Isometric representations of totally ordered semigroups

In this chapter, we discuss several special types of modules that will make frequent appearances later. These include semisimple modules, artinian modules, and torsionfree modules, which have prototypes in vector spaces, finite abelian groups, and torsionfree abelian groups, respectively. That special types of modules have useful roles in the study of arbitrary modules may be seen already in the case of abelian groups (i.e., ℤ-modules). In studying an arbitrary abelian group, an almost reflexive first step is to look at its torsion part (i.e., the torsion subgroup) and its torsionfree part (i.e., the factor group modulo the torsion subgroup), since entirely different techniques are available (and needed) for dealing with torsion groups and torsionfree groups. On the torsionfree side, vector spaces make an appearance due to the facts that the torsionfree divisible abelian groups are exactly the vector spaces over ℚ and that every torsionfree abelian group can be embedded in a divisible one. On the torsion side, many questions can be reduced to the case of finite abelian groups, since every torsion abelian group is a directed union of finite subgroups. In studying torsion abelian groups, one also reduces to the case of p -groups for various primes p . Vector spaces make another appearance here, since in an abelian p -group the set of elements of order p (together with 0) forms a vector space over ℤ/ p ℤ.

A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group FA has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that FA is faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and FA is a free A-groupp of Lyndon' type, then FA has a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then FA has a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, FA has a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if FQ has a faithful Magnus representation in Q[[Y]].

We present a calculus for first-order theorem proving in the presence of the axioms of totally ordered divisible abelian groups. The calculus extends previous superposition or chaining calculi for divisible torsion-free abelian groups and dense total orderings without endpoints. As its predecessors, it is refutationally complete and requires neither explicit inferences with the theory axioms nor variable overlaps. It offers thus an efficient way of treating equalities and inequalities between additive terms over, e. g., the rational numbers within a first-order theorem prover.

the residue class fields of these rings are always totally ordered and, over noncompact spaces, may be non-Archimedean. Kohls [22] showed that the residue class domains of these rings are also totally ordered. In some recent researches [1 ; 2] the author has exploited a fact, known to Baer [3] and Krull [23]: that a totally ordered non-Archimedean field has a natural valuation. It is also clear that a totally ordered non-Archimedean integral domain has a natural valuation sorts. These valuations, although derived from the total order on these domains, can be defined for integral domains that are not ordered; thus an attempt at an application of this valuation theory to the residue class domains of the ring of continuous complex valued functions is natural. The solution of certain integral algebraic equations in these domains will be treated in ?1, together with the proof that these domains are integrally closed in their quotient fields (Theorem 1.8). In ?2 a place-like mapping will be defined on these domains, which has many of the algebraic (Theorem 2.2) as well as the topological (Theorem 2.5) properties of a place. In ?3 an abstract discussion will be given ofAbelian groups and integral domains with valuation, followed by an application of these ideas to the natural valuation on the residue class domains of these real and complex ftunction algebras (Theorem 3.6). It will be shown that these valuations are given by the place-like mappings defined in ?2 and can be used to analyze the structure of the prime ideals in these domains in terms of the order and algebraic properties of the value semigroup. In ?4 order properties akin to that of being an inZ-set will be studied and will be shown to be inherited, in modified form, by the value set of an Abelian group with valuation from the group. Application will then be made to the domains in question (Theorem 4.4).

The study of Hopficity in Abelian groups has been largely motivated by the fundamental results of Baumslag, who proved that torsion groups are always Hopfian regardless of their cardinality, but left several questions open concerning torsion-free groups. Later, Corner addressed some of these questions by providing counterexamples showing that a direct sum of two Hopfian groups can be non-Hopfian, and that a group with an automorphism group of order two does not guarantee Hopficity. These results highlighted the need for new constructions to explore Hopficity in torsion-free Abelian groups. Our work introduces a new approach based on divisibility techniques, as it contributes to the understanding of free-torsion groups with respect to the Hopficity property, providing new insights into their structural properties and implications within group theory. Our analysis also demonstrates how divisibility properties, as well as the introduction of totally invariant subgroups and homomorphisms, can be used to establish Hopficity in specific Abelian groups, particularly those that are free-torsion. In order to reach all of this, we start by taking a group defined as an infinite direct sum of cyclic groups; then we construct a specific subgroup generated by two particular families of elements; and finally we show that this group is Hopfian through results from the theory of divisible subgroups.

In [Todor Tsankov, The additive group of the rationals does not have an automatic presentation, May 2009, arXiv:0905.1505v1], it was shown that the group of rational numbers is not FA-presentable, i.e. it does not admit a presentation by a finite automaton. More generally, any torsion-free abelian group that is divisible by infinitely many primes is not of this kind. In this article we extend the result from [13] and prove that any torsion-free FA-presentable abelian group G is an extension of a finite rank free group by a finite direct sum of Prüfer groups ℤ(p∞).

An associative ring R is called a unique addition ring (UA-ring) if its multiplicative semigroup (R, · ) can be equipped with a unique binary operation+ transforming the triple (R, ·, +) to a ring. An R-module A is said to be an End-UA-module if the endomorphism ring EndR(A) of A is a UA-ring. In the paper, the torsion-free End-UA-modules over commutative Dedekind domains are studied. In some classes of Abelian torsion-free groups, the Abelian groups having UA-endomorphism rings are found.

For every Tikhonov space X the free abelian topological group A(X) and the free locally convex vector space L(X) admit a topologically faithful unitary representation. For compact spaces X this is due to Jorge Galindo.

Given a finite metacyclic group G G , a central extension F F having the projective lifting property over all fields is constructed. This extension and its group rings are used to investigate the faithful irreducible projective representations of G G and the fields over which they can be realized. A full description of the finite metacyclic groups having central simple twisted group rings over fields is given.

On a Noether Normalization for a mod-2 Cohomology Ring

We extend Euler’s totient function (from arithmetic) to any irreducible subfactor planar algebra, using the Möbius function of its biprojection lattice, as Hall did for the finite groups. We prove that if it is nonzero, then there is a minimal 2 2 -box projection generating the identity biprojection. We explain a relation with a problem of K.S. Brown. As an application, we define the dual Euler totient of a finite group and we show that if it is nonzero, then the group admits a faithful irreducible complex representation. We also get an analogous result at depth 2 2 , involving the central biprojection lattice.

Let (S, *) be a ring with involution *. The involution is positive definite if, for all finite subsets {r,-} of S, £ w * = 0 implies all the r{ are zero. Then the set of self-adjoint elements of S, denoted S, possesses a natural partial ordering, with positive cone consisting of elements of the form Z l w * ; with this ordering, S is a directed partially ordered abelian group. Let Sb denote the set of bounded elements, that is, the set of elements s such that ss* is less than an integral multiple of 1 in this ordering. Then Sb is a *-subalgebra of S whenever S is an algebra over the rationals. We will be studying the objects Sb9 S , and (Sb) , from the point of view of their ordered structures. For instance, suppose S is a field, and * is the identity. Then S is formally real, and Sb must be Prufer domain, all of whose residue fields are themselves formally real (and in fact, are embeddable in the reals). Viewing Sb as a partially ordered abelian group with order unit 1 (indeed, Sb is the convex subgroup of S generated by 1), Sb has the Riesz decomposition property, and its normalized extremal states are precisely the ring homomorphisms into the reals. There is a natural mapping from the collection of total orderings of S to the set of extremal states of Sb9 and this in turn maps to Spec Sb (the prime ideal space of Sb); when S is even a real algebra much more can be said. If either S is a field and * is not the identity, or S is a quaternionic division algebra with the natural involution, essentially the same properties hold, with the appropriate modifications. A useful tool here is an involutory version of the Artin Schreier Theorem, about the existence of sufficiently many total orderings finer than the natural ordering. Studies are made of several specific bounded subrings. For instance, if S is the rational function field in one variable over the reals, then Sb is a Dedekind domain with class group of order 2, with spectrum the unit circle (in the point-open topology), and all of its maximal ideals are not principal. Expanding the scope of S somewhat, we next allow S to be a division

In the work of Herden and Strüngmann (In Models, modules and Abelian groups (2008) 169–186 de Gruyter), an embedding problem for torsion-free Abelian groups was considered. It was shown for a large class of such groups, including the class of all bounded extensions of completely decomposable groups, that any member of the class can be purely embedded into some completely decomposable group. Moreover, an algorithm was given that determines explicitly the pure embedding and the completely decomposable overgroup. We continue the approach from the work of Herden and Strüngmann (In Models, modules and Abelian groups (2008) 169–186 de Gruyter) improving the algorithm and extending the main theorem to a broader class of torsion-free Abelian groups including some Hawaiian groups from the article of Mader and Strüngmann (J. Algebra 229 (2000) 205–233) and thus complementing the main result from the article of Strüngmann (Proc. Amer. Math. Soc. 137 (2009) 3657–3668). A byproduct and starting point for this generalization will be a discussion of the following group class problem: Which groups $G$ have the property that for any cardinal $\kappa$ any subgroup $U$ of the direct sum $G^{(\kappa)}$ is the kernel of some endomorphism of $G^{(\kappa)}$?

Let T be an abelian group and λ an uncountable regular cardinal. The question of whether there is a λ-universal group U among all torsion-free abelian groups G of cardinality less than or equal to λ satisfying Exta(G,T) = 0 is considered. U is said to be λ-universal for T if, whenever a torsion-free abelian group G of cardinality at most λ satisfies Ext(G,T)=0, there is an embedding of G into U. For large classes of abelian groups T and cardinals λ, it is shown that the answer is consistently no, that is to say, there is a model of ZFC in which, for pairs T and λ, there is no universal group. In particular, for T torsion, this solves a problem by Kulikov.

We shall characterize the complete first order theories of torsion free abelian groups having algebraically prime models, elementarily prime models, minimal models, etc., using the well known invariants from [9], [3]. This study was suggested by the questions raised in [2]. Group here will always mean abelian group. For reference on groups see Fuchs [5], [6], for model theory Chang-Keisler [1]. All theories considered are theories in the first order language of group theory. A group G is minimal (strictly minimal) if H < G implies H = G (H-= G), and elementarily prime (algebraically prime) if G is elementarily (algebraically) embeddable in every H G. p, q always denote prime numbers, P the set of all primes. The p-rank of a group G, p-rk(G), is the dimension of G/pG as a vector space over Z/pZ. We define tf(p, G): = min {p-rk(G), ~o}. By the results of [9], [3], two nontrivial torsion free abelian groups G and H are elementarily equivalent iff for all primes p tf(p, G) = tf(p, H). This gives a natural 1-1 correspondence between the complete theories of nontrivial torsion free abelian groups and the functions /3 : P --~ ~o + 1. /3 will always stand for such a function, and the corresponding complete theory will be denoted by T 0. S o is the set of primes p for which/3(p) ~ 0. For more definitions see section 3. Let 13 be a function from P to o) + 1. We shall study the following questions: Does T o have an elementarily (algebraically) prime model? Does T o have a minimal (strictly minimal) model, and if this is the case, how many such models does T o have? Do the minimal and strictly minimal models of T o coincide? These questions have been studied for the special case of the theory of the