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equations [2013/05/29 09:08]
jipsen
equations [2017/10/02 10:57] (current)
jipsen
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|1  |trivial equations:  | $x = y$ $\quad f(x) = y$ $\quad x*y = z$  | $\Rightarrow$ one-element algebras  || |1  |trivial equations:  | $x = y$ $\quad f(x) = y$ $\quad x*y = z$  | $\Rightarrow$ one-element algebras  ||
|2  |identity operation:  | $f(x) = x$  ||| |2  |identity operation:  | $f(x) = x$  |||
-|3  |self-inverse operation:  | $f(f(x)) = x$  |||+|3  |involutive operation:  | $f(f(x)) = x$  |||
|4  |inverse operations:  | $f(g(x)) = x$  ||| |4  |inverse operations:  | $f(g(x)) = x$  |||
|5  |inside absorption:  | $f(g(x)) = f(x)$  ||| |5  |inside absorption:  | $f(g(x)) = f(x)$  |||
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|29  |left subtraction:  | $x*(x+y) = y$  | right subtraction:  | $(y+x)*x = y$  | |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:  | $(x+y)*z = (x*z)+(y*z)$  | |30  |left distributive:  | $x*(y+z) = (x*y)+(x*z)$  | right distributive:  | $(x+y)*z = (x*z)+(y*z)$  |
-|31  |$f$-commutative:  | $f(x)*f(y) = f(y)*f(x)$  ||| +|31 |left self-distributive:  | $x*(y*z) = (x*y)*(x*z)$  | right distributive:  | $(x*y)*z = (x*z)*(y*z)$  | 
-|32 |$f$-involutive:  | $f(x*y) = f(y)*f(x)$  ||| +|32 |$f$-commutative:  | $f(x)*f(y) = f(y)*f(x)$  ||| 
-|33 |$f$-interdistributive:  | $f(x*y) = f(x)+f(y)$  ||| +|33 |$f$-involutive:  | $f(x*y) = f(y)*f(x)$  ||| 
-|34 |$f$-distributive:  | $f(x*y) = f(x)*f(y)$  | also $f$-linear  || +|34 |$f$-interdistributive:  | $f(x*y) = f(x)+f(y)$  ||| 
-|35 |left $f$-constant multiple:  | $f(1*x) = 1*f(x)$  | right $f$-constant multiple:  | $f(x*1) = f(x)*1$  | +|35 |$f$-distributive:  | $f(x*y) = f(x)*f(y)$  | also $f$-linear  || 
-|36 |left twisted:  | $f(x*y)*x = x*f(y)$  | right twisted:  | $x*f(y*x) = f(y)*x$  | +|36 |left $f$-constant multiple:  | $f(1*x) = 1*f(x)$  | right $f$-constant multiple:  | $f(x*1) = f(x)*1$  | 
-|37 |left locality:  | $f(f(x)*y) = f(x*y)$  | right locality:  | $f(x*f(y)) = f(x*y)$  | +|37 |left twisted:  | $f(x*y)*x = x*f(y)$  | right twisted:  | $x*f(y*x) = f(y)*x$  | 
-|38 |left $f$-distributive:  | $f(f(x)*y) = f(x)*f(y)$  | right $f$-distributive:  | $f(x*f(y)) = f(x)*f(y)$  | +|38 |left locality:  | $f(f(x)*y) = f(x*y)$  | right locality:  | $f(x*f(y)) = f(x*y)$  | 
-|39 |left $f$-absorbtive:  | $f(x)*f(x*y) = f(x*y)$  | right $f$-absorbtive:  | $f(x*y)*f(y)) = f(x*y)$  | +|39 |left $f$-distributive:  | $f(f(x)*y) = f(x)*f(y)$  | right $f$-distributive:  | $f(x*f(y)) = f(x)*f(y)$  | 
-|40 |entropic:  | $(x*y)*(z*w) = (x*z)*(y*w)$  ||| +|40 |left $f$-absorbtive:  | $f(x)*f(x*y) = f(x*y)$  | right $f$-absorbtive:  | $f(x*y)*f(y)) = f(x*y)$  | 
-|41 |paramedial:  | $(x*y)*(z*w) = (w*y)*(z*x)$  |||+|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  
 + 
 +<code> 
 +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 
 +</code>