For finitely axiomatized sequential theories F and reflexive theories R, we give a characterization of the relation ’F interprets R’ in terms of provability of restricted consistency statements on cuts. This characterization is used in a proof that the set of ${\prod}_{1}$ (as well as ${\sum}_{1}$) sentences π such that GB interprets ZF+π is ${\Sigma}_{3}^{0}$-complete.

We show that the class of groups which
have monoid presentations by means of finite special
[λ]-confluent string-rewriting systems strictly contains the class of plain groups
(the groups which are free products of a finitely generated free
group and finitely many finite groups),
and that any group
which has an infinite cyclic central subgroup
can be presented by such a string-rewriting system if and only if it is the
direct product of an infinite cyclic group and a finite cyclic group.

We show that the class of groups which have monoid presentations by means of finite special $\left[\lambda \right]$-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.

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