Formations of lattice ordered groups and of GMV-algebras

2008 ◽  
Vol 58 (5) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Infinite Distributivity ◽  
Subdirect Products

AbstractA class of lattice ordered groups is called a formation if it is closed with respect to homomorphic images and finite subdirect products. Analogously we define the formation of GMV-algebras. Let us denote by ℱ1 and ℱ2 the collection of all formations of lattice ordered groups or of GMV-algebras, respectively. Both ℱ1 and ℱ2 are partially ordered by the class-theoretical inclusion. We prove that ℱ1 satisfies the infinite distributivity law

scholarly journals Tight Riesz groups

1972 ◽  
Vol 13 (2) ◽  
pp. 224-240 ◽  
Author(s):  
R. J. Loy ◽  
J. B. Miller
Keyword(s):  
General Theory ◽  
Open Interval ◽  
Discrete Topology ◽  
Topological Groups ◽  
Ordered Group ◽  
Partially Ordered ◽  
Interval Topology ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Positive Elements

The theory of partially ordered topological groups has received little attention in the literature, despite the accessibility and importance in analysis of the group Rm. One obstacle in the way of a general theory seems to be, that a convenient association between the ordering and the topology suggests that the cone of all strictly positive elements be open, i.e. that the topology be at least as strong as the open-interval topology U; but if the ordering is a lattice ordering and not a full ordering then U itself is already discrete. So to obtain in this context something more interesting topologically than the discrete topology and orderwise than the full order, one must forego orderings which make lattice-ordered groups: in fact, the partially ordered group must be an antilattice, that is, must admit no nontrivial meets or joins (see § 2, 10°).

On a cancellation rule for subdirect products of lattice ordered groups and of GMV-algebras

2007 ◽  
Vol 57 (3) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Present Note ◽  
Variety Of Algebras ◽  
Subdirect Decomposition ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Subdirect Products

AbstractThe notion of internal subdirect decomposition can be defined in each variety of algebras. In the present note we prove the validity of a cancellation rule concerning such decompositions for lattice ordered groups and for GMV-algebras. For the case of groups, this cancellation rule fails to be valid.

Torsion classes of generalized Boolean algebras

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Analogous Result ◽  
Boolean Algebras ◽  
Radical Class ◽  
Torsion Class ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Brouwerian Lattice ◽  
Generalized Boolean Algebras

AbstractTorsion classes and radical classes of lattice ordered groups have been investigated in several papers. The notions of torsion class and of radical class of generalized Boolean algebras are defined analogously. We denote by T g and R g the collections of all torsion classes or of all radical classes of generalized Boolean algebras, respectively. Both T g and R g are partially ordered by the class-theoretical inclusion. We deal with the relation between these partially ordered collection; as a consequence, we obtain that T g is a Brouwerian lattice. W. C. Holland proved that each variety of lattice ordered groups is a torsion class. We show that an analogous result is valid for generalized Boolean algebras.

Weak relatively uniform convergences on MV-algebras

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Štefan Černák ◽  
Ján Jakubík
Keyword(s):  
Present Article ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebra ◽  
Brouwerian Lattice ◽  
Mv Algebras ◽  
Natural Way

AbstractWeak relatively uniform convergences (wru-convergences, for short) in lattice ordered groups have been investigated in previous authors’ papers. In the present article, the analogous notion for MV-algebras is studied. The system s(A) of all wru-convergences on an MV-algebra A is considered; this system is partially ordered in a natural way. Assuming that the MV-algebra A is divisible, we prove that s(A) is a Brouwerian lattice and that there exists an isomorphism of s(A) into the system s(G) of all wru-convergences on the lattice ordered group G corresponding to the MV-algebra A. Under the assumption that the MV-algebra A is archimedean and divisible, we investigate atoms and dual atoms in the system s(A).

scholarly journals Compatible tight Riesz orders and prime subgroups

1973 ◽  
Vol 14 (2) ◽  
pp. 145-160 ◽  
Author(s):  
N. R. Reilly
Keyword(s):  
Interpolation Property ◽  
Finite Width ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Lattice Order ◽  
Partially Ordered ◽  
Partially Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Strengthened Form

A tight Riesz group is a partially ordered group which satisfies a strengthened form of the Riesz interpolation property. The term “tight” was introduced by Miller in [8], and the tight interpolation property has been considered in papers by Fuchs [3], Miller [8, 9], Loy and Miller [7] and Wirth [12]. If the closure of the cone P, in the interval topology, of such a partially ordered group G contains no pseudozeros, then is itself the cone of a partial order on G. Loy and Miller found of particular interest the case in which this associated partial orderis a lattice order. This situation was then considered in reverse by A. Wirth [12] who investigated under what circumstances a lattice ordered group would permit the existence of a tight Riesz order (called a compatible tight Riesz order) for which the initial lattice order is the order defined by the closure of the cone of the tight Riesz order.Wirth gave two fundamental anduseful characterizations of those subsets of the cone of a lattice ordered group that canbe the strict cone of a compatible tight Riesz order; one is in terms of archimedean classes and the other is an elementwise characterization. Although Loy, Miller and Wirth restricted their attention to abelian groups, much of what they do carries over verbatim to nonabelian groups. In the main result of this paper (Theorem 2.6) a description of the strict cone of a compatible tight Riesz order on a lattice ordered group Gis given in terms of the prime subgroups of G.This is particularly useful when one is attempting to identify the compatible tight Riesz orders on some particular lattice ordered group or class of lattice ordered groups, since it narrows down to a convenient family of subsets the possible candidates for strict cones of compatible tight Riesz orders. These can then be tested under Wirth's criteria. This technique is illustrated in § 5, where the compatible tight Riesz orders are determined o a lattice ordered group of the type V(Γ, Gγ), where Γ is of finite width, and in § 6, where two examples are considered.

Torsion classes of abelian cyclically ordered groups

2012 ◽  
Vol 62 (4) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Distributive Lattice ◽  
Partial Order ◽  
Torsion Class ◽  
Proper Class ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups

AbstractWe introduce the notion of torsion class of abelian cyclically ordered groups; the definition is analogous to that used in the theory of lattice ordered groups. The collection T of all such classes is partially ordered by the class-theoretical inclusion. Though T is a proper class, we can apply the usual terminology for this partial order. We prove that T is a complete, infinitely distributive lattice having infinitely many atoms.

K-Radical classes and product radical classes of MV-algebras

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Analogous Result ◽  
Radical Class ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebra ◽  
One To One ◽  
Abelian Lattice ◽  
Mv Algebras

AbstractFor an MV-algebra let J 0() be the system of all closed ideals of ; this system is partially ordered by the set-theoretical inclusion. A radical class X of MV-algebras will be called a K-radical class iff, whenever ∈ X and is an MV-algebra with J 0() ≅ J 0(), then ∈ X. An analogous notation for lattice ordered groups was introduced and studied by Conrad. In the present paper we show that there is a one-to-one correspondence between K-radical classes of MV-algebras and K-radical classes of abelian lattice ordered groups. We also prove an analogous result for product radical classes of MV-algebras; product radical classes of lattice ordered groups were studied by Ton.

Generalized Commutativity of Lattice-Ordered Groups

2015 ◽  
Vol 65 (2) ◽  
Author(s):  
M. R. Darnel ◽  
W. C. Holland ◽  
H. Pajoohesh
Keyword(s):  
Theorem Proving ◽  
Natural Generalization ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Weak Commutativity

AbstractIn this paper we explore generalizations of Neumann’s theorem proving that weak commutativity in ordered groups actually implies the group is abelian. We show that a natural generalization of Neumann’s weak commutativity holds for certain Scrimger ℓ-groups.

A theorem and a question about epicomplete archimedean lattice-ordered groups

2009 ◽  
Vol 62 (2-3) ◽  
pp. 165-184 ◽  
Author(s):  
R. N. Ball ◽  
A. W. Hager ◽  
D. G. Johnson ◽  
A. Kizanis
Keyword(s):  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice
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