论文标题
从$ a $到$ b $到$ z $
From $A$ to $B$ to $Z$
论文作者
论文摘要
Brandt Semigroup $ {\ bf B} _2 $生成的品种可以在由单个身份$ x^2y^2y^2 \ y y^2x^2 $的Semigroup $ {\ bf a} _2 $生成的品种中定义。埃德蒙·李（Edmond Lee）询问Monoids $ {\ bf b} _2^1 $和$ {\ bf a} _2^1 $是否相同。我们采用了超图的同态理论的编码，以表明实际上有一个$ {\ bf a} _2^1 $的连续性，满足$ x^2y^2y^2 \ y y^2x^2 $，并且包含$ X^2y^2 \ of $ x^2y^2 $，并且包含$ {\ bf bf bf b} _2 _2^_2^1 $。另一个结果是，通过任何有限的身份系统，都不能在$ {\ bf a} _2^1 $的品种中定义$ {\ bf b} _2^1 $的品种。继续向下，我们转向$ {\ bf b} _2^1 $的子视界。 We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity $x^2y\approx yx^2$ and containing the monoid $M({\bf z}_\infty)$, where ${\bf z}_\infty$ denotes the infinite limit of the Zimin words ${\bf z} _0 = x_0 $，$ {\ bf z} _ {n+1} = {\ bf z} _n x_ x_ {n+1} {\ bf z} _n $。
The variety generated by the Brandt semigroup ${\bf B}_2$ can be defined within the variety generated by the semigroup ${\bf A}_2$ by the single identity $x^2y^2\approx y^2x^2$. Edmond Lee asked whether or not the same is true for the monoids ${\bf B}_2^1$ and ${\bf A}_2^1$. We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of ${\bf A}_2^1$ that satisfy $x^2y^2\approx y^2x^2$ and contain ${\bf B}_2^1$. A further consequence is that the variety of ${\bf B}_2^1$ cannot be defined within the variety of ${\bf A}_2^1$ by any finite system of identities. Continuing downward, we then turn to subvarieties of ${\bf B}_2^1$. We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity $x^2y\approx yx^2$ and containing the monoid $M({\bf z}_\infty)$, where ${\bf z}_\infty$ denotes the infinite limit of the Zimin words ${\bf z}_0=x_0$, ${\bf z}_{n+1}={\bf z}_n x_{n+1}{\bf z}_n$.
