Author: Eiko

Tags: algebra, associative algebra, diamond lemma

Time: 2024-11-04 16:50:31 - 2024-11-04 16:52:53 (UTC)

We discuss in more detail about the Diamond Lemma in rings, following The Diamond Lemma for Ring Theory by Bergman.

Bases For Associative $k$-Algebras

Let $X$ be a set, $⟨X⟩$ the free semigroup, $k⟨X⟩$ the free associative algebra.

$S$ be the set $\{(w_i,f_i)\}$ of pairs of leading terms $w_i$ and relations $f_i$, for $A,B$ in $⟨X⟩$, the map $r_{A{\sigma }_{i}B}$ is the $k$-linear reduction mapping of $\mathrm{End}(k⟨X⟩)$ that maps $A{w}_{i}B\mapsto A{f}_{i}B$, and fixes all other basis elements.

• An element $a\in k⟨X⟩$ is reduction-finite if for every infinite sequence of reductions, there is a point from which the rest of the reductions act trivially.

• reduction-unique if it is reduction finite and all final sequences of reductions give the same result which we denote by $r_S(a)$.

Reduction-Unique Elements Form A Submodule

• The set of reduction-finite elements form a submodule. This is obvious since the reduction is linear and has to stop on each component.

• An element $a$ is reduction-unique, iff starting from any finite sequence of reductions $r$, $ra$ is reduction-unique.

• Reduction-finite means there exists a final sequence. The final result might not be unique though.

• The set of reduction-unique elements is a $k$-submodule of $k⟨X⟩$, and $r_S$ can be defined on this submodule, into the set of irreducible elements (elements on which all reductions act trivially).

  $$r_S:\mathrm{RedUni}(k⟨X⟩)\to \mathrm{Irred}(k⟨X⟩)$$

  Because starting from any sequence we can choose a final sequence $r$ of $a+b$, from which we can find $r^{\prime }r(a)$ final and $r^{″}{r}^{\prime }r(b)$ final, so $r^{″}{r}^{\prime }r(a+b)=r^{″}{r}^{\prime }r(a)+r^{″}{r}^{\prime }r(b)$.

• Let $a,b,c\in k⟨X⟩$ and assume that for all monomials $A,B,C$ inside $a,b,c$ respectively, the product $ABC$ is reduction-unique, then $abc$ is reduction-unique and this is true for all $ar(b)c$, for which they reach the same results $r_S(ar(b)c)=r_S(abc)$.

Overlap Ambiguities

For reductions $\alpha ,\beta$ and words $A,B,C$ that $w_{\alpha }=AB$, $w_{\beta }=BC$, we say that this is an overlap ambiguity. It is resolvable if exists sequence $r$ such that they reduce $ABC$ to the same result,

$$r\alpha (ABC)=r(f_{\alpha }C)=r\beta (ABC)=r(A{f}_{\beta }).$$

Inclusion Ambiguities

Similarly if you want to reduce $ABC$ where you can also reduce $B$, we have an inclusion ambiguity. It is resolvable if there exists a sequence $r$ such that $r\alpha (ABC)=r(A{f}_{\alpha }C)=r\beta (ABC)=r(g_{\beta })$.

Semigroup Partial Order

A semigroup partial order $\le$ is a partial order that is compatible with multiplications.

We say it is compatible with reduction system $S=\{(w_i,f_i)\}$ if for all $\sigma \in S$, $\mathrm{Terms}(f_{\sigma })<w_{\sigma }$. i.e. reduction always strictly reduces the leading term, or equivalently say that $w_{\sigma }$ is the leading term of $f_{\sigma }$.

Let $I$ be the ideal of $k⟨X⟩$ generated by relations $w_i-f_i$, i.e. a $k$-module generated by $A(w_i-f_i)B$. As an approximation we can form the submodule (not ideal) $I_A$ denoting the submodule of $k⟨X⟩$ spanned by elements of the form $B(w_i-f_i)C$ such that $B{w}_{i}C<A$. These are the relations whose leading term $<A$.

Diamond Lemma

The Diamond Lemma states that given a semigroup partial order (partial orders that are compatible with multiplications) $\le$ and compatible with the reduction system $S$,

The following are equivalent:

• All ambiguities of $S$ are resolvable.

• All ambiguities of $S$ are resolvable relative to $\le$.

• All elements of $k⟨X⟩$ are reduction-unique under $S$.

• A set of representatives in $k⟨X⟩$ for the elements of $R=k⟨X⟩/I$ is given by $\mathrm{Irred}(k⟨X⟩)$, spanned by $S$-irreducible elements.