Partial algebra

In abstract algebra, a partial algebra is a pair <A, P> where A is a set and P is a collection of partial operations on A. In universal algebra, when P consists of operations that are defined on all arguments taken from A, then the algebra is a total algebra. Frequently the adjective total is omitted when there are no partial operations.

As a motivation for the study of partial algebras, George Grätzer considered the case of "subsets of an algebra and properties of operations on these subsets, even if the subsets are not closed under all operations."

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