The variety generated by the Brandt semigroup \(B_2\) can be defined within the variety generated by the semigroup \(A_2\) by the single identity \(x^2y^2\approx y^2x^2\). Edmond Lee asked whether or not the same is true for the monoids \(B_2^1\) and \(A_2^1\). We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of \(A_2^1\) that satisfy \(x^2y^2\approx y^2x^2\) and contain \(B_2^1\). A further consequence is that the variety of \(B_2^1\) cannot be defined within the variety of \(A_2^1\) by any finite system of identities. Continuing downward, we then turn to subvarieties of \(B_2^1\). 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 \(M(\mathbf {z}_\infty )\), where \(\mathbf {z}_\infty \) denotes the infinite limit of the Zimin words \(\mathbf {z}_0=x_0\), \(\mathbf {z}_{n+1}=\mathbf {z}_n x_{n+1}\mathbf {z}_n\).

布兰特半群\（B_2 \）生成的变体可以在半群\（A_2 \）生成的变体中通过单个标识\（x ^ 2y ^ 2 \ approx y ^ 2x ^ 2 \）进行定义。爱德蒙·李（Edmond Lee）询问对等分体\（B_2 ^ 1 \）和\（A_2 ^ 1 \）是否成立。我们采用超图的同态理论的编码表明，有实际上的不同子簇的连续\（A_2 ^ 1 \）满足\（X ^ 2Y ^ 2 \约Y 1 2×^ 2 \）和含有\（B_2 ^ 1 \）。进一步的结果是\（B_2 ^ 1 \）的种类不能在\（A_2 ^ 1 \）的种类内定义通过任何有限的身份系统。继续向下，然后我们转到\（B_2 ^ 1 \）的子变量。我们通过显示存在一个连续的不同子变量集来满足李的另一个问题，这些子变量都满足更强的同一性\（x ^ 2y \ approx yx ^ 2 \）并包含单等式\（M（\ mathbf {z} _ \ infty）\），其中\（\ mathbf {z} _ \ infty \）表示Zimin单词\（\ mathbf {z} _0 = x_0 \）的无穷大，\（\ mathbf {z} _ {n +1} = \ mathbf {z} _n x_ {n + 1} \ mathbf {z} _n \）。