Interassociates of monogenic semigroups. (Q1434115)

A semigroup \((S,*)\) is said to be an `interassociate' of the semigroup \((S,\cdot)\) if \(x\cdot(y*z)=(x\cdot y)*z\) and \(x*(y\cdot z)=(x*y)\cdot z\) for all \(x,y,z\) in \(S\). Two problems that arise very naturally in the study of interassociates is their determination and whether they are isomorphic. In this paper these problems are considered in the case of monogenic semigroups. Let \(S\) be generated by \(a\) and define an operation \(*_k\) on \(S\) by \(a^x*_k a^y=a^{x+k+y-2}\), then the interassociates of \(S\) are of the form \((S,*_k)\) for \(k\geq 1\). These are all interassociates one of another, and there are \(|S|\) of them. If \(S\) has infinite order then no two of these are isomorphic. If \(S\) has index \(i\) and period \(p\), then the `nexus', \(n\), of \(S\) is the least residue of \(i\) modulo \(p\), and conditions for isomorphism are given in terms of \(p\), \(i\) and \(n\).