Subvariety intervals and limit varieties of zero extensions of mirror-word semirings

Abstract

For the mirror word $\mathbf{m}_n=a_1\cdots a_na_n\cdots a_1$, let $\mathcal{V}_n=\mathrm{V}(S(\mathbf{m}_n))$ and $\mathcal{W}_n=\mathrm{V}(S(\mathbf{m}_n)^0)$. We show that $[\mathcal{W}_n,\mathcal{W}_{n+1}]$ ($n\geq 1$) and $[\mathcal{V}_n,\mathcal{W}_n]$ ($n\geq 2$) have cardinality $2^{\aleph_0}$. Every variety in $[\mathcal{W}_2,\mathcal{W}_\infty]$ has infinite axiomatic rank. The unique limit subvariety of $\mathcal{W}_1$ is $\mathrm{V}(SR_6)$. For $n\geq 2$, we construct a distinct directed-path limit variety contained in $\mathcal{W}_n$ but not in $\mathcal{V}_n$.