Product

Product may refer to:

Business

Sciences

Mathematics

In linear algebra:

In abstract algebra:

Product breakdown structure

In project management, a product breakdown structure (PBS) is a tool for analysing, documenting and communicating the outcomes of a project, and forms part of the product based planning technique.

The PBS provides ''an exhaustive, hierarchical tree structure of deliverables (physical, functional or conceptual) that make up the project, arranged in whole-part relationship'' (Duncan, 2015).

This diagrammatic representation of project outputs provides a clear and unambiguous statement of what the project is to deliver.

The PBS is identical in format to the work breakdown structure (WBS), but is a separate entity and is used at a different step in the planning process. The PBS precedes the WBS and focuses on cataloguing all the desired outputs (products) needed to achieve the goal of the project. This feeds into creation of the WBS, which identifies the tasks and activities required to deliver those outputs. Supporters of product based planning suggest that this overcomes difficulties that arise from assumptions about what to do and how to do it by focusing instead on the goals and objectives of the project - an oft-quoted analogy is that PBS defines where you want to go, the WBS tells you how to get there.

Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

Definition

Let C be a category with some objects X₁ and X₂. An object X is a product of X₁ and X₂, denoted X₁ × X₂, if it satisfies this universal property:

The unique morphism f is called the product of morphisms f₁ and f₂ and is denoted < f₁, f₂ >. The morphisms π₁ and π₂ are called the canonical projections or projection morphisms.

Above we defined the binary product. Instead of two objects we can take an arbitrary family of objects indexed by some set I. Then we obtain the definition of a product.

An object X is the product of a family (X_i)_i∈I of objects iff there exist morphisms π_i : X → X_i, such that for every object Y and a I-indexed family of morphisms f_i : Y → X_i there exists a unique morphism f : Y → X such that the following diagrams commute for all i∈I: