Definition of "semigroup"

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.

1961, Alfred Hoblitzelle Clifford, G. B. Preston, The Algebraic Theory of Semigroups, page 70

It follows naturally that various classes of ordered sets can be characterized by semigroup properties of endomorphism semigroups.

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

If one considers the variety of semigroups, one has the binary operation of multiplication defined on every semigroup.

2012, Jorge Almeida, Benjamin Steinberg, “Syntactic and Global Subgroup Theory: A Synthesis Approach”, in Jean-Camille Birget, Stuart Margolis, John Meakin, Mark V. Sapir, editors, Algorithmic Problems in Groups and Semigroups, page 5