Partially 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.

Scalar extensions for algebraic structures of Łukasiewicz logic

In 1974, J. Martinez introduced the variety ${\mathcal W}$ of weakly Abelian lattice-ordered groups; it is defined by the identity \[ x^{-1}(y\vee 1)x\vee (y\vee 1)^2=(y\vee 1)^2.\]

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.

Topological characterization of filter topological MV-algebras

The purpose of this study is to derive data that can implement a specific design and appearance through experiments to create a virtual sample as close to the original as possible. A literature review was conducted on the 3D Virtual Fashion CAD program and CLO, which is the most widely used program in fashion companies. We then conducted a survey of practitioners on the use of 3D Virtual Fashion CAD programs. This was done to demonstrate the fact that thin fabrics are implemented differently from the actual, and that hoods, sleeves, and flaps are applied with the most variation in regard to structural aspects. The subject of the experimental study, anorak jumpers with a hood of thin, rustling, woven fabric most commonly made of softshell fabric was created in two designs. The experiment identified the material of the sample which was made of three fabrics and created a virtual sample property value based on the designated basic property value. This was done to determine what property value was most similar to the sample after consultation with an expert. After applying it to the virtual sample as the final property value, an expert survey was conducted to evaluate its appearance. The results revealed that the basic property value input to CLO is light and thin and that it is not possible to accurately apply a material with a rustling appearance. The difference between the actual sample and the one shown in the virtual sample was quite noticeable with the virtual sample demonstrating problems including the collapse of the shape, amount of wrinkles being tilted downward, and the flap lifting on the body plate. This problem was found by adjusting stretch, bending strength, buckling, and buckling strength, as well as using rules for controlling physical property values according to fabric and design.

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.

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.

Free lattice-ordered Abelian groups

Characteristic values of the mechanical properties of radiata pine plywood and the derivation of basic values of the layers for a calculation method

We provide a generalization of Mundici’s equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation x mapsto -x. The primitive operations are +, vee , wedge , 0, 1, -1. A prime example of these structures is mathbb {R}, with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation x mapsto lnot x. The primitive operations are oplus , odot , vee , wedge , 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra [0, 1]subseteq mathbb {R}. We obtain the original Mundici’s equivalence as a corollary of our main result.

In the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups, equipped with a translation invariant metric, we define the boundedness, the norm, the modulus of injectivity and the spectral radius of endomorphisms. Beyond the investigation of their properties, our first main goal is an extension of the celebrated Rådström cancellation theorem. Another result generalizes the Neumann invertibility theorem. Next we define the convexity of sets with respect to a family of endomorphisms, and we describe the set-theoretical and algebraic structure of the class of such sets. Given a subset, we also consider the family of endomorphisms that make this subset convex, and we establish the basic properties of this family. Our first main result establishes conditions which imply midpoint convexity. The next main result, using our extension of the Rådström cancellation theorem, presents further structural properties of the family of endomorphisms that make a subset convex.

This paper is focused on the functional renormalization group applied to the ${T}_{5}^{6}$ tensor model on the Abelian group $U(1)$ with closure constraint. For the first time, we derive the flow equations for the couplings and mass parameters in a suitable truncation around the marginal interactions with respect to the perturbative power counting. For the second time, we study the behavior around the Gaussian fixed point, and show that the theory is nonasymptotically free. Finally, we discuss the UV completion of the theory. We show the existence of several nontrivial fixed points, study the behavior of the renormalization group flow around them, and point out evidence in favor of an asymptotically safe theory.

Every semifield of non-zero characteristic is either a field of prime characteristic or a semifield of characteristic 1. Semifields of characteristic 1 are equivalent to abelian lattice-ordered groups. It is proved that such a semifield A is algebraically closed if and only if the pure equations xn = a and certain quadratic equations are solvable in A. Using a sheaf representation for z-projectable abelian l-groups on the co-Zariski space of minimal primes, a sheaf-theoretic characterization of algebraic closedness in characteristic 1 is obtained. Concerning the solvability of quadratic equations, the criterion consists in a topological condition for the base space. The results are built upon an analysis of rational functions in characteristic 1. While polynomials satisfy the “fundamental theorem of algebra”, the multiplicative structure of rational functions is determined by means of “divisors”, extracted from the additive structure of A modulo parallelogram identities.

Physical and optical properties of terbium doped lithium aluminum borate glass