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I've just come across the definition of the n-ary derived operation, namely that starting with an operational type $(\Omega, \alpha)$, set $ X_n = (x_1, ... , x_n) $ and $ \Omega$-structure $A$, we can define a function $ t_A : A^n \to A $ for each $t \in FX_n $ inductively as follows:

if $ t = x_i \ (1 \leq i \leq n) $, then $t_A$ is the projection map onto the ith factor.

if $ t = \omega t_1 ... t_m$ where $ \alpha(\omega) = m $, then $t_A$ is the composite map $ \omega_A \circ ((t_1)_A, ... , (t_m)_A) : A^n \to A^m \to A $.

In particular, if $t$ is the term $\omega x_1 ... x_n $ where $ n = \alpha(\omega) $, then $ t_A = \omega_A$.

Now I'm trying to make this definition more concrete, by interpreting it in a group theoretical context. Take $ \Omega = (m, i, e) $, where $m, i, e$ are the binary, unary and nullary group operations respectively. Now, I'm getting confused about the roles that $ X_n$ and $A$ play here. Which should I take to be the set of elements of the group? What should the other be interpreted as (if applicable)?


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See the OP's prior question for the notation. – Bill Dubuque Aug 26 '11 at 16:26
Thanks - I'd assumed it was standard notation. – TRY Aug 26 '11 at 16:52
The notation is motley - some quite obfuscated. – Bill Dubuque Aug 26 '11 at 17:01

This is simply the recursive definition of the $n$-ary operations induced by $\Omega$-terms in the "variables" $x_1,\ldots,x_n$. For example, in groups the commutator $\:[x_1,x_2]\: :=\: x_1\:x_2\:x_1^{-1}\:x_2^{-1}\:$ is a derived binary operation, and $x_1\:x_2^{-1}\:x_3\:$ is a derived ternary operation, and the $n$-ary product $\:x_1\:x_2\:\cdots\:x_n\:$ is a derived $n$-ary operation. In other words, every term in the free $\Omega$-algebra $F_\Omega\{x_1,\ldots,x_n\}$ induces an $n$-ary function by simply evaluating the term on the supplied arguments. It's just the analog in general equational algebras (varieties) of the well-known fact that formal $n$-variable polynomials over a ring induce $n$-ary polynomial functions on the ring.

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In the context of groups, $A$ is the underlying set, $m$ is group multiplication, $i$ is the group inverse, and $e$ is the identity. The axioms of groups are:

  1. $m(a,m(b,c)) = m(m(a,b),c)$ (associativity).
  2. $m(a,i(a)) = m(i(a),a))$ (inverse).
  3. $m(a,e) = m(e,a) = a$ (identity element).
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What are the $ x_i$? – TRY Aug 26 '11 at 10:53
The $x_i$ are the arguments of the function you're trying to construct. For example, in the language of groups you can construct the function $f(x_1,x_2,x_3) = m(x_1,e)$, using the projection $x_1$ and composition with $e$ and $m$. – Yuval Filmus Aug 26 '11 at 12:15

In this context, "an $\Omega$-structure $A$" just means a particular group $\mathbf{G}$. "$FX_n$" means the free group on $X_n=\{x_1,\ldots,x_n\}$. The function $t_A\colon A^n\to A$ is a set-theoretic function from the $n$th fold cartesian power of (the underlying set of) $A$ to the set $A$, and is defined in terms of an element $t\in FX_n$.

For example, say $A=S_3$, the symmetric group in three letters; $n=4$, so $FX_4$ is the free group in $x_1,x_2,x_3,x_4$. For each element $t\in FX_4$ we get a function from $A^4$ to $A$.

  • The function associated to $t=x_3$, for example, is just the function that sends $(\sigma_1,\sigma_2,\sigma_3,\sigma_4)$ to $\sigma_3$ for all $\sigma_i\in G$.

  • The function associated to $t=x_1x_2x_3^{-1}$ is the function that sends $(\sigma_1,\sigma_2,\sigma_3,\sigma_4)$ to $\sigma_1\sigma_2\sigma_3^{-1}$.

  • The function associated to $t=x_3x_2^{-1}x_4^2x_1x_3^{-1}$ is the function that sends $(\sigma_1,\sigma_2,\sigma_3,\sigma_4)$ to $\sigma_3\sigma_2^{-1}\sigma_4^2\sigma_1\sigma_3^{-1}$.

etc. Each element of the free group gives you an operation on $A$.

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