论文标题
单粒和环中的分解
Factorization in monoids and rings
论文作者
论文摘要
令$ h^\ times $为多个书面单体$ h $的单位组。我们说，如果$ xyz \ ne y $为$ x，y，z \ in H $，则$ x \ n n $ a $ x \ notin h^\ times $或$ z \ notin h^\ times $;如果$ yx \ ne x \ ne xy $ for ahl $ x，y \ in h $，则$ y \ y \ notin h^\ times $; F.G.U.如果有一个有限的集合$ a \ subseteq h $，以便每个非单元$ h $都是$ u，uav $的元素的有限产物，其中$ u，v \ in h^\ times $ in h^\ times $和$ a \ in a $; L.F.G.U.如果对于H $中的每个$ x \，则最小的分隔夹子模式包含$ x $的$ h $是f.g.u;并且原子如果每个非单元都可以作为原子的有限产物写入，那么原子是一种非单位，不会因两个非单元的产物而构成。 我们概括为L.F.G.U.或无环L.F.G.U. MONOIDS到目前为止的一些结果仅是单位现成的L.F.G.U.交换性单体（取消单体是单位临界的，并且仅在无环时才是单位c的，是单位c的）。特别是，我们证明了以下内容： $ \ bullet $如果$ h $是原子L.F.G.U.单肌，然后每个非单元只有有限的多个因素化（分解为原子），这些因素是“最小”和“成对的非等效性”（就原子“ Atoms”上的“字母”上的自由型单体上的某些自然定义关系）。 $ \ bullet $如果$ h $是一个环l.f.g.u.单型，然后是原子；而且，如果我们还假设$ h $是可交换的，那么每个元素都只有许多“成对非等效”因素化的有限程度。
Let $H^\times$ be the group of units of a multiplicatively written monoid $H$. We say $H$ is acyclic if $xyz \ne y$ for all $x, y, z \in H$ with $x \notin H^\times$ or $z \notin H^\times$; unit-cancellative if $yx \ne x \ne xy$ for all $x, y \in H$ with $y \notin H^\times$; f.g.u. if there is a finite set $A \subseteq H$ such that every non-unit of $H$ is a finite product of elements of the form $uav$ with $u, v \in H^\times$ and $a \in A$; l.f.g.u. if, for each $x \in H$, the smallest divisor-closed submonoid of $H$ containing $x$ is f.g.u; and atomic if every non-unit can be written as a finite product of atoms, where an atom is a non-unit that does not factor into a product of two non-units. We generalize to l.f.g.u. or acyclic l.f.g.u. monoids a few results so far only known for unit-cancellative l.f.g.u. commutative monoids (cancellative monoids are unit-cancellative, and a commutative monoid is unit-cancellative if and only if it is acyclic). In particular, we prove the following: $\bullet$ If $H$ is an atomic l.f.g.u. monoid, then every non-unit has only finitely many factorizations (into atoms) that are "minimal" and "pairwise non-equivalent" (with respect to some naturally defined relations on the free monoid over the "alphabet" of atoms). $\bullet$ If $H$ is an acyclic l.f.g.u. monoid, then it is atomic; and moreover, each element has only finitely many "pairwise non-equivalent" factorizations if we additionally assume $H$ to be commutative.
