Semiautomorphic inverse property loops

We define a variety of loops called semiautomorphic, inverse property loops that generalize Moufang and Steiner loops. We first show an equivalence between a previously studied variety of loops. Next we extend several known results for Moufang and Steiner loops. That is, the commutant is a subloop and if a is in the commutant, then a² is a Moufang element, a³ is a c-element and a⁶ is in the center. Finally, we give two constructions for semiautomorphic inverse property loops based on Chein’s and de Barros and Juriaans’ doubling constructions.