Zappa-Szép products of bands and groups. (Q1006332)

If \(A,S\) are semigroups and mappings \((s,a)\to s\cdot a\in A\), \((s,a)\to s^a\in S\) satisfy \(s\cdot (t\cdot a)=st\cdot a\), \(s\cdot ab=(s\cdot a)(s^a\cdot b)\), \(s^{ab}=(s^a)^b\), \((st)^a=s^{t\cdot a}t^a\) for every \(a,b\in A\), \(s,t\in S\), then the set \(A\times S\) with operation \((a,s)(b,t)=(a(s\cdot b),s^bt)\) is called the Zappa-Szép product of \(A,S\). Here are investigated Zappa-Szép products of groups and bands; e.g. it is shown, that if the band is a semilattice, the Zappa-Szép product is orthodox and \(\mathcal L\)-unipotent.