The k-abelian equivalence of words, counting the numbers of occurrences of factors of length at most k, has been analyzed in recent years from several different directions. We continue this analysis. The k-abelian equivalence classes are known to constitute a monoid. Hence, equations over these monoids are well defined. We show that these monoids satisfy a compactness property: each system of equations with a finite number of unknowns is equivalent to some of its finite subsystems.

We give two proofs for this compactness result. One is based the fact that the monoid can be embedded into the (multiplicative) monoid of matrices, and the other directly on linear algebra. The former method allows the application of Hilbert's basis theorem. The latter one, in turn, allows to conclude an upper bound for the size of the finite subsystem.

该ķ的话-abelian等价，最多计数长度的因素的出现次数ķ，在最近几年进行了分析，从几个不同的方向。我们继续进行此分析。已知k- abelian等价类构成一个单面体。因此，关于这些类半体的方程式得到了很好的定义。我们证明了这些单线态满足紧凑性：每个具有有限数量的未知数的方程组等效于其某些有限子系统。

对于这种紧凑性结果，我们给出了两个证明。一个基于一个事实，即mono半群可以嵌入到矩阵的（乘法）mono半群中，而另一个则直接基于线性代数。前一种方法允许应用希尔伯特基本定理。反过来，后者允许得出有限子系统大小的上限。