A GG NOT FH SEMISTAR OPERATION ON MONOIDS

Abstract

Let  $S$  be a  g-monoid with
quotient group  q$(S)$. Let  $\bar {\rm F}(S)$ (resp., F$(S)$,
f$(S)$)  be the  $S$-submodules of  q$(S)$ (resp., the  fractional
ideals of  $S$,  the finitely generated  fractional ideals of
$S$). Briefly, set  f := f$(S)$, g := F$(S)$, h := $\bar{\rm
F}(S)$, 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, we
prove that  a  gg-cancellative semistar operation
need not be  fh-cancellative.

Keywords

• Semistar operation,monoid

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