Synthetic closure refers to a method in mathematics and logic where a set is extended to include all elements necessary to satisfy certain properties or operations. This process ensures that the set remains closed under specific operations, meaning that applying these operations to elements within the set will yield results that are also within the set. In the context of algebra, synthetic closure can be seen in structures like groups or fields, where operations such as addition or multiplication are defined. By including all necessary elements, synthetic closure helps maintain the integrity of mathematical systems and allows for consistent results across various calculations.