“Plethory” is the name introduced by Borger and Wieland for a structure first introduced by Tall and Wraith. It is a biring equipped with an extra monoid structure, the plethysm product.
D. Tall? and G. Wraith?, Representable functors and operations on rings, Proc. London Math. Soc. 3, pages 619–643, 1970.
James Borger? and Ben Wieland?, Plethystic algebra, Advances in Mathematics 194 (2005), 246–283. (web)
William Mycroft? and Sarah Whitehouse?, The plethory of operations in complex topological K-theory?, preprint, 2020. arXiv:2001.01608