Structures

Definition 1.  A structure is a set X together with an n-ary operation O:X^n\to X.

We can characterize an operation on a structure: let O be an n-ary operation on X.  O is associative if

\begin{array}{lcl}O(O(x_1,...,x_n),x_{n+1},...,x_{2n-1})&=&O(x_1,O(x_2,...,x_{n+1}),...,x_{2n-1})\\&\vdots&\\&=& O(x_1,...,x_{n-1},O(x_n,...,x_{2n-1}))\end{array}

where x_i\in X for 1\leq i\leq 2n-1.  The statement is trivial in the nullary and unary case.  In the binary case we have

(x_1x_2)x_3=x_1(x_2x_3)

or in the ternary case

[[x_1,x_2,x_3],x_4,x_5]=[x_1,[x_2,x_3,x_4],x_5]=[x_1,x_2,[x_3,x_4,x_5]].

More generally if m\leq n, P is an m-ary operation, and O is an n-ary operation, we say P associates with O if

\begin{array}{lcl}O(P(x_1,...,x_m),x_{m+1},...,x_{n+m-1})&=&O(x_1,P(x_2,...,x_{m+1}),...,x_{m+n-1})\\&\vdots&\\&=&O(x_1,...,x_{n-1},P(x_n,...,x_{n+m-1}))\end{array}.

Thus an associative operation is one that associates with itself.

Definition 2.  A nullary operation N(\,)=1\in X that associates with an n-ary operation O is called an identity with respect to O.

Definition 3.  Let X have an identity 1 with respect to an n-ary operation O.  We say an (n-1)-ary operation U is unital with respect to O if:

O(U(x_1,...,x_{n-1}),x_1,...,x_{n-1})=O(x_1,...,x_{n-1},U(x_1,...,x_{n-1}))=1.

Example 4.  The unary inverse operation in a group is unital with respect to the binary product.

We say an n-ary operation O is commutative if

O(x_1,...,x_n)=O(x_{\sigma(1)},...,x_{\sigma(n)})

where \sigma is a permutation of \{1,...,n\}.

Now let P be an m-ary operation.  O is left-distributive with respect to P if

O(x_1,...,x_{n-1},P(y_1,...,y_m))=P(O(x_1,...,x_{n-1},y_1),...,O(x_1,...,x_{n-1},y_m))

and right-distributive with respect to P if

O(P(y_1,...,y_m),x_1,...,x_{n-1})=P(O(y_1,x_1,...,x_{n-1}),...,O(y_m,x_1,...,x_{n-1})).

O is distributive with respect to P if it is left and right distributive with respect to P.

Example 5.  In a ring, multiplication is distributive with respect to addition ((a+b)c=ac+bc and c(a+b)=ca+cb).  Addition does not distribute with respect to multiplication though, since that would mean a+bc=(a+b)(a+c) and ab+c=(a+c)(b+c). (Note it does hold in quotient rings: (a+I)(b+I)=ab+I).

Definition 6.  The signature of a structure is the sequence \sigma(X)=(n_k)_{k\geq 0} where n_k is the number of k-ary operations on X.  Let n_k be treated as an ordinal and f:n_k\to\{O:O\mbox{~is a~}k\mbox{-ary operation}\} be a bijection.  We say f(i)=O_i for an ordinal i\leq n_k and k-ary operation O_i.  If we wish to specify what arity of operation we are discussing, we may write O_{k_i} instead of O_i to clarify that we mean the ith k-ary operation on X.  We call the map f a k-ary operational ordering.  A sequence \{f_k\}_{k\geq 0} of k-ary operational orderings on a structure X is called an operational ordering.

If X and Y are structures with the same signature and each have operational orderings, then there is bijective map \Omega between each sets’ operations.  It simply sends the ith k-ary operation of X to the ith k-ary operation of Y.  To distinguish between operations in the structure and costructure, we write

\displaystyle \Omega(O_{k_i})=O^{k_i}.

Specifically if \{f_k\},\{g_k\} are the operational orderings of X and Y (i.e. f_k(i)=O_{k_i} and g_k(i)=O^{k_i}) we may write \Omega as a map

\displaystyle \Omega:\bigoplus_{k\geq 0} O_k\to\bigoplus_{k\geq 0} O^k

where O_k is the set of k-ary operations on X and O^k are those on Y with the map defined by

\displaystyle\Omega=\bigoplus_{k\geq 0}g_k\circ f_k^{-1}

since if O_\alpha is some arbitrary operation in X, we have

\displaystyle\Omega(O_\alpha)=\left(\bigoplus_{k\geq 0}g_k\circ f_k^{-1}\right)(O_\alpha)=g_k(i)=O^{k_i}

for some i\leq n_k.

Definition 7.  Let X and Y be structures with the same signature and possess operational orderings \{f_k\} and \{g_k\} respectively.  A map \varphi:X\to Y is an (f_k,g_k)homomorphism, or homomorphism for short, if

\displaystyle \varphi(O_{k_i}(x_1,...,x_k))=O^{k_i}(\varphi(x_1),...,\varphi(x_k))

for all k and all i\leq n_k.  \varphi is a strong homomorphism if

\displaystyle \varphi(O_k(x_1,...,x_k))=O^k(\varphi(x_1),...,\varphi(x_k))

where O_k is any k-ary operation on X and O^k is any k-ary operation on Y.  An (f_k,g_k)-homomorphism (or strong) is an (f_k,g_k)isomorphism (or strong isomorphism) if it is bijective.

The latter type of homomorphism implies the former (for any choice of operational orderings).  Most homomorphisms studied in algebra are the former type.

Example 8.  A ring homomorphism \varphi:R\to S is a homomorphism where \Omega(+_R)=+_S, \Omega(\cdot_R)=\cdot_S, \Omega(-_R)=-_S, and \Omega(0_R)=0_S.  A strong homomorphism between rings would thus additionally require \varphi(x+y)=\varphi(x)\varphi(y), \varphi(xy)=\varphi(x)+\varphi(y).  Hence

\varphi(x)=\varphi(0+x)=\varphi(0)\varphi(x)=0\cdot\varphi(x).

So 0 is an identity with respect to multiplication on the range.  Note that on the range we therefore have

x=x+x-x=0x+0x-x=(0+0)x-x=0x-x=x-x=0.

Thus in particular a strong isomorphism of rings implies R=S=0.  Such a triviality does not always occur in general with strong isomorphisms since an arbitrary structure may not have any identities.  But we do have the following.

Proposition 9.  If \varphi:X\to Y is a strong isomorphism between structures, then O_{k_i}=O_{k_j} and O^{k_i}=O^{k_j} for any k and choice of i,j\leq n_k.

Proof.  We have

\varphi(O_{k_i}(x_1,...,x_k))=O^{k_i}(\varphi(x_1),...,\varphi(x_k))

as well as

\varphi(O_{k_i}(x_1,...,x_k))=O^{k_j}(\varphi(x_1),...,\varphi(x_k)),

which implies O^{k_i}=O^{k_j} on the image of \varphi.  But since \varphi is bijective, we have O^{k_i}=O^{k_j} for all elements in Y.  For the first case we simply have O_{k_i}=O^{k_i}=O_{k_j} on X for all i,j\leq n_k.

Thus if \varphi:X\to Y is a strong isomorphism, then X and Y have a reduced signature where n_k\in\{0,1\} for all k.

Proposition 10.  Let \varphi:X\to Y be an isomorphism.

  1. If O_{k_i} associates with O_{m_j}, then O^{k_i} associates with O^{m_j}.
  2. If O_{k_i} is commutative, then O^{k_i} is commutative.
  3. If O_{k_i} is left (right) distributive with respect to O_{m_j}, then O^{k_i} is left (right) distributive with respect to O^{m_j}.
  4. If O_{0_i} is an identity with respect to O_{m_j}, then O^{0_i} is an identity with respect to O^{m_j}.  In particular if O_{(m-1)_k} is unital with respect to O_{m_j}, then O^{(m-1)_k} is unital with respect to O^{m_j}.

These are easy to show and just require a bit of index swapping and parentheses galore, so I leave them to the reader.

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