Totalization

Totalization is a term used in several disciplines to describe the process of making something complete or aggregating components into a single whole. In mathematics and computer science it most often refers to converting a partial function into a total function by providing values for inputs where the original function is undefined. This is commonly done by introducing a bottom element or a default value, yielding a function that is defined for all inputs. The concept is central in domain theory and in the design of total, terminating programs. An example is a partial function f: A → B that is defined only for some a in A, which can be extended to a total function f†: A → B⊥ by mapping undefined inputs to the bottom value, and then possibly to a distinguished output.

In homological algebra, totalization (or taking the total complex) refers to a construction that converts a double complex into a single complex by summing along diagonals; this produces a chain complex whose components combine information from both directions with appropriate signs. The total complex is foundational in the study of spectral sequences and derived categories.

In international law and social policy, a totalization agreement (or totalization treaty) coordinates the social security systems of two countries to prevent gaps or double coverage in retirement, disability, or survivor benefits. Workers’ credited service earned abroad can be counted toward eligibility under certain rules, facilitating cross-border work.

Other uses include general data processing contexts in which totals are computed across datasets or multiple streams are aggregated into a single metric. See also related topics such as partial functions, total functions, and total complexes.