The root system of prime subgroups of a free lattice-ordered group (without G.C.H.)

Order ◽  
1989 ◽  
Vol 6 (3) ◽  
pp. 305-309 ◽  
Author(s):  
Manfred Droste ◽  
Stephen H. McCleary
Keyword(s):  
Root System ◽  
Free Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group

Countable lattice-ordered groups

1983 ◽  
Vol 94 (1) ◽  
pp. 29-33 ◽  
Author(s):  
A. M. W. Glass
Keyword(s):  
Free Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Recursively Enumerable Set ◽  
Ordered Groups ◽  
Recursively Enumerable ◽  
Lattice Ordered Groups ◽  
Natural Analogues ◽  
Group Words ◽  
Combinatorial Group

A lattice-ordered group is a group and a lattice such that the group operation distributes through the lattice operations (i.e. f(g ∨ h)k = fgk ∨ fhk and dually). Lattice-ordered groups are torsion-free groups and distributive lattices. They further satisfy f ∧ g = (f−1 ∨ g−1)−1 and f ∨ g = (f−1 ∧ g−1)−1. Since the lattice is distributive, each lattice-ordered group word can be written in the form ∨A ∧B ωαβ where A and B are finite and each ωαβ is a group word in {xi: i ∈ I}. Unfortunately, even for free lattice-ordered groups, this form is not unique. We will use the prefix l- for maps between lattice-ordered groups that preserve both the group and lattice operations, and e for the identity element. A presentation (xi;rj(x) = e)i∈I, j∈J is the quotient of the free lattice-ordered group F on {xi: i∈I} by the l-ideal (convex normal sublattice subgroup) generated by its subset {rj(x): j ∈ J}. {xi: i ∈ I} is called a generating set and {ri(x):j∈J} a defining set of relations. If I and J are finite we have a finitely presented lattice-ordered group. If we can effectively enumerate all lattice-ordered group words r1(x), r2(x),… in xi; i∈I}. If I is finite and J (for this enumeration) is a recursively enumerable set, we say that we have a recursively presented lattice-ordered group. Throughout Z denotes the group of integers and ℝ the real line.Our purpose in this paper is to prove the natural analogues of three theorems from combinatorial group theory (5), chapter IV, theorems 4·9, 3·1 and 3·5-in particular, theorem C is a natural analogue of an unpublished theorem of Philip Hall (4).

scholarly journals Minimal vector lattice covers

1971 ◽  
Vol 5 (3) ◽  
pp. 331-335 ◽  
Author(s):  
Roger D. Bleier
Keyword(s):  
Vector Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

The π-Full Tight Riesz Orders on A(Ω)

1981 ◽  
Vol 24 (2) ◽  
pp. 137-151
Author(s):  
Gary Davis ◽  
Stephen H. McCleary
Keyword(s):  
Permutation Group ◽  
Ordered Set ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Totally Ordered Set

Let G be a lattice-ordered group (l-group), and let t, u∈ G+. We write tπu if t ∧ g = 1 is equivalent to u ∧ g = 1, and say that a tight Riesz order T on G is π-full if t ∈ T and t π U imply u∈T. We study the set of π-full tight Riesz orders on an l-permutation group (G, Ω), Ω a totally ordered set.

scholarly journals Generalized discrete l-groups

1975 ◽  
Vol 20 (3) ◽  
pp. 281-289 ◽  
Author(s):  
Joe L. Mott
Keyword(s):  
Lattice Ordered Group ◽  
Ordered Group ◽  
Abelian Lattice Ordered Group ◽  
Group A ◽  
Abelian Lattice

Let G be an abelian lattice ordered group (an l-group). If G is, in fact, totally ordered, we say that G is an 0–group. A subgroup and a sublattice of G is an l-subgroup. A subgroup C of G is called convex if 0 ≦ g ≦ c ∈ C and g ∈ G imply g ∈ C, C is an l-ideal if C is a convex l-subgroup of G. If C is an l-ideal of G, then G/C is also an l-group under the canonical ordering inherited from G. If, in fact, G/C is an 0–group, then C is said to be a prime subgroup of G.

Archimedean Closures in Lattice-Ordered Groups

1969 ◽  
Vol 21 ◽  
pp. 1004-1012 ◽  
Author(s):  
Richard D. Byrd
Keyword(s):  
Partial Answer ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Regular Subgroup

Conrad (10) and Wolfenstein (15; 16) have introduced the notion of an archimedean extension (a-extension) of a lattice-ordered group (l-group). In this note the class of l-groups that possess a plenary subset of regular subgroups which are normal in the convex l-subgroups that cover them are studied. It is shown in § 3 (Corollary 3.4) that the class is closed with respect to a-extensions and (Corollary 3.7) that each member of the class has an a-closure. This extends (6, p. 324, Corollary II; 10, Theorems 3.2 and 4.2; 15, Theorem 1) and gives a partial answer to (10, p. 159, Question 1). The key to proving both of these results is Theorem 3.3, which asserts that if a regular subgroup is normal in the convex l-subgroup that covers it, then this property is preserved by a-extensions.

scholarly journals Groups of complexes of a representable lattice-ordered group

1978 ◽  
Vol 19 (2) ◽  
pp. 135-139 ◽  
Author(s):  
R. D. Byrd ◽  
J. T. Lloyd ◽  
J. W. Stepp
Keyword(s):  
Maximal Subgroup ◽  
Maximal Subgroups ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Lattice Order

In 1954 N. Kimura proved that each idempotent in a semigroup is contained in a unique maximal subgroup of the semigroup and that distinct maximal subgroups are disjoint [13] (or see [6, pp. 21–23]). This generalized earlier results of Schwarz [14] and Wallace [15]. These maximal subgroups are important in the study of semigroups. If G is a group, then the collection S(G) of nonempty complexes of G is a semigroup and it is natural to inquire what properties of G are inherited by the maximal subgroups of S(G). There seems to be very little literature devoted to this subject. In [5, Theorem 2], with certain hypotheses placed on an idempotent, it was shown that if G is a lattice-ordered group (“1-group”) then a maximal subgroup of S(G) containing an idempotent satisfying these conditions admits a natural lattice-order. The main result of this note (Theorem 1) is that if Gis a representable 1-group and E is a normal idempotent of S(G) and a dual ideal of the lattice G, then the maximal subgroup of S(G) containing E admits a representable lattice-order.

Unital topology on a unital l-group

2020 ◽  
Vol 70 (5) ◽  
pp. 1189-1196
Author(s):  
Mahmood Pourgholamhossein ◽  
Mohammad Ali Ranjbar
Keyword(s):  
Vector Lattice ◽  
Riesz Space ◽  
Lattice Ordered Group ◽  
Order Unit ◽  
Ordered Group ◽  
Strong Link ◽  
Fundamental Properties ◽  
Essential Properties ◽  
Unit Norm

AbstractIn this paper we investigate some fundamental properties of unital topology on a lattice ordered group with order unit. We show that some essential properties of order unit norm on a vector lattice with order unit, are valid for unital l-groups. For instance we show that for an Archimedean Riesz space G with order unit u, the unital topology and the strong link topology are the same.

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