Some closure properties in monoids (Q1206783)

A subset \(U\) of a semigroup \(S\) is called resultant-closed if \(axb\in U\) for \(a,x,b\in S\) whenever \(x,ax,bx\in U\). A subset \(U\) of nonnegative reals is called semiaffine if \(2q-p\in U\) whenever \(q,p\in U\) with \(p<q\). Any semiaffine set \(U\subset Z^ +\) is resultant-closed, and \(0,q,r\in U\) with \(0<q,r\) implies either \(q+r-3d\in U\), or all nonnegative multiples of \(d\) are in \(U\) where \(d\) is the greatest common divisor of \(q\) and \(r\). Other properties of semiaffine sets are stated. We say that a monoid is bitotal if both left and right divisibility preorders are total. A submonoid of a bitotal monoid is an equalizer if and only if it is resultant-closed. The equalizers in the monoid of natural numbers are just semiaffine submonoids.