Basic 𝕋-Spaces in the Relatively Free Grassmann Algebra Without Unity

In this paper, we consider the 𝕋-space structure of the relatively free Grassmann algebra 𝔽(3) without unity over an infinite field of prime and zero characteristic. Our work is focused on 𝕋-spaces 𝕎n generated by all n-words. A question about connections between 𝕎r and 𝕎n 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 𝕋-spaces produce infinite strictly descending chains of inclusions in the algebra 𝔽(3).