-
Notifications
You must be signed in to change notification settings - Fork 40
Tour of selected equations
Equation 1 x = x: the trivial law
This is the weakest law, satisfied by all magmas. No other law is equivalent to this law.
Equation 2 x = y: the singleton law
This is the strongest law, satisfied only by the trivial magmas: the singleton and empty magmas. Many laws are equivalent to this law; informally, they are so ``overdetermined'' that they can only be satisfied by trivial magmas. In fact, from our list of 4694 laws, exactly 1495 other laws are equivalent to this one.
Equations 14 x = y ◇ (x ◇ y), 29 x = (y ◇ x) ◇ y: 2001 Putnam laws
Problem A1 of the 2001 Putnam asked to show (in our language) that Equation 14 implies Equation 29. In fact, the two laws are equivalent.
Equation 43 x ◇ y = y ◇ x: the commutative law
One of the most famous laws in all of algebra.
Equation 46 x ◇ y = z ◇ w: the constant law
A very strong law, that makes the entire multiplication table constant.
Equations 65 x = y ◇ (x ◇ (y ◇ x)), 1491 x = (y ◇ x) ◇ (y ◇ (y ◇ x)): "Asterix and Oberlix"
Equation 65 is one of the simplest equations for which several outgoing implications remain unresolved, see this ongoing discussion. In particular, it is unknown whether either of these equations implies the other, although 1491 implies all seven equations for which it is unknown whether 65 implies it. One of the issues is that there seems to be a difference of behavior between the finite and infinite models of these equations.
Equation 168 x = (y ◇ x) ◇ (x ◇ z): the central groupoid law
This law defines central groupoids, which are magmas with some interesting combinatorial structure. For instance, all finite central groupoids have a cardinality that is necessarily a square number. This paper of Knuth establishes many of the basic properties of these objects.
Equations 381 x ◇ y = (x ◇ z) ◇ y, 3722 x ◇ y = (x ◇ y) ◇ (x ◇ y), 3744 x ◇ y = (x ◇ z) ◇ (w ◇ y): 1978 Putnam laws
Problem A4 of the 1978 Putnam asked to show (in our language) that Equation 3744 implies Equations 3722 and 381. This Putnam question referred to 3744 as a "bypass law".
Equation 387 x ◇ y = (y ◇ y) ◇ x: a law from MathOverflow
This response by pastebee to a MathOverflow question established that there was an equational law strictly between the constant law 46 and the commutative law 43, namely Equation 387.
Equation 477 x = y ◇ (x ◇ (y ◇ (y ◇ y))): a confluent law
This law is a non-trivial example of a confluent law: a law in which every word has a unique shortest reduction using the law. This makes it possible to easily determine which other equations are implied by this law, giving anti-implications that were not obtainable by other means. See this discussion.
Equation 1571 x = (y ◇ z) ◇ (y ◇ (x ◇ z)): a law for abelian groups of exponent 2
It was shown by Mendelsohn and Padmanabhan that this law characterizes abelian groups of exponent 2.
Equation 1689 x = (y ◇ x) ◇ ((x ◇ z) ◇ z): a non-trivially singleton law
This law was identified by Kisielewicz as a law that collapses to the singleton law 2, but all known proofs are surprisingly lengthy.
Equation 4512 x ◇ (y ◇ z) = (x ◇ y) ◇ z: the associative law
One of the most famous laws in all of algebra.
Equations 3588 x ◇ y = z ◇ ((x ◇ y) ◇ z), 3994 x ◇ y = (z ◇ (x ◇ y)) ◇ z: An Austin pair
It was shown as part of this project that Equation 3944 implies Equation 3588 for finite magmas, but not for infinite magmas.
An Austin law is a law which has infinite models, but no non-trivial finite models. Kisielewicz showed that this law has no non-trivial finite models, but it remains open whether there is an infinite model.
It was shown by Knuth that this law characterizes "natural central groupoids", which, up to isomorphisms, are Cartesian squares S × S with magma operation (a,b) ∘ (c,d) = (b,c). These are special cases of central groupoids (Equation 168).
Kisielewicz showed that this law is an Austin law: it has no non-trivial finite models, but it has an infinite model. He also showed there is no shorter Austin law.
It was shown by McCune, Veroff, Fitelson, Harris, Feist, and Wos that this law defines the Sheffer stroke.
It was shown by Kisielewicz that this law is an Austin law: it has no non-trivial finite models, but it has an infinite model.