algebra

Noun

algebra (countable and uncountable, plural algebras)

1. (uncountable) Elementary algebra: A system for representing and manipulating unknown quantities (variables) in equations.
   • Also the rule of false position, with dyuers examples not only vulgar, but some appertaynyng to the rule of Algeber.1551, James A.H. Murray, editor, A New English Dictionary on Historical Principles: Founded Mainly on the Materials Collected by the Philological Society., volume 1, Oxford: Clarendon Press...
   • Let us conceive, then, of an Algebra in which the symbols x, y, z, &c. admit indifferently of the values 0 and 1, and of these values alone.1854, George Boole, “Signs and their Laws”, in An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities, London: Walton and Maberly, page 37
2. (capitalized, sometimes, uncountable) Abstract algebra: A broad field of study in modern mathematics (often mentioned alongside analysis) loosely characterized by its concern for abstraction and symmetry, dealing with the behavior, classification, and application of a large class of objects (called algebraic structures) and the maps between them (called, most generally, morphisms).
3. A universal algebra.
4. An algebraic structure consisting of a module over a commutative ring (or a vector space over a field) along with an additional binary operation that is bilinear over module (or vector) addition and scalar multiplication.

   Synonyms:

   algebra over a field, algebra over a ring

5. (countable) A collection of subsets of a given set, such that this collection contains the empty set, and the collection is closed under unions and complements (and thereby also under intersections and differences).

   Synonyms:

   field of sets, algebra of sets

6. (figuratively) A system or process (especially one that is complex or convoluted) that substitutes one thing for another, or uses signs or symbols to represent concepts or ideas.
   • Fly ! Fly ! avaunt with that base cowardly gibbrish ; That Algebra of honour ; which had never Been nam'd, if all had equal courage—what?1663, William Clark, edited by William Hugh Logan, Marciano; or, The discovery: A tragi-comedy, Edinburgh: Reprinted for Private Circulation, published 1871, →ISBN, page 13
7. (historical, rare, uncountable) The surgical treatment of a dislocated or fractured bone. Also (countable): a dislocation or fracture.
   • Ne take noon hede to brynge togidere þe parties of þe boon þat is to-broken or dislocate, til viij. daies ben goon in þe wyntir, & v. in þe somer; for þanne it schal make quytture, and be sikir from swellynge; & þanne brynge togidere þe brynkis eiþer þe disiuncture after þe techynge þat schal be...a1420, The British Museum Additional MS, 12,056, “Wounds complicated by the Dislocation of a Bone”, in Robert von Fleischhacker, editor, Lanfranc's "Science of cirurgie.", London: K. Paul, Trench...
   • Algebra is used today by surgeons to mean bone-setting, i.e. the restoration of bones, and the idea of restoration is present in the mathematical context, too.1987, John Newsome Crossley, “Latency”, in The emergence of number, Singapore: World Scientific, →ISBN, Al-Khwarizwi, page 65