Semigroup Algebra

The semigroup algebra K[S], where K is a field and S a semigroup, is formally defined in the same way as the group algebra K[G]. Similarly, a semigroup ring R[S] is a variation of the group ring R[G], where the group G is replaced by a semigroup S. Usually, it is required that S have an identity element e so that R[S] is a unit ring and R=Re is a subring of R[S].

The group algebra K[N] is the set of all formal expressions


where r_n in K for all n and a_n=0 for all but finitely many indices n so that a_n=0 for sufficiently large n (say, n>N). Hence, we can write the general element as




defines an isomorphism of K-algebras between K[N] and the polynomial ring K[x].

More generally, if S is the subsemigroup of N^r generated by the elements alpha_i=(a_(i1),...,a_(ir)), for i=1,...,t, the semigroup algebra K[S] is isomorphic to the subalgebra of the polynomial ring K[x_1,...,x_r] generated by the monomials m_(alpha_i)=x_1^(a_(i1))...x_n^(a_(ir)).

See also

Group Algebra

This entry contributed by Margherita Barile

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Okniński, J. Semigroup Algebras. New York: Dekker, 1991.

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Semigroup Algebra

Cite this as:

Barile, Margherita. "Semigroup Algebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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