Irredundans

Irredundans, or irredundance, is a term used in several areas of mathematics to describe a property of a collection of objects where no member is dispensable for maintaining a given effect or consequence. The precise meaning depends on the context, but a common core is that removing any element lowers the strength of the collection in some sense.

In algebra, an irredundant generating set of an algebra A is a subset G such that G generates A and for every g in G the subalgebra generated by G \ {g} is strictly smaller than A. Equivalently, no element of G is a consequence of the others; each generator contributes something essential to the structure.

In logic, a set of formulas Γ is irredundant if for every φ in Γ, Γ \ {φ} does not entail φ. In other words, removing any formula from Γ reduces its deductive power, and every member adds new information.

In graph theory, an irredundant set is a subset S of vertices with the property that each vertex in S has a private witness with respect to S—typically a vertex w adjacent to that vertex but not adjacent to any other vertex in S. The irredundance number ir(G) is the maximum size of an irredundant set in the graph. This concept is related to, but distinct from, domination and independence notions.

Overall, irredundance captures a minimality idea: a collection is irredundant if every member plays a necessary role for achieving the intended property, whether generating a structure, sustaining deductive power, or ensuring unique witnesses in a graph.