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Syntax | Terms | Equations | Quasiequations

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: $(x*y)*(z*w) = (w*y)*(z*x)$