Presentatrices

Presentatrices is a rarely used plural form of presentatrix, a term that can denote a matrix used to present an algebraic structure, most commonly a module or a group, by describing its generators and relations. In this sense, a presentatrix encodes the defining relations among generators in a compact, linear-algebraic object.

In the language of module theory, a presentation of a finitely generated module M over a ring R is given by a surjective homomorphism R^n → M with kernel generated by the columns of an m×n matrix A with entries in R. The matrix A is called a presentation matrix (also known as a relation matrix). The rows of A encode relations among the chosen generators, while the columns correspond to the generators themselves. The term presentatrices would be the corresponding plural for discussions that adopt a Latinized naming convention.

Usage and terminology vary. Most texts prefer the terms presentation matrix or relation matrix rather than presentatrix. The plural presentatrices may appear in some older or more formulaic writings that adopt the -atrix/-atrices pattern, but it is not standard in contemporary textbooks.

Relation to invariants: For finitely presented abelian groups, the Smith normal form of a presentation matrix yields the invariant factor decomposition, providing a complete description of the group. More generally, the study of presentation matrices connects to homological algebra, where the matrix defines a chain complex whose homology captures the structure of the presented module or group.

See also: presentation, presentation matrix, relation matrix, group presentation, module presentation.