algebraic structure

== English == === Noun === algebraic structure (plural algebraic structures) (algebra, universal algebra) Any one of the numerous types of mathematical object studied in algebra and especially in universal algebra. (more formally) A mathematical object comprising a carrier set (aka underlying set or domain), an optional scalar set, a set of operations (typically binary operations, but otherwise each of finite arity) and a set of identities (axioms) which the operations must satisfy. 1975, Ronald James Williams, Algebraic Structures Up to Homotopy, University of California, San Diego, page 16, In this chapter we propose a general means for describing algebraic structures, both strict and up to homotopy, and we apply the results of the preceding chapter to the study of these […] . 1995, George R. Kempf, Algebraic Structures, Bertelsmann (Vieweg), page 129, Thus the algebraic structures which we have been studying are part of category theory, but the important theorem is not generally categorical nonsense, although there are theorems in category theory which we will not study. ==== Usage notes ==== While algebraic structures are the principal subject of study in universal algebra, the term algebra is preferred within that field. Outside universal algebra, the term algebra is itself reserved for the specific structures algebra over a field (“vector space over a field”) and algebra over a ring (“module over a commutative ring”). ==== Synonyms ==== (formal structure of a carrier set, operations and axioms): algebra (in the context of universal algebra), algebraic system, universal algebra ==== Hyponyms ==== setoid group vector space module ring division ring integral domain field linear algebra, associative algebra lattice category ==== Translations ==== === Further reading === Universal algebra on Wikipedia.Wikipedia Operation (mathematics) on Wikipedia.Wikipedia Model theory on Wikipedia.Wikipedia Category:Algebraic structures on Wikipedia.Wikipedia Universal algebra on Encyclopedia of Mathematics Algebraic system on Encyclopedia of Mathematics