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In abstract algebra

Partial algebra generalizes the notion of universal algebra to partial operations. An example would be a field, in which the multiplicative inversion is the only proper partial operation (because division by zero is not defined).

The set of all partial functions (partial transformations) on a given base set, X, {\displaystyle X,} forms a regular semigroup called the semigroup of all partial transformations (or the partial transformation semigroup on X {\displaystyle X}), typically denoted by P T X. {\displaystyle {\mathcal {PT}}_{X}.} The set of all partial bijections on X {\displaystyle X} forms the symmetric inverse semigroup.