Undecidability of the elementary theory of the semilattice of GLP-words

The Lindenbaum algebra of Peano PA can be enriched by the $n$-consistency operators which assign, to a given formula, the statement that the formula is compatible with the theory PA extended by the set of all true $\Pi_n$-sentences. In the Lindenbaum algebra of PA, a lower semilattice is generated from $\mathbf{1}$ by the $n$-consistency operators. We prove the undecidability of the elementary theory of this semilattice and the decidability of the elementary theory of the subsemilattice (of this semilattice) generated by the $0$-consistency and $1$-consistency operators only.
Bibliography: 16 titles.