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: | $(x*y)*(z*x) = x*((y*z)*x)$ |
Here are the identities in the syntax of the Lean Theorem Prover
section identities
variables {α: Type u} {β: Type v}
variables f g: α → α → α
variables h k: α → α
variable c: α
local notation a⬝b := f a b
local notation a+b := g a b
local notation a⁻¹ := h a
local notation 1 := c
local notation 0 := c
def involutive := ∀x, h(h x) = x
def inverse_operations := ∀x, h(k x) = x
def left_absorption := ∀x, h(k x) = k x
def right_absorption := ∀x, h(k x) = h x
def unary_idempotent := ∀x, h(h x) = h x
def idempotent := ∀x, x⬝x = x
def left_identity := ∀x, 1⬝x = x
def right_identity := ∀x, x⬝1 = x
def left_zero := ∀x, 0⬝x = 0
def right_zero := ∀x, x⬝0 = 0
def left_inverse := ∀x, x⁻¹⬝x = 1
def right_inverse := ∀x, x⬝x⁻¹ = 1
def left_const_mult := ∀x, c⬝x = h x
def right_const_mult := ∀x, x⬝c = h x
def square_constant := ∀x, x⬝x = c
def square_unary := ∀x, x⬝x = h x
def left_unary_identity := ∀x, (h x)⬝x = x
def right_unary_identity := ∀x, x⬝(h x) = x
def left_unary_const_mult := ∀x, h(c⬝x) = c⬝(h x)
def right_unary_const_mult := ∀x, h(x⬝c) = (h x)⬝c
def commutative := ∀x y, x⬝y = y⬝x
def left_unary_projection := ∀x y, x⬝y = h x
def right_unary_projection := ∀x y, x⬝y = h y
def left_idempotent := ∀x y, x⬝(x⬝y) = x⬝y
def right_idempotent := ∀x y, (x⬝y)⬝y = x⬝y
def left_rectangular := ∀x y, (x⬝y)⬝x = x
def right_rectangular := ∀x y, x⬝(y⬝x) = x
def left_absorption1 := ∀x y, (x⬝y)+y = y
def right_absorption1 := ∀x y, y+(x⬝y) = y
def left_absorption2 := ∀x y, (x⬝y)+x = x
def right_absorption2 := ∀x y, x+(y⬝x) = x
def left_subtraction := ∀x y, x⬝(x+y) = y
def right_subtraction := ∀x y, (y+x)⬝x = y
def unary_commutative := ∀x y, (h x)⬝(h y) = (h y)⬝(h x)
def unary_involutive := ∀x y, h(x⬝y) = (h y)⬝(h x)
def interdistributive := ∀x y, h(x⬝y) = (h x)+(h y)
def unary_distributive := ∀x y, h(x⬝y) = (h x)⬝(h y)
def left_twisted := ∀x y, (h(x⬝y))⬝x = x⬝(h y)
def right_twisted := ∀x y, x⬝(h(y⬝x)) = (h y)⬝x
def left_locality := ∀x y, h((h x)⬝y) = h(x⬝y)
def right_locality := ∀x y, h(x⬝(h y)) = h(x⬝y)
def left_unary_distributive := ∀x y, h((h x)⬝y) = (h x)⬝(h y)
def right_unary_distributive:= ∀x y, h(x⬝(h y)) = (h x)⬝(h y)
def left_absorbtive := ∀x y, (h x)⬝(h(x⬝y)) = h(x⬝y)
def right_absorbtive := ∀x y, (h(x⬝y))⬝(h y) = h(x⬝y)
def flexible := ∀x y, (x⬝y)⬝x = x⬝(y⬝x)
def associative := ∀x y z, x⬝(y⬝z) = (x⬝y)⬝z
def left_commutative := ∀x y z, x⬝(y⬝z) = y⬝(x⬝z)
def right_commutative := ∀x y z, (x⬝y)⬝z = (x⬝z)⬝y
def interassociative1 := ∀x y z, x⬝(y+z) = (x⬝y)+z
def interassociative2 := ∀x y z, x⬝(y+z) = (x+y)⬝z
def left_distributive := ∀x y z, x⬝(y+z) = (x⬝y)+(x⬝z)
def right_distributive := ∀x y z, (x+y)⬝z = (x⬝z)+(y⬝z)
def left_self_distributive := ∀x y z, x⬝(y⬝z) = (x⬝y)⬝(x⬝z)
def right_self_distributive := ∀x y z, (x⬝y)⬝z = (x⬝z)⬝(y⬝z)
def Moufang1 := ∀x y z, ((x⬝y)⬝x)⬝z = x⬝(y⬝(x⬝z))
def Moufang2 := ∀x y z, ((x⬝y)⬝z)⬝y = x⬝(y⬝(z⬝y))
def Moufang3 := ∀x y z, (x⬝y)⬝(z⬝x) = (x⬝(y⬝z))⬝x
def Moufang4 := ∀x y z, (x⬝y)⬝(z⬝x) = x⬝((y⬝z)⬝x)
def entropic := ∀x y z w, (x⬝y)⬝(z⬝w) = (x⬝z)⬝(y⬝w)
def paramedial := ∀x y z w, (x⬝y)⬝(z⬝w) = (w⬝y)⬝(z⬝x)
end identities