Square reader monad ($T(A)=A^2,\eta (a)=(a,a),\mu ((a,b),(c,d))=(a,d)$)

The unit repeats an element and multiplication selects the two outer diagonal entries. Both unit laws hold directly, and the associativity law selects the same two diagonal entries of an eight-entry array. Its algebras for a monad are precisely rectangular bands, including the empty algebra.