2 Selected laws
In this project we study the 4694 laws (up to symmetry and relabeling) of total order at most \(4\).
Selected laws of interest are listed below, as well as in this file.
Equation 1 is the law \(0 \simeq 0\) (or the equation \(x=x\)).
This is the trivial law, satisfied by all magmas. It is self-dual.
Equation 2 is the law \(0 \simeq 1\) (or the equation \(x=y\)).
This is the singleton law, satisfied only by the empty and singleton magmas. It is self-dual.
Equation 3 is the law \(0 \simeq 0 \diamond 0\) (or the equation \(x = x \diamond x\)).
This is the idempotence law. It is self-dual.
Equation 4 is the law \(0 \simeq 0 \diamond 1\) (or the equation \(x = x \diamond y\)).
This is the left absorption law.
Equation 5 is the law \(0 \simeq 1 \diamond 0\) (or the equation \(x = y \diamond x\)).
This is the right absorption law (the dual of Definition 2.4).
Equation 6 is the law \(0 \simeq 1 \diamond 1\) (or the equation \(x = y \diamond y\)).
This law is equivalent to the singleton law.
Equation 7 is the law \(0 \simeq 1 \diamond 2\) (or the equation \(x = y \diamond z\)).
This law is equivalent to the singleton law.
Equation 8 is the law \(0 \simeq 0 \diamond (0 \diamond 0)\) (or the equation \(x = x \diamond (x \diamond x)\)).
Equation 14 is the law \(0 \simeq 1 \diamond (0 \diamond 1)\) (or the equation \(x = y \diamond (x \diamond y))\).
Appears in Problem A1 from Putnam 2001. See Theorem 5.2.
Equation 16 is the law \(0 \simeq 1 \diamond (1 \diamond 0)\) (or the equation \(x = y \diamond (y \diamond x))\).
Equation 23 is the law \(0 \simeq (0 \diamond 0) \diamond 0\) (or the equation \(x = (x \diamond x) \diamond x\)).
This is the dual of Definition 2.8.
Equation 29 is the law \(0 \simeq (1 \diamond 0) \diamond 1\) (or the equation \(x = (y \diamond x) \diamond y)\).
Appears in Problem A1 from Putnam 2001. Dual to Definition 2.9. See Theorem 5.2.
Equation 38 is the law \(0 \diamond 0 \simeq 0 \diamond 1\) (or the equation \(x \diamond x = x \diamond y\)).
This law asserts that the magma operation is independent of the second argument.
Equation 39 is the law \(0 \diamond 0 \simeq 1 \diamond 0\) (or the equation \(x \diamond x = y \diamond x\)).
This law asserts that the magma operation is independent of the first argument (the dual of Definition 2.13).
Equation 40 is the law \(0 \diamond 0 \simeq 1 \diamond 1\) (or the equation \(x \diamond x = y \diamond y\)).
This law asserts that all squares are constant. It is self-dual.
Equation 41 is the law \(0 \diamond 0 \simeq 1 \diamond 2\) (or the equation \(x \diamond x = y \diamond z\)).
This law is equivalent to the constant law, Definition 2.20.
Equation 42 is the law \(0 \diamond 1 \simeq 0 \diamond 2\) (or the equation \(x \diamond y = x \diamond z\)).
Equivalent to Definition 2.13.
Equation 43 is the law \(0 \diamond 1 \simeq 1 \diamond 0\) (or the equation \(x \diamond y = y \diamond x\)).
The commutative law. It is self-dual.
Equation 45 is the law \(0 \diamond 1 \simeq 2 \diamond 1\) (or the equation \(x \diamond y = z \diamond y\)).
This is the dual of Definition 2.17.
Equation 46 is the law \(0 \diamond 1 \simeq 2 \diamond 3\) (or the equation \(x \diamond y = z \diamond w\)).
The constant law: all products are constant. It is self-dual.
Equation 63 is the law \(0 \simeq 1 \diamond (0 \diamond (0 \diamond 1))\) (or the equation \(x = y \diamond (x \diamond (x \diamond y))\)).
The “Dupont” law, studied further in Section 7.6.
Equation 65 is the law \(0 \simeq 1 \diamond (0 \diamond (1 \diamond 0))\) (or the equation \(x = y \diamond (x \diamond (y \diamond x))\)).
The “Asterix” law, studied further in Section 7.2.
Equation 168 is the law \(0 \simeq (1 \diamond 0) \diamond (0 \diamond 2)\) (or the equation \(x = (y \diamond x) \diamond (x \diamond z)\)).
The law of a central groupoid. It is self-dual.
Equation 206 is the law \(0 \simeq (0 \diamond (0 \diamond 1)) \diamond 1\) (or the equation \(x = (x \diamond (x \diamond y)) \diamond y\)).
Our project located this law as one member of an “Austin pair”; see Chapter 3. The infinite counterexample is constructed using the infinite 3-regular tree.
Equation 381 is the law \(0 \diamond 1 \simeq (0 \diamond 2) \diamond 1\) (or the equation \(x \diamond y = (x \diamond z) \diamond y\)).
Appears in Putnam 1978, Problem A4, part (b).
Equation 387 is the law \(0 \diamond 1 \simeq (1 \diamond 1) \diamond 0\) (or the equation \(x \diamond y = (y \diamond y) \diamond x\)).
Introduced in MathOverflow. See Theorem 5.1
Equation 477 is the law \(0 \simeq 1 \diamond (0 \diamond (1 \diamond (1 \diamond 1)))\) (or the equation \(x = y \diamond (x \diamond (y \diamond (y \diamond y)))\)).
An example of a confluent law; see Theorem 10.8.
Equation 854 is the law \(0 = 0 \diamond ((1 \diamond 2) \diamond (0 \diamond 2))\) (or the equation \(x = x \diamond ((y \diamond z) \diamond (x \diamond z))\)).
Studied in Chapter 14
Equation 953 is the law \(0 = 1 \diamond ((2 \diamond 0) \diamond (2 \diamond 2))\) (or the equation \(x = y \diamond ((z \diamond x) \diamond (z \diamond z))\)).
An example of a trivial law; see Theorem 5.7.
Equation 1485 is the law \(0 \simeq (1 \diamond 0) \diamond (0 \diamond (2 \diamond 1))\) (or the equation \(x = (y \diamond x) \diamond (x \diamond (z \diamond y))\)).
The “Obelix” law, studied further in Section 7.2.
Equation 1491 is the law \(0 \simeq (1 \diamond 0) \diamond (1 \diamond (1 \diamond 0))\) (or the equation \(x = (y \diamond x) \diamond (y \diamond (y \diamond x))\)).
The “Obelix” law, studied further in Section 7.2.
Equation 1571 is the law \(0 \simeq (1 \diamond 2) \diamond (1 \diamond (0 \diamond 2))\) (or the equation \(x = (y \diamond z) \diamond (y \diamond (x \diamond z))\)).
Introduced in [ 10 ] . As shown in Theorem 5.6, this law characterizes abelian groups of exponent two.
Equation 1648 is the law \(0 \simeq (0 \diamond 1) \diamond ((0 \diamond 1) \diamond 1)\) (or the equation \(x = (x \diamond y) \diamond ((x \diamond y) \diamond y)\)).
The golden ratio is a coefficient of the linearization of this law.
Equation 1657 is the law \(0 \simeq (0 \diamond 1) \diamond ((1 \diamond 1) \diamond 0)\) (or the equation \(x = (x \diamond y) \diamond ((y \diamond y) \diamond x)\)).
Equation 1659 is the law \(0 \simeq (0 \diamond 1) \diamond ((1 \diamond 1) \diamond 2)\) (or the equation \(x = (x \diamond y) \diamond ((y \diamond y) \diamond z)\)).
Equation 1661 is the law \(0 \simeq (0 \diamond 1) \diamond ((1 \diamond 2) \diamond 1)\) (or the equation \(x = (x \diamond y) \diamond ((y \diamond z) \diamond y)\)).
These two laws admit infinite models on the natural numbers arising from the modified base model construction. See Section 7.5.
Equation 1689 is the law \(0 \simeq (1 \diamond 0) \diamond ((0 \diamond 2) \diamond 2)\) (or the equation \(x = (y \diamond x) \diamond ((x \diamond z) \diamond z)\)).
Mentioned in [ 4 ] . See Theorem 5.5.
Equation 1701 is the law \(0 \simeq (1 \diamond x) \diamond ((2 \diamond 0) \diamond 0)\) (or the equation \(x = (y \diamond x) \diamond ((z \diamond x) \diamond x)\)).
This law admits infinite models on the natural numbers arising from the modified base model construction. See Section 7.5.
Equation 2662 is the law \(0 \simeq ((0 \diamond 1) \diamond (0 \diamond 1)) \diamond 0\) (or the equation \(x = ((x \diamond y) \diamond (x \diamond y)) \diamond x\)).
Appears in [ 10 ] .
Equation 3167 is the law \(0 \simeq (((1 \diamond 1) \diamond 2) \diamond 2) \diamond 0\) (or the equation \(x = (((y \diamond y) \diamond z) \diamond z) \diamond x\)).
Equation 3588 is the law \(0 \diamond 1 \simeq 2 \diamond ((0 \diamond 1) \diamond 2)\) (or the equation \(x \diamond y = z \diamond ((x \diamond y) \diamond z)\)).
Our project located this law as one member of an “Austin pair”; see Chapter 3.
Equation 3722 is the law \(0 \diamond 1 \simeq (0 \diamond 1) \diamond (0 \diamond 1)\) (or the equation \(x \diamond y = (x \diamond y) \diamond (x \diamond y)\)).
Appears in Putnam 1978, Problem A4, part (a). It is self-dual.
Equation 3744 is the law \(0 \diamond 1 \simeq (0 \diamond 2) \diamond (3 \diamond 1)\) (or the equation \(x \diamond y = (x \diamond z) \diamond (w \diamond y)\)).
This law is called a “bypass operation” in Putnam 1978, Problem A4. It is self-dual. See Theorem 5.4.
Equation 3994 is the law \(0 \diamond 1 \simeq (2 \diamond (0 \diamond 1)) \diamond 2\) (or the equation \(x \diamond y = (z \diamond (x \diamond y)) \diamond z\)).
Our project located this law as one member of an “Austin pair”; see Chapter 3.
Equation 4315 is the law \(0 \diamond (1 \diamond 0) \simeq 0 \diamond (1 \diamond 2)\) (or the equation \(x \diamond (y \diamond x) = x \diamond (y \diamond z)\)).
Equation 4512 is the law \(0 \diamond (1 \diamond 2) \simeq (0 \diamond 1) \diamond 2\) (or the equation \(x \diamond (y \diamond z) = (x \diamond y) \diamond z\)).
The associative law. It is self-dual.
Equation 4513 is the law \(0 \diamond (1 \diamond 2) \simeq (0 \diamond 1) \diamond 3\) (or the equation \(x \diamond (y \diamond z) = (x \diamond y) \diamond w\)).
Equation 4522 is the law \(0 \diamond (1 \diamond 2) \simeq (0 \diamond 3) \diamond 4\) (or the equation \(x \diamond (y \diamond z) = (x \diamond w) \diamond u\)).
Dual to Definition 2.50.
Equation 4564 is the law \(0 \diamond (1 \diamond 2) \simeq (3 \diamond 1) \diamond 2\) (or the equation \(x \diamond (y \diamond z) = (w \diamond y) \diamond z\)).
Dual to Definition 2.47.
Equation 4579 is the law \(0 \diamond (1 \diamond 2) \simeq (3 \diamond 4) \diamond 2\) (or the equation \(x \diamond (y \diamond z) = (w \diamond u) \diamond z\)).
Dual to Definition 2.48.
Equation 4582 is the law \(0 \diamond (1 \diamond 2) \simeq (3 \diamond 4) \diamond 5\) (or the equation \(x \diamond (y \diamond z) = (w \diamond u) \diamond v\)).
This law asserts that all triple constants (regardless of bracketing) are constant.
2.1 Equations of order greater than \(4\)
We note some selected laws of order more than \(5\), which are used in some later chapters of the blueprint.
Equation 5093 is the law \(0 \simeq 1 \diamond (1 \diamond (1 \diamond (0 \diamond (2 \diamond 1))))\) (or the equation \(x = y \diamond (y \diamond (y \diamond (x \diamond (z \diamond y))))\)).
This law of order \(5\) was mentioned in [ 4 ] . See Theorem 3.3.
Equation 26302 is the law \(0 \simeq (1 \diamond ((2 \diamond 0) \diamond 3)) \diamond (0 \diamond 3)\) (or the equation \(x = (y \diamond ((z \diamond x) \diamond w)) \diamond (x \diamond w)\)).
A law that characterizes natural central groupoids; see Theorem 5.9.
Equation 28770 is the law \(0 \simeq (((1 \diamond 1) \diamond 1) \diamond 0) \diamond (1 \diamond 2)\) (or the equation \(x = (((y \diamond y) \diamond y) \diamond x) \diamond (y \diamond z)\)).
This law of order \(5\) was introduced by Kisielewicz [ 5 ] . See Theorem 3.2.
Equation 345169 is the law \(0 \simeq (1 \diamond ((0 \diamond 1) \diamond 1)) \diamond (0 \diamond (2 \diamond 1))\) (or the equation \(x = (y \diamond ((x \diamond y) \diamond y)) \diamond (x \diamond (z \diamond y))\)).
This law of order \(6\) was shown in [ 9 ] to characterize the Sheffer stroke in a boolean algebra; see Theorem 5.8.
Equation 374794 is the law \(0 \simeq (((1 \diamond 1) \diamond 1) \diamond 0) \diamond ((1 \diamond 1) \diamond 2)\) (or the equation \(x = (((y \diamond y) \diamond y) \diamond x) \diamond ((y \diamond y) \diamond z)\)).
This law of order \(6\) was introduced by Kisielewicz [ 5 ] ; see Theorem 3.1.