Elementarily

Elementarily is an adverb formed from the adjective elementary. It denotes doing something in a simple, basic, or elementary manner, at an elementary level, or in a straightforward or fundamental way.

In ordinary usage, elementarily is relatively rare. More common phrasing is to say something is done “in an elementary way” or “at an elementary level.” The term is often reserved for formal or careful writing when the emphasis is on simplicity, basic concepts, or accessibility, as opposed to advanced methods.

In mathematics and logic, elementarily has a specialized sense. Phrases such as elementarily equivalent describe two structures that satisfy the same first-order sentences; elementarily embedded describes an inclusion that preserves all such sentences. An elementary substructure is a substructure that shares the same first-order theory as the larger structure. These concepts are central to model theory, a branch of mathematical logic, where elementarity captures a precise preservation of logical properties under structural relationships.

Etymology traces elementarily to elementary, from Latin elementarius meaning basic or fundamental. In both everyday and technical use, elementarily retains the sense of simplicity or fundamental nature, though its technical meaning is confined largely to logic and related fields.

See also: elementary, elementarity, elementary substructure, elementary equivalence.