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

 1 trivial equations: $x = y$ $\quad f(x) = y$ $\quad x*y = z$ $\Rightarrow$ one-element algebras 2 identity operation: $f(x) = x$ 3 involutive operation: $f(f(x)) = x$ 4 inverse operations: $f(g(x)) = x$ 5 inside absorption: $f(g(x)) = f(x)$ 6 outside absorption: $f(g(x)) = g(x)$ 7 order-$n$ operation: $f^n(x) = x$ 8 $f$-idempotent $f(f(x)) = f(x)$ 9 constant operations: $f(x) = 1$ $\quad f(x) = f(y)$ $\quad x*y = 1$ $x*y = f(z)$ $x*y = z*w$ 10 left projection: $x*y = x$ right projection: $x*y = y$ 11 idempotent: $x*x = x$ 12 $n$-potent: $x^{n+1} = x^n$ 13 left identity: $1*x = x$ right identity: $x*1 = x$ 14 left zero: $0*x = 0$ right zero: $x*0 = 0$ 15 left $f$-projection: $x*y = f(x)$ right $f$-projection: $x*y = f(y)$ 16 square constant: $x*x = 1$ 17 square definition: $x*x = f(x)$ 18 left constant multiple: $1*x = f(x)$ right constant multiple: $x*1 = f(x)$ 19 commutative: $x*y = y*x$ 20 left inverse: $f(x)*x = 1$ right inverse: $x*f(x) = 1$ 21 left $f$-identity: $f(x)*x = x$ right $f$-identity: $x*f(x) = x$ 22 interassociative: $x*(y+z) = (x+y)*z$ 23 associative: $x*(y*z) = (x*y)*z$ 24 left commutativity: $x*(y*z) = y*(x*z)$ right commutativity: $(x*y)*z = (x*z)*y$ 25 left idempotent: $x*(x*y) = x*y$ right idempotent: $(x*y)*y = x*y$ 26 left rectangular: $(x*y)*x = x$ right rectangular: $x*(y*x) = x$ 27 left absorption: $(x*y)+x = x$ right absorption: $x+(y*x) = x$ 28 left absorption1: $(x*y)+y = y$ right absorption1: $y+(x*y) = y$ 29 left subtraction: $x*(x+y) = y$ right subtraction: $(y+x)*x = y$ 30 left distributive: $x*(y+z) = (x*y)+(x*z)$ right distributive: $(x+y)*z = (x*z)+(y*z)$ 31 left self-distributive: $x*(y*z) = (x*y)*(x*z)$ right distributive: $(x*y)*z = (x*z)*(y*z)$ 32 $f$-commutative: $f(x)*f(y) = f(y)*f(x)$ 33 $f$-involutive: $f(x*y) = f(y)*f(x)$ 34 $f$-interdistributive: $f(x*y) = f(x)+f(y)$ 35 $f$-distributive: $f(x*y) = f(x)*f(y)$ also $f$-linear 36 left $f$-constant multiple: $f(1*x) = 1*f(x)$ right $f$-constant multiple: $f(x*1) = f(x)*1$ 37 left twisted: $f(x*y)*x = x*f(y)$ right twisted: $x*f(y*x) = f(y)*x$ 38 left locality: $f(f(x)*y) = f(x*y)$ right locality: $f(x*f(y)) = f(x*y)$ 39 left $f$-distributive: $f(f(x)*y) = f(x)*f(y)$ right $f$-distributive: $f(x*f(y)) = f(x)*f(y)$ 40 left $f$-absorbtive: $f(x)*f(x*y) = f(x*y)$ right $f$-absorbtive: $f(x*y)*f(y)) = f(x*y)$ 41 flexible: $(x*y)*x = x*(y*x)$ 42 entropic: $(x*y)*(z*w) = (x*z)*(y*w)$ 43 paramedial: $(x*y)*(z*w) = (w*y)*(z*x)$ 44 Moufang1: $((x*y)*x)*z = x*(y*(x*z))$ Moufang2: $((x*y)*z)*y = x*(y*(z*y))$ 45 Moufang3: $(x*y)*(z*x) = (x*(y*z))*x$ Moufang4: $(x*y)*(z*x) = x*((y*z)*x)$