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    $\begingroup$ This is really nice! I am doing the following and it seems to be assuming that things associate and commute. Can we stop that? $\endgroup$ Commented Mar 12, 2021 at 1:13
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    $\begingroup$ @BillyRubina What is $\theta$? You can edit you question and add the new code about $\theta$ $\endgroup$ Commented Mar 12, 2021 at 2:22
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    $\begingroup$ It is a function but I'd like to leave it unevaluated in the form $\theta(x)$. The trouble is that Mathematica changes the order, I don't know why. $\endgroup$ Commented Mar 12, 2021 at 2:29
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    $\begingroup$ @ SetAttributes[CircleDot, {Orderless, Flat, OneIdentity}]; CircleDot[{x_, y_}, {a_, b_}] := {x*θ[y]*a, y*b} {g1, h1}⊙{g2, h2}⊙{g3, h3} ? $\endgroup$ Commented Mar 12, 2021 at 2:46
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    $\begingroup$ @BillyRubina SetAttributes[CircleDot, {Flat, OneIdentity}]; CircleDot[{x_, y_}, {a_, b_}] := {x ** \[Theta][y] ** a, y ** b}; {g1, h1}\[CircleDot]{g2, h2}\[CircleDot]{g3, h3} $\endgroup$ Commented Mar 12, 2021 at 3:44