A GG NOT FH SEMISTAR OPERATION ON MONOIDS

Abstract

Let $S$ be a g-monoid withquotient group q$(S)$. Let $\bar {\rm F}(S)$ (resp., F$(S)$,f$(S)$) be the $S$-submodules of q$(S)$ (resp., the fractionalideals of $S$, the finitely generated fractional ideals of$S$). Briefly, set f := f$(S)$, g := F$(S)$, h := $\bar{\rmF}(S)$, and let $\{\rm{x,y}\}$ be a subset of the set $\{$f,g, h$\}$ of symbols. For a semistar operation $\star$ on $S$,if $(E + E_1)^\star = (E + E_2)^\star$ implies ${E_1}^\star ={E_2}^\star$ for every $E \in$ x and every $E_1, E_2 \in$ y,then $\star$ is called xy-cancellative. In this paper, weprove that a gg-cancellative semistar operationneed not be fh-cancellative.