On the variety generated by all nilpotent lattice-ordered groups

This paper introduces the notion of a functorial torsion class (FTC): in a concrete category $\mathfrak{C}$ which has image factorization, one considers monocoreflective subcategories which are closed under formation of subobjects. Here the interest is in FTCs in the category of abelian lattice-ordered groups with designated strong order unit. The FTCs $\mathfrak{T}$ consisting of archimedean latticeordered groups are characterized: for each subgroup A of the rationals with the identity 1, either $\mathfrak{T} = \mathfrak{S}\left( A \right)$, the class of all lattice-ordered groups of functions on a set X which have finite range in A, or $$\mathfrak{T} = \mathbb{T}\left( A \right)$$, the class of all subgroups of A with 1. As for FTCs possessing non-archimedean groups, it is shown that if $\mathfrak{T}$ is an FTC containing a subgroup A of the reals with 1, of rank two or greater, then $\mathfrak{T}$ contains all ℓ-groups of the form $A\vec \times G$, for all abelian lattice-ordered groups G. Finally, the least FTC that contains a non-archimedean group is the class of all $\mathbb{Z}\vec \times G$, for all abelian lattice-ordered groups G.

Let π be a set of primes. We will call a group π-separated if it can be decomposed as a central product of a π-torsion group of bounded exponent and a π-radicable group. It is easy to see that an abelian π-separated group can in fact be decomposed as a direct product of bounded and π-radicable factors (Lemma 1.1 below). A poly-π-separated group is one which can be obtained from π-separated groups by forming a finite series of group extensions. We will show here:Theorem 1. Every poly-π-separated nilpotent group is π-separated.The need for such a result arises in model theory in the case that π is the set of all primes, in which case we refer simply to separated and poly-separated groups. In connection with the conjecture that ω-stable simple groups are algebraic, it is useful to have a structure theory for solvable ω-stable groups analogous to the theory available in the algebraic case over algebraically closed fields. In particular the structure of nilpotent ω-stable groups is of interest, for example in connection with the known result [Zi1], [Ne] that the derived subgroup of a connected solvable ω-stable group of finite Morley rank is nilpotent.As a model-theoretic application of Theorem 1 we obtain:Theorem 2. If G is an ω-stable nilpotent group then G may be decomposed as a central product B * D with B and D 0-definable subgroups, B torsion of bounded exponent, and D radicable. In particular, B and D are ω-stable. Furthermore, ω · rk(B ∩ D) ≤ rk D.Corollary. The ω-stable groups of finite Morley rank are exactly the central products B * D of ω-stable nilpotent groups of finite Morley rank with B torsion of bounded exponent, D radicable, and B ∩ D finite.In addition to the purely algebraic Theorem 1, these results depend on Macintyre's characterization [Mac] of the ω-stable abelian groups as exactly the separated ones.

MV-algebras and Riesz MV-algebras are categorically equivalent to abelian lattice-ordered groups with strong unit and, respectively, with Riesz spaces (vector-lattices) with strong unit. A standard construction in the literature of lattice-ordered groups is the vector-lattice hull of an archimedean latticeordered group. Following a similar approach, in this paper we define the Riesz hull of a semisimple MV-algebra.

Scalar extensions for algebraic structures of Łukasiewicz logic

Definitions and examples basic properties values, primes and polars Abelian and normal-valued lattice-ordered groups Archimediean function groups soluble right partially ordered groups and generalizations permutations applications completions varieties of lattice-ordered groups unsolved problems.

Introduced by Chang in the late fifties, MV-algebras stand to Łukasiewicz’s infinite-valued propositional logic as boolean algebras stand to the classical propositional calculus. As stated by Chang in his original paper, “MV is supposed to suggest many-valued logics … for want of a better name”. The name has stuck. After some decades of relative quiescency, MV-algebras are today intensely investigated. On the one hand, these algebras find applications in such diverse fields as error-correcting feedback codes and logic-based control theory. On the other hand, MV-algebras are interesting mathematical objects in their own right. The main aim of this paper is to show that the interaction between MV-algebras and lattice-ordered abelian groups — including the timehonored theory of magnitudes — has much to offer, not only to specialists in these two fields, but also to people interested in the fan-theoretic description of toric varieties, and in the K0-theory of AF C*-algebras.

In this paper, we study the capacity/entropy region of finite, directed, acyclic, multiple-sources, and multiple-sinks network by means of group theory and entropy vectors coming from groups. There is a one-to-one correspondence between the entropy vector of a collection of \(n\) random variables and a certain group-characterizable vector obtained from a finite group and \(n\) of its subgroups. We are looking at nilpotent group characterizable entropy vectors and show that they are all also abelian group characterizable, and hence they satisfy the Ingleton inequality. It is known that not all entropic vectors can be obtained from abelian groups, so our result implies that to get more exotic entropic vectors, one has to go at least to soluble groups or larger nilpotency classes. The result also implies that Ingleton inequality is satisfied by nilpotent groups of bounded class, depending on the order of the group.

In this chapter we shall prove Chang’s completeness theorem stating that if an equation holds in the unit real interval [0,1], then the equation holds in every MV-algebra. Thus, intuitively, the two element structure {0,1} stands to boolean algebras as the interval [0,1] stands to MV-algebras. Our proof is elementary, and makes use of tools (such as “good sequences”) that shall also find applications in a subsequent chapter to show the equivalence between MV-algebras and lattice-ordered abelian groups with strong unit.

On terraces for abelian groups

A group G is called semi-n-abelian, if for every g ∈ G there exists at least one a(g) ∈ G-which depends only on g-such that (gh)n = a-1(g)gnhna(g) for all h ∈ G; a group G is called n-abelian, if a(g) = e for all g ∈ G. According to Durbin the following holds for n-abelian groups: If G is n-abelian for at lesast 3 consecutive integers, then G in n-abelian for all integers and these groups are exactly the abelian groups. In this paper this problem is generalized to the semi-n-abelian case: If a finite group G is semi-n-abelian for at least 4 consecutive integers then G is semi-n-abelian for all integers and these groups are exactly the nilpotent groups, where the Sylow-2-subgroup is abelian, the Sylow-3-subgroup is any element of the Levi-variety ([[g, h], h] = e ∀ g, h ∈ G) and the Sylow-p-subgroup (p < 3) is of class <2. As a consequence we get a description of all finite (3-)groups, which are elements of the Levi-variety.

Herrlich, Salicrup, and Strecker [HSS] have shown that Kuratowski’s Theorem, namely, that a space X is compact if and only if for every space Y, the projection π2X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider settin than the category of topological spaces. The first author, in an earlier paperj [Fl] , applied this categorical interpretation of compactness in categories of R-modules, obtaining a theory of compactness for each torsion theory T. In the case of the category of abelian groups and a hereditary torsion theory T, a group G is T-compact provided G/TG is a T-injective. In this note, the notion of compact is extended to the categories of hypercentral groups, nilpotent groups, and of FC-groups; it is shown that if T π denotes the π-torsion subgroup functor for a set of primes π, then a group G is T π-compact provided G/T πG is π-complete, extending the abelian group result in a natural way.

The article continues the series of papers on the absolute of finitely generated groups. The absolute of a group with a fixed system of generators is defined as the set of ergodic Markov measures for which the system of cotransition probabilities is the same as for the simple (right) random walk generated by the uniform distribution on the generators. The absolute is a new boundary of a group, generated by random walks on the group. We divide the absolute into the Laplacian part and degenerate part and describe the connection between the absolute, homogeneous Markov processes, and the Laplace operator; prove that the Laplacian part is preserved under tak- ing some central extensions of groups; reduce the computation of the Laplacian part of the absolute of a nilpotent group to that of its abelinization; consider a number of fundamental examples (free groups, commutative groups, the discrete Heisenberg group). Keywords: absolute, Laplace operator, dynamic Cayley graph, nilpotent groups, Laplacian part of absolute.

Homotopy decompositions of polyhedra into products with the second factor [formula omitted

Hamiltonian cycles and paths in vertex-transitive graphs with abelian and nilpotent groups

We consider two algorithmic problems concerning sub-semigroups of Heisenberg groups and, more generally, two-step nilpotent groups. The first problem is Intersection Emptiness, which asks whether a finite number of given finitely generated semigroups have empty intersection. This problem was first studied by Markov in the 1940s. We show that Intersection Emptiness is PTIME decidable in the Heisenberg groups H_n(𝕂) over any algebraic number field 𝕂, as well as in direct products of Heisenberg groups. We also extend our decidability result to arbitrary finitely generated 2-step nilpotent groups. The second problem is Orbit Intersection, which asks whether the orbits of two matrices under multiplication by two semigroups intersect with each other. This problem was first studied by Babai et al. (1996), who showed its decidability within commutative matrix groups. We show that Orbit Intersection is decidable within the Heisenberg group H₃(ℚ).