Biased definitions

Idea

Often in mathematics, when requiring some structure/operation/property/… to exist at every finite arity?, it suffices to require only the binary ($2$-ary) and nullary ($0$-ary) forms, from which the others follow. A definition in which only these are required is called biased.

For example, in defining a category, one could use an “unbiased” definition in which composites of all finite sequences of morphisms are directly postulated, with corresponding associativity laws, but it suffices to require only binary composites and nullary composites (i.e., identity morphisms) and some particular corresponding associativity laws.

As a special case of this, we have perhaps the prototypical example of a binary/nullary pair sufficing to generate all finite instances of some structure: the natural numbers themselves (the arities of the operations that we are considering) are the free monoid on one generator, and thus are freely associatively produced from that one generator (aka, $1$) using only binary and nullary addition.

When a nullary operation does not exist

Sometimes a nullary operation does not exist but one still wants to decompose a n-ary operation into binary operations. For example, consider the reals, $\mathbb{R}$, as an unbounded lattice (top, $\top$, and bottom, $\bot$, do not exist) where

$\wedge =$ product $=$ meet $= infinum = min$

and

$\vee =$ coproduct $=$ join $= supremum = max$.

Here $\bigwedge(\{\})$ does not exist while

$\bigwedge(\{a\}) = a$

$\bigwedge(\{a,b\}) = a \wedge b$

$\bigwedge(\{a,b,c\}) = a \wedge (b \wedge c) = (a \wedge b) \wedge c = a \wedge b \wedge c$.

One approach is to compute in the extended reals, $\mathbb{R}_{\pm \infty}$ ($\mathbb{R}$ enlarged with $+\infty$ and $-\infty$.) Here

$\top = +\infty =$ terminal object

and

$\bot = -\infty =$ initial object.

In $\mathbb{R}_{\pm \infty}$ we have the nullary $\wedge() = +\infty$ which gives:

$\bigwedge(\{\}) = +\infty$

$\bigwedge(\{a\}) = \bigwedge(\{a, +\infty\}) = a$.

Another approach is to define a special scheme for composition for when a nullary operator does not exist that instead uses a unary operator that is an identity map (or factored through one).

This approach generalizes to any semigroup or to any category with binary (co) products. A yet more general context (possibly not fully worked out) would be a binary operation in any associative operad.

Revised on September 29, 2015 22:11:15 by Mike Shulman (76.88.99.98)