A Note on Almost GCD Monoids

A commutative cancellative monoid H (with 0 adjoined) iscalled an almost GCD (AGCD) monoid if for x,y in H, there exists a naturalnumber n = n(x,y) so that xⁿ and yⁿ have an LCM, that is, xⁿH cap yⁿH is principal. We relate AGCD monoids to therecently introduced inside factorial monoids (there is a subset Q of Hso that the submonoid F of H generated by Q and the units of H isfactorial and some power of each element of H is in F). For example, weshow that an inside factorial monoid H is an AGCD monoid if and onlyif the elements of Q are primary in H, or equivalently, H is weaklyKrull, distinct elements of Q are v-coprime in H, or the radical ofeach element of Q is a maximal t-ideal of H. Conditions are given foran AGCD monoid to be inside factorial and the results are put in thecontext of integral domains.