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Here we list equations, with the shorter term on the right (if possible).

 trivial equations: $x = y$ $\quad f(x) = y$ $\quad x*y = z$ $\Rightarrow$ one-element algebras identity operation: $f(x) = x$ self-inverse operation: $f(f(x)) = x$ inverse operations: $f(g(x)) = x$ order-$n$ operation: $f^n(x) = x$ $f$-idempotent $f(f(x)) = f(x)$ constant operations: $f(x) = 1$ $\quad f(x) = f(y)$ $\quad x*y = 1$ $x*y = f(z)$ $x*y = z*w$ left projection: $x*y = x$ right projection: $x*y = y$ idempotent: $x*x = x$ $n$-potent: $x^{n+1} = x^n$ left identity: $1*x = x$ right identity: $x*1 = x$ left zero: $0*x = 0$ right zero: $x*0 = 0$ left $f$-projection: $x*y = f(x)$ right $f$-projection: $x*y = f(y)$ square constant: $x*x = 1$ square definition: $x*x = f(x)$ left constant multiple: $1*x = f(x)$ right constant multiple: $x*1 = f(x)$ commutative: $x*y = y*x$ left inverse: $f(x)*x = 1$ right inverse: $x*f(x) = 1$ left $f$-identity: $f(x)*x = x$ right $f$-identity: $x*f(x) = x$ interassociative: $x*(y+z) = (x+y)*z$ associative: $x*(y*z) = (x*y)*z$ left commutativity: $x*(y*z) = y*(x*z)$ right commutativity: $(x*y)*z = (x*z)*y$ left idempotent: $x*(x*y) = x*y$ right idempotent: $(x*y)*y = x*y$ left rectangular: $(x*y)*x = x$ right rectangular: $x*(y*x) = x$ left distributive: $x*(y+z) = (x*y)+(x*z)$ right distributive: $(x+y)*z = (x*z)+(y*z)$ $f$-commutative: $f(x)*f(y) = f(y)*f(x)$ $f$-involutive: $f(x*y) = f(y)*f(x)$ $f$-interdistributive: $f(x*y) = f(x)+f(y)$ $f$-distributive: $f(x*y) = f(x)*f(y)$ also $f$-linear left $f$-constant multiple: $f(1*x) = 1*f(x)$ right $f$-constant multiple: $f(x*1) = f(x)*1$ left twisted: $f(x*y)*x = x*f(y)$ right twisted: $x*f(y*x) = f(y)*x$ left locality: $f(f(x)*y) = f(x*y)$ right locality: $f(x*f(y)) = f(x*y)$ left $f$-distributive: $f(f(x)*y) = f(x)*f(y)$ right $f$-distributive: $f(x*f(y)) = f(x)*f(y)$ left $f$-absorbtive: $f(x)*f(x*y) = f(x*y)$ right $f$-absorbtive: $f(x*y)*f(y)) = f(x*y)$ entropic: $(x*y)*(z*w) = (x*z)*(y*w)$ paramedial: $(x*y)*(z*w) = (w*y)*(z*x)$