Grassroots platforms aim to offer an egalitarian alternative to global platforms. Whereas global platforms can have only a single instance, grassroots platforms can have multiple instances that emerge and operate independently of each other and of any global resource except the network, and can interoperate and coalesce into ever-larger instances once interconnected. Key grassroots platforms include grassroots social networks, grassroots cryptocurrencies, and grassroots democratic federations. Previously, grassroots platforms were defined formally and proven grassroots using unary distributed transition systems, in which each transition is carried out by a single agent. However, grassroots platforms cater for a more abstract specification using transactions carried out atomically by multiple agents, something that cannot be expressed by unary transition systems. As a result, their original specifications and proofs were unnecessarily cumbersome and opaque. We enhance the notion of a distributed transition system to include atomic transactions and revisit the notion of grassroots platforms within this new foundation; present crisp specifications of key grassroots platforms using atomic transactions: befriending and defriending for grassroots social networks, coin swaps for grassroots cryptocurrencies, and communities forming, joining, and leaving a federation for grassroots democratic federations; prove a general theorem that a platform specified by atomic transactions that are so-called interactive is grassroots; show that the atomic transactions used to specify all three platforms are interactive; and conclude that the platforms thus specified are indeed grassroots. We thus provide a crisp mathematical foundation for grassroots platforms and a solid and clear starting point from which their implementation can commence.