DECIDABILITY OF FIRST-ORDER THEORIES FOR GROUPS AND MONOIDS OF INTEGRAL MATRICES

Let G be a semilinearly ordered group with a positive cone P. Denote by n(G) the greatest convex directed noʹrrnal subgroup of G. by o(G) the greatest convex right-ordered subgroup of G. and by r(G) a set of all dements x of G such that x and x ^ are comparable with any element of P+ (the collection of all group elements comparable with an identity element). Previously, it was proved that r(G) is a convex right-ordered subgroup of G. and n(G) < r(G) < o(G). Here. we establish a new property of r(G), and show that the inequalities in the given system of inclusions are, generally, strict.