This paper proposes a new algebraic system, Test Algebra (TA), for identifying faults in combinatorial testing for SaaS (Software-as-a-Service) applications. SaaS as a part of cloud computing is a new software delivery model, and mission-critical applications are composed, deployed, and executed in cloud platforms. Testing SaaS applications is a challenging task because new applications need to be tested when they are composed before they can be deployed for execution. Combinatorial testing algorithms can be used to identify faulty configurations and interactions from 2-way all the way to k-way where k is the number of components in the application. The TA defines rules to identify faulty configurations and interactions. Using the rules defined in the TA, a collection of configurations can be tested concurrently in different servers and in any order and the results obtained will be still same due to the algebraic constraints.