Linear algebra and operator theory without vector spaces: a classification of involutions on arbitrary associative rings and semigroups

With the objective of finding ways to extend some standard topics in Linear Algebra and operator theory so as to be applicable in the more general context of arbitrary semigroups with an involution *:S→ S ∗ : S → S , nine possible properties of involutions are compared, and shown to have new inter-connections, including new results connecting the Moore–Penrose generalized inverse, the $$*$$ ∗ -order and the singular value decomposition. Several interesting questions remain open.