rizable

Rizable is an adjective used in some mathematical writings to indicate that an abstract object can be made concrete or represented in a particular ambient space. The term is not widely standardized and is often considered nonstandard or informal; in most contexts authors prefer realizable or representable, depending on the specific setting.

In geometry and graph theory, rizable can describe whether an abstract structure can be realized as a geometric object in Euclidean space. For example, a graph might be described as rizable in the plane if there exists a straight-line embedding of the graph in the plane without edge crossings. More generally, a simplicial complex or polyhedral surface may be called rizable when it can be realized as a polyhedron in some dimension.

In matroid theory, a matroid is said to be rizable over a field F if it can be represented by a configuration of vectors in a vector space over F. The standard terminology in this area is that the matroid is F-representable or representable over F. In formal writing, the use of rizable here is rare and can create ambiguity.

Usage notes: because realizable and representable are the conventional terms, rizable is typically avoided in rigorous publications. When used, it is advisable to define the intended ambient space and the precise sense of realization at the outset to prevent confusion.

See also: realizability; representable; realizable set; geometric realization.