algebraic structure

English

Noun

algebraic structure (plural algebraic structures)

1. (algebra, universal algebra) Any one of the numerous types of mathematical object studied in algebra and especially in universal algebra.
2. (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.

   The definition of algebraic structure establishes a framework through which to compare otherwise very different abstract objects, and to study similarities and patterns made apparent by the definition.

   • 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.

   • 2019, Palash B. Pal, A Physicist's Introduction to Algebraic Structures, Cambridge University Press, page 37:

     Any algebraic structure is defined by a non-empty set of elements, and a set of rules or conditions that the elements satisfy. Thus, even a set equipped with a relation, as described in Section 2.4, is an algebraic structure. This structure is sometimes called a setoid, a name that we won't have much use for.

   • 2023, Michael Pilquist, Rúnar Bjarnason, Paul Chiusano, Functional programming with Scala, Second edition, Shelter Island: Manning Publications Co, →ISBN:

     A monoid is a purely algebraic structure consisting of an associative binary operation and an identity element for that operation.

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

formal structure of a carrier set, operations and identities

• Czech: algebraická struktura f
• Danish: algebraisk struktur c
• Finnish: algebrallinen rakenne (fi)
• French: structure algébrique f
• German: algebraische Struktur f
• Hungarian: algebrai struktúra (hu)
• Icelandic: algebrumynstur n, algebrulegt mynstur n
• Macedonian: алгебарска структура f (algebarska struktura)
• Norwegian:

  Bokmål: algebraisk struktur (no) m

  Nynorsk: algebraisk struktur m

• Portuguese: estrutura algébrica f
• Romanian: structură algebrică f
• Swedish: algebraisk struktur c

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