Narrow Results

Given two numerical semigroups S and S', the distance between S and S' is the cardinality of S\S' plus the cardinality of S'\S. In this paper we study those numerical semigroups S for which there is a symmetric numerical semigroup whose distance to S is one.

In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups $G$ are fully characterized in the class of all groups by the set $tp(G)$ of types realized in $G$. Furthermore, it turns out that these groups $G$ are fully characterized already by some particular rather restricted fragments of the types from $tp(G)$. In particular, every finitely generated nilpotent group is completely defined by its ${\exists }^{+}$-types, while a finitely generated rigid group is completely defined by its $\forall$-types, and a finitely generated metabelian or polycyclic group is completely defined by its $\forall \exists$-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.

We give two algorithmic procedures to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number and type, and the whole set of almost symmetric numerical semigroups with fixed Frobenius number. Our algorithms allow to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number with similar or even higher efficiency that the known ones. They have been implemented in the GAP [The GAP Group, GAP — Groups, Algorithms and Programming, Version 4.8.6; 2016, https://www.gap-system.org] package NumericalSgps [M. Delgado and P. A. García-Sánchez and J. Morais, “numericalsgps”: A GAP package on numerical semigroups, https://github.com/gap-packages/numericalsgps].

Let L_S denote the language of (right) S-acts over a monoid S and let Σ_S be a set of sentences in L_S which axiomatises S-acts. A general result of model theory says that Σ_S has a model companion, denoted by T_S, precisely when the class of existentially closed S-acts is axiomatisable and in this case, T_S axiomatises . It is known that T_S exists if and only if S is right coherent. Moreover, by a result of Ivanov, T_S has the model-theoretic property of being stable.

In the study of stable first order theories, superstable and totally transcendental theories are of particular interest. These concepts depend upon the notion of type: we describe types over T_S algebraically, thus reducing our examination of T_S to consideration of the lattice of right congruences of S. We indicate how to use our result to confirm that T_S is stable and to prove another result of Ivanov, namely that T_S is superstable if and only if S satisfies the maximal condition for right ideals. The situation for total transcendence is more complicated but again we can use our description of types to ascertain for which right coherent monoids S we have that T_S is totally transcendental and is such that the U-rank of any type coincides with its Morley rank.