Regular Semantics is a term which describes the guarantees provided by a data register shared by several processors in a parallel machine or in a network of computers working together
Atomic semantics are defined formally in Lamport's "On Interprocess Communication" Distributed Computing 1, 2 (1986), 77-101. (Also appeared as SRC Research Report 8).
Regular semantics are defined for a variable with a single writer but multiple readers. These semantics are stronger than safe semantics but weaker than atomic semantics: they guarantee that there is a total order to the write operations which is consistent with real-time and that read operations return either the value of the last completed write or that of one of the writes which are concurrent with the read.