Finite simple abelian algebras are strictly simple

A finite universal algebra is called strictly simple if it is simple and has no nontrivial subalgebras. An algebra is said to be Abelian if for every term t(x,y¯)t(x,\bar y) and for all elements a,b,c¯,d¯a,b,\bar c,\bar d, we have the following implication: t(a,c¯)=t(a,d¯)→t(b,c¯)=t(b,d¯)t(a,\bar c) = t(a,\bar d) \to t(b,\bar c) = t(b,\bar d). It is shown that every finite simple Abelian universal algebra is strictly simple. This generalizes a …