A Note on Semigroup Algebras of Permutable Semigroups
Attila Balázs Nagy, Márton Zubor · Communications in Algebra · 2016
Let S be a semigroup and 𝔽 be a field. For an ideal J of the semigroup algebra 𝔽[S] of S over 𝔽, let ϱJ denote the restriction (to S) of the congruence on 𝔽[S] defined by the ideal J. A semigroup S is called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β of S. In this paper we show that if S is a semilattice or a rectangular band then φ{S; 𝔽}: J → ϱJ is a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (ℬS; ○ ) if and only if S is a permutable semigroup.