n. (Logic) 1a. a mapping, the domain and range of which are supersets of those of the original mapping, and such that the restriction of the larger mapping to the original domain agrees with the original mapping. For example, the principal complex square root is an extension of the positive square root defined for positive numbers.
b. a function or operator defined on a superspace of the domain of a given function and coinciding for arguments for which both are defined. See also Hahn-Banach theorem and Tietze's extension theorem. 2. also called conservative extension. a formal theory whose primitive terms, formation rules, and axioms include those of a given theory, and that contains the given theory in the sense that everything that is true in the given theory is also true in the extended theory. For example, Zermelo-Frankel set theory is a conservative extension of Peano arithmetic, and first-order predicate calculus of sentential calculus. Compare reduct. 3. (Algebra) a. (of a ring) a ring H, of which a given ring, G, is an ideal such that the factor ring H/G is isomorphic to N, where G is extended by N.
b. (of a group) a group H, of which a given group, G, is a normal subgroup such that the factor group H/G is isomorphic to N, where G is extended by N. 4. (Logic) the class of entities to which a given expression correctly applies. For example, the extension of the phrase satellite of Mars is the set of which the only members are Phobos and Deimos. Compare intension.