A shelf is a set with a binary operation that distributes over itself. The most widely studied class of shelves are the racks, and among those the quandles, but the axioms for shelves are simpler.
A left shelf is a set with a binary operation obeying the left self-distributive law
Similarly a set with a binary operation obeying the right self-distributive law is called a right shelf.
A unital left shelf (meaning a left shelf together with an element that acts as an identity on both the left and the right) is the same as a graphic monoid: for a proof see graphic category.
Of course all the usual examples of racks and quandles are a fortiori shelves. But there are notable examples not of this type.
Let be the braid group. With the usual inclusion by appending a strand to the end of a braid on strands, the colimit of the chain is the infinite braid group . Let be the homomorphism that sends to . Then there is a left distributive operation on where
One may verify the left distributivity by a string diagram calculation (which appears on page 29 of this Google book, Dehornoy3).
Shelves make an appearance in set theory via large cardinal axioms. Let be a model of ZFC, and let be the collection of elements of rank less than an ordinal of . One (rather strong) large cardinal axiom on (limit ordinals) is:
There exists an elementary embedding on the structure that is not the identity.
Then, for , put
If we regard as an unary relation on , then induces an elementary embedding . In particular, if is any elementary embedding , which as a set of ordered pairs we may regard as a subset of , then as a set of ordered pairs is also an elementary self-embedding of . We get in this way a binary operation on elementary embeddings, which we denote as , and it is not difficult to verify that is left self-distributive.
Let denote the free left shelf generated by 1 element. If denotes the collection of elementary embeddings on the structure , then the preceding observations imply that is a left shelf, so any induces a shelf homomorphism
(Laver) If is not the identity, then is injective.
The famous Laver tables (derived from set-theoretic considerations which we omit for now) describe certain finite quotients of . Letting denote the generator of , define by and . The quotient of by the single relation is a shelf of cardinality , the largest power of dividing ; it is denoted . It can be described alternatively as the unique left shelf on the set such that (here represents the image of under the quotient ).
The “multiplication table” of an is called a Laver table. The behavior of Laver tables is largely not understood, but we mention a few facts. The first row consisting of entries is periodic (of some order dividing ). Under the large cardinal assumption that a nontrivial elementary self-embedding on a exists, this period tends to as does, but whether it does as a consequence of ZFC is not known. What is known is that this period, even if it increases to , does so quite slowly: if we define to be the smallest such that , then grows more quickly than say the Ackermann function.
Let be the monoid of positive braids, which as a monoid is presented by generators subject to the braid relations
If is a shelf, then there is a monoid homomorphism whose transform to an action is described by the equations
conversely, if is any binary operation, then these equations describe an action of only if is left distributive.
These are some general references:
Alissa Crans, Lie 2-algebras, Chapter 3.1: Shelves, Racks, Spindles and Quandles, Ph.D. thesis, U.C. Riverside, 2004. (pdf).
Patrick Dehornoy, Braids and self-distributivity, Progress in Mathematics 192, Birkhäuser Verlag, 2000.
These develop the connection between the free shelf on one generator and elementary embeddings in set theory:
Richard Laver, The left distributive law and the freeness of an algebra of elementary embeddings, Adv. Math. 91 (1992), 209–231.
Richard Laver, On the algebra of elementary embeddings of a rank into itself, Adv. Math. 110 (1995), 334–346.
Randall Dougherty and Thomas Jech, Finite left distributive algebras and embedding algebras, Adv. Math. 130 (1997), 201–241.
Randall Dougherty, Critical points in an algebra of elementary embeddings, Ann. Pure Appl. Logic 65 (1993), 211–241.
Randall Dougherty, Critical points in an algebra of elementary embeddings, II
For a popularized account of this material, see:
Last revised on November 1, 2024 at 04:24:12. See the history of this page for a list of all contributions to it.