Lattice-ordered Groups Research Articles

MV-frames

Abstract Each complete MV-algebra is naturally equipped with a structure of a frame. For this reason, we shall call such MV-algebras MV-frames and investigate some of their main frame properties. We completely characterize algebraic MV-frames as well as regular MV-frames. In addition, we consider nuclei on MV-frames in general and on MV-frames of ideals of Łukasiewicz rings in particular. Finally, we use the Chang–Mundici functor to explore the frame structures of complete unital lattice-ordered groups. 2010 Mathematics Subject Classification. 06D35, 06E15, 06D50

Hereditarily projectable Archimedean lattice-ordered groups with unit

Let W be the category of Archimedean lattice-ordered groups with distinguished weak unit and unit-preserving ℓ-group homomorphisms. We denote by P, wP, Loc, HA, respectively, the collections of projectable, weakly projectable, local, and hyperarchimedean W-objects. For any C ⊆ W, we say that a W-object H is hereditarily C, denoted H ∈ hC, if G ∈ C whenever G is a W-subobject of H. Our focus is hP. Taking off from the known items P = wP ∩ Loc, h HA = HA ⊆ P, and C (X) ∈ P iff X is basically disconnected (BD), we show For C (X), equivalent are: hP; hLoc and X BD; X finite. If the Yosida space Y G is BD, then (1) If G is hLoc, then G is bounded; (2) If G is hP, then G is HA. hP = hwP, and every G ∈ hP with G ∉ HA has Yosida space Y G = αN, the one-point compactification of N. There are various G ∈ hP with Y G = αN, including: G bounded not HA; G not bounded with bounded part HA; G not bounded with bounded part not HA.

Separable MV-algebras and lattice-ordered groups

The theory of extensive categories determines in particular the notion of separable MV-algebra (equivalently, of separable unital lattice-ordered Abelian group). We establish the following structure theorem: An MV-algebra is separable if, and only if, it is a finite product of algebras of rational numbers—i.e., of subalgebras of the MV-algebra [0,1]∩Q. Beyond its intrinsic algebraic interest, this research is motivated by the long-term programme of developing the algebraic geometry of the opposite of the category of MV-algebras, in analogy with the classical case of commutative K-algebras over a field K.

Finite intersection of valuation overrings of polynomial rings in at most three variables

The group of divisibility of an integral domainis the multiplicative group of nonzero principal fractional ideals ofthe domain and is a partially ordered group under reverse inclusion. We study the group of divisibility of a finite intersection of valuation overrings of polynomial rings in at most three variables and we classify all semilocal lattice-ordered groups which are realizable over k[x1,x2,...,xn] for n≤3.

Spectral subspaces of spectra of Abelian lattice-ordered groups in size aleph one

It is well known that the lattice $${{\,\mathrm{Id_c}\,}}{G}$$ of all principal $$\ell $$ -ideals of any Abelian $$\ell $$ -group G is a completely normal distributive 0-lattice; yet not every completely normal distributive 0-lattice is a homomorphic image of some $${{\,\mathrm{Id_c}\,}}{G}$$ , via a counterexample of cardinality $$\aleph _2$$ . We prove that every completely normal distributive 0-lattice with at most $$\aleph _1$$ elements is a homomorphic image of some $${{\,\mathrm{Id_c}\,}}{G}$$ . By Stone duality, this means that every completely normal generalized spectral space with at most $$\aleph _1$$ compact open sets is homeomorphic to a spectral subspace of the $$\ell $$ -spectrum of some Abelian $$\ell $$ -group.

Quasi-Engel varieties of lattice-ordered groups

Quasi-Engel varieties of lattice-ordered groups

Realizations of semilocal ℓ-groups over k[x 1,x 2,…x n

Doering and Lequain in 1999 introduced a weak approximation theorem for dependent valuation rings and they proved that every finitely generated lattice-ordered group can be realized as the group of divisibility of a semilocal Bézout overring of a polynomial ring over a field k in infinitely many variables, where each of the valuation rings appearing in the finite intersection has residue field k. Moreover, they proved that every semilocal lattice-ordered group admits a lexico-cardinal decomposition form. In this work, we focus on realizing the semilocal -group over a polynomial ring in finitely many variables. We prove that every semilocal lattice-ordered group having a finite rational rank can be realized as the group of divisibility of a Bézout overring of up to lexico-cardinal decomposition, where k is a field and are indeterminates over k and n depends on the group. As a corollary, we prove that every semilocal -group either finitely generated or divisible with finite rational rank is realizable over where each of the valuation rings appearing in the finite intersection has residue field k.

Some Modifications of Hull Operators in Archimedean Lattice-Ordered Groups with Weak Unit

Some Modifications of Hull Operators in Archimedean Lattice-Ordered Groups with Weak Unit

Radically principal MV-algebras

Abstract An MV-algebra A is radically principal if every prime ideal P of A is radically principal, i.e., there exists a principal ideal I of A such that Rad ( P ) = Rad ( I ) $ \text{Rad}(P)=\text{Rad}(I) $ . We investigate radically principal MV-algebras and provide some characterizations as well as some classes of examples. We prove a Cohen-like theorem, precisely, an MV-algebra is radically principal if and only if every maximal ideal is radically principal. It is also shown that the radically principal hyperarchemedian MV-algebras are the weakly finite ones and the radically principal Boolean algebras are the finite ones. Radically principal MV-algebras are also studied from the perspective of lattice-ordered groups.

Adjunction of a strong unit to a hyper-archimedean lattice-ordered group

Adjunction of a strong unit to a hyper-archimedean lattice-ordered group

Connected topological lattice-ordered groups

Connected topological lattice-ordered groups

New axioms for the lattice-ordered groups existentially closed in $\mathbf W^+$

New axioms for the lattice-ordered groups existentially closed in $\mathbf W^+$

From partially ordered monoids to partially ordered groups via free nuclear preimages

Two fundamental constructions operating on residuated lattices and partially ordered monoids (pomonoids) are nuclear images and conuclear images. Nuclear images allow us to construct many of the ordered algebras which arise in non-classical logic (such as pomonoids, semilattice-ordered monoids, and residuated lattices) from cancellative ordered algebras. Conuclear images then allow us to construct some of these cancellative algebras from partially ordered or lattice-ordered groups. Among other things, we show that finite (commutative) integral residuated lattices are precisely the finite nuclear images of (commutative) cancellative integral residuated lattices and that (commutative) integrally closed pomonoids are precisely the nuclear images of subpomonoids of partially ordered (Abelian) groups. The key construction is the free nuclear preimage of a pomonoid. As a by-product of our study of free nuclear preimages, we obtain a syntactic characterization of quasivarieties of pomonoids and semilattice-ordered monoids closed under nuclear images.

The Archimedean kernel of a lattice-ordered group

The Archimedean kernel of a lattice-ordered group

Projective classes as images of accessible functors

Abstract We are dealing with projective classes (in short $\textrm {PC}$) over first-order vocabularies with no restrictions on the (possibly infinite) arities of relation or operation symbols. We verify that $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {RPC}(\mathbin {\mathscr {L}}_{\infty \lambda })$ for any infinite cardinal $\lambda $, and that if $\lambda $ is singular, then $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda ^+})$. If $\lambda $ is regular, then a class of structures over a $\lambda $-ary vocabulary is $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })$-definable iff it is the image of a $\lambda $-continuous functor on a $\lambda $-accessible category; we also provide separating counterexamples for the non $\lambda $-ary case. We prove that many $\textrm {PC}$ classes of structures, previously known not to be closed under elementary equivalence over any $\mathbin {\mathscr {L}}_{\infty \lambda }$, are not even $\textrm {co}\textrm {-}\textrm {PC}$ over $\mathbin {\mathscr {L}}_{\infty \infty }$. Those classes arise from diverse contexts including convex $\ell $-subgroup lattices of lattice-ordered groups, ideal lattices of rings, nonstable K$_0$-theory of rings, coordinatization of sectionally complemented modular lattices and real spectra of commutative unital rings. For example, the class of posets of finitely generated two-sided ideals of all unital rings is $\textrm {PC}$ but not $\textrm {co}\textrm {-}\textrm {PC}$ over $\mathbin {\mathscr {L}}_{\infty \infty }$. We also provide a negative solution to a problem, raised in 2011 by Gillibert and the author, asking whether essential surjectivity of a ‘well-behaved’ functor on objects entails its essential surjectivity on diagrams indexed by arbitrary finite posets.

The ideal structure of $$C^{*}$$-algebras related to lattice-ordered groups

The ideal structure of $$C^{*}$$-algebras related to lattice-ordered groups

From distributive ℓ-monoids to ℓ-groups, and back again

We prove that an inverse-free equation is valid in the variety LG of lattice-ordered groups (ℓ-groups) if and only if it is valid in the variety DLM of distributive lattice-ordered monoids (distributive ℓ-monoids). This contrasts with the fact that, as proved by Repnitskiĭ, there exist inverse-free equations that are valid in all Abelian ℓ-groups but not in all commutative distributive ℓ-monoids, and, as we prove here, there exist inverse-free equations that are valid in all totally ordered groups but not in all totally ordered monoids. We also prove that DLM has the finite model property and a decidable equational theory, establish a correspondence between the validity of equations in DLM and the existence of certain right orders on free monoids, and provide an effective method for reducing the validity of equations in LG to the validity of equations in DLM.

Nilpotency and the Hamiltonian property for cancellative residuated lattices

This paper studies nilpotent and Hamiltonian cancellative residuated lattices and their relationship with nilpotent and Hamiltonian lattice-ordered groups. In particular, results about lattice-ordered groups are extended to the domain of residuated lattices. The two key ingredients that underlie the considerations of this paper are the categorical equivalence between Ore residuated lattices and lattice-ordered groups endowed with a suitable modal operator; and Malcev’s description of nilpotent groups of a given nilpotency class c in terms of a semigroup equation.

On minimum proper essential extensions in a category

In a concrete category A satisfying certain natural conditions, we investigate minimum proper essential extenstions (A-mpee’s) and how this relates to atoms in the complete lattice hoA of hull operators on A (if A # is the category with objects the A-objects but morphisms just the essential extensions in A, then hoA consists exactly of the reflections in A # ). A further hypothesis inserts itself: We call an A-mpee µ strong just in case the domain and codomain of µ are not isomorphic; [S] is the assumption that every A-mpee is strong. We show (1) if µ is strong, then an atom of hoA is determined by µ, and (2) if [S], then every id ≠ h ∈ hoA lies above an atom and every atom is determined by an A-mpee µ. Examples are presented in various categories of lattice-ordered groups.

$C(\beta\mathbb{N}\setminus\mathbb{N})$ among the Archimedean $\ell$-groups with strong unit

Let $\mathcal{W}^{+}$ be the class of Archimedean lattice-ordered groups with distinguished strong unit $1 \gt 0$. This paper studies the $\ell$-groups existentially closed (e.c.) in $\mathcal{W}^{+}$ and those infinitely generic (i.g.) in $\mathcal{W}^{+