Syntax | Terms | Equations | Horn formulas | Universal formulas | First-order formulas | Theories

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