Hindmanʼs coloring theorem in arbitrary semigroups

Hindmanʼs Theorem asserts that, for each finite coloring of the natural numbers, there are distinct natural numbers a1,a2,…a1,a2,… such that all of the sums ai1+ai2+⋯+aimai1+ai2+⋯+aim (m⩾1m⩾1, i1