P}$-semirings like $C^+(X)$. The structure of $z$-ideals and $z^\circ$-ideals are obtained in $\mathcal{P}$-semirings via the $z$-ideals and $z^\circ$-ideals of its difference ring. We show that each $k$-ideal of a $\mathcal{P}$-semiring is a $z$-ideal if and only if it is a von Neumann regular semiring.