Abstract:
In this paper, we consider the $\mathbb{T}$-space structure of the relatively free Grassmann algebra $\mathbb{F}^{(3)}$ without unity over an infinite field of prime and zero characteristic. Our work is focused on $\mathbb{T}$-spaces $\mathbb{W}_n$ generated by all $n$-words. A question about connections between $\mathbb{W}_r$ and $\mathbb{W}_n$ for different natural numbers $r$ and $n$ is investigated. The proved theorem on these connections allows us to construct the diagrams of inclusions that, to some extent, clarify the structure of the algebra: the basic $\mathbb{T}$-spaces produce infinite strictly descending chains of inclusions in the algebra $\mathbb{F}^{(3)}$.
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