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From semi- +‎ group, reflecting the fact that not all the conditions required for a group are required for a semigroup. (Specifically, the requirements for the existence of identity and inverse elements are omitted.)


semigroup (plural semigroups)

  1. (mathematics) Any set for which there is a binary operation that is closed and associative.
    • 1961, Alfred Hoblitzelle Clifford, ‎G. B. Preston, The Algebraic Theory of Semigroups (page 70)
      If a semigroup S contains a zeroid, then every left zeroid is also a right zeroid, and vice versa, and the set K of all the zeroids of S is the kernel of S.
    • 1988, A. Ya Aǐzenshtat, Boris M. Schein (translator), On Ideals of Semigroups of Endomorphisms, Ben Silver (editor), Nineteen Papers on Algebraic Semigroups, American Mathematical Society Translations, Series 2, Volume 139, page 11,
      It follows naturally that various classes of ordered sets can be characterized by semigroup properties of endomorphism semigroups.
    • 2012, Jorge Almeida, Benjamin Steinberg, Syntactic and Global Subgroup Theory: A Synthesis Approach, Jean-Camille Birget, Stuart Margolis, John Meakin, Mark V. Sapir (editors), Algorithmic Problems in Groups and Semigroups, page 5,
      If one considers the variety of semigroups, one has the binary operation of multiplication defined on every semigroup.


  • (set for which a closed associative binary operation is defined): magma


  • (set for which a closed associative binary operation is defined): group, monoid

Derived terms[edit]