scholarly journals LEFT-ORDERABLE COMPUTABLE GROUPS

2018 ◽  
Vol 83 (1) ◽  
pp. 237-255
Author(s):  
MATTHEW HARRISON-TRAINOR
Keyword(s):  
Orderable Group ◽  
Left Order

AbstractDowney and Kurtz asked whether every orderable computable group is classically isomorphic to a group with a computable ordering. By an order on a group, one might mean either a left-order or a bi-order. We answer their question for left-orderable groups by showing that there is a computable left-orderable group which is not classically isomorphic to a computable group with a computable left-order. The case of bi-orderable groups is left open.

scholarly journals Formal language convexity in left-orderable groups

2020 ◽  
Vol 30 (07) ◽  
pp. 1437-1456
Author(s):  
Hang Lu Su
Keyword(s):  
Formal Language ◽  
Hyperbolic Group ◽  
Infinite Family ◽  
Positive Cone ◽  
Finitely Generated ◽  
Orderable Group ◽  
Language Representation ◽  
Left Order ◽  
Finite Index Subgroups ◽  
Special Case

We propose a criterion for preserving the regularity of a formal language representation when passing from groups to subgroups. We use this criterion to show that the regularity of a positive cone language in a left-orderable group passes to its finite index subgroups, and to show that there exists no left order on a finitely generated acylindrically hyperbolic group such that the corresponding positive cone is represented by a quasi-geodesic regular language. We also answer one of Navas’ questions by giving an example of an infinite family of groups which admit a positive cone that is generated by exactly [Formula: see text] generators, for every [Formula: see text]. As a special case of our construction, we obtain a finitely generated positive cone for [Formula: see text].

Right Invariant Right Hereditary Rings

1974 ◽  
Vol 26 (5) ◽  
pp. 1186-1191 ◽  
Author(s):  
H. H. Brungs
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Maximal Orders ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Left Order ◽  
Principal Ideal Domains

Let R be a right hereditary domain in which all right ideals are two-sided (i.e., R is right invariant). We show that R is the intersection of generalized discrete valuation rings and that every right ideal is the product of prime ideals. This class of rings seems comparable with (and contains) the class of commutative Dedekind domains, but the rings considered here are in general not maximal orders and not Dedekind rings in the terminology of Robson [9]. The left order of a right ideal of such a ring is a ring of the same kind and the class contains right principal ideal domains in which the maximal right ideals are two-sided [6].

scholarly journals PALINDROMES AND ORDERINGS IN ARTIN GROUPS

2010 ◽  
Vol 19 (02) ◽  
pp. 145-162 ◽  
Author(s):  
FLORIAN DELOUP
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Type A ◽  
Reverse Order ◽  
Artin Groups ◽  
Left Order ◽  
Group B ◽  
Half Twist

The braid group Bn, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev : Bn →Bn, [Formula: see text], defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x ↦ Δ-1xΔ by the generalized half-twist (Garside element). More generally, the involution rev is defined for all Artin groups (equipped with Artin's presentation) and the involution τ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We study palindromes and palindromes invariant under τ in Artin groups of finite type. Our main results are the injectivity of the map [Formula: see text] in all finite-type Artin groups, the existence of a left-order compatible with rev for Artin groups of type A, B, D, and the existence of a decomposition for general palindromes. The uniqueness of the latter decomposition requires that the Artin groups carry a left-order.

scholarly journals On a problem in the theory of ordered groups

1972 ◽  
Vol 6 (3) ◽  
pp. 435-438 ◽  
Author(s):  
Colin D. Fox
Keyword(s):  
Cambridge Philos ◽  
Orderable Group ◽  
Counter Example ◽  
Ordered Groups ◽  
Defining Relation

The group G presented on two generators a, c with the single defining relation a−1c2a = c2a2c2 [proposed by B.H. Neumann in 1949 (unpublished), discussed by Gilbert Baumslag in Proc. Cambridge Philos. Soc. 55 (1959)] has been considered as a possible example of an orderable group which can not be embedded in a divisible orderable group, contrary to the conjecture that no such examples exist. It is known from Baumslag's discussion that G can not be embedded in any divisible orderable group. However, it is shown in this note that G is not orderable, and thus is not a counter-example to the conjecture.

Non-amalgamation of ordered groups

1984 ◽  
Vol 95 (2) ◽  
pp. 191-195 ◽  
Author(s):  
A. M. W. Glass ◽  
D. Saracino ◽  
C. Wood
Keyword(s):  
Positive Integer ◽  
Total Order ◽  
Ordered Group ◽  
Orderable Group ◽  
Ordered Groups

An ordered group (o-group for short) is a group endowed with a linear (i.e. total) order such that for all x, y, z, xz ≤ yz and zx ≤ zy whenever x ≤ y. A group for which such an order exists is called an orderable group. A group G is said to be divisible if for each positive integer m and each g ε G, there is x ε G such that xm = g.

Commutative orders

1996 ◽  
Vol 126 (6) ◽  
pp. 1201-1216 ◽  
Author(s):  
David Easdown ◽  
Victoria Gould
Keyword(s):  
Homomorphic Image ◽  
Commutative Semigroup ◽  
Permutation Identity ◽  
Left Order ◽  
Right Order

A subsemigroup S of a semigroup Q is a left (right) order in Q if every q ∈ Q can be written as q = a*b(q = ba*) for some a, b ∈S, where a* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. If S is both a left order and a right order in Q, we say that S is an order in Q. We show that if S is a left order in Q and S satisfies a permutation identity xl…xn = x1π…xnπ where 1 < 1π and nπ<n, then S and Q are commutative. We give a characterisation of commutative orders and decide the question of when one semigroup of quotients of a commutative semigroup is a homomorphic image of another. This enables us to show that certain semigroups have maximum and minimum semigroups of quotients. We give examples to show that this is not true in general.

scholarly journals Left orders in regular ℋ-Semigroups II

1990 ◽  
Vol 32 (1) ◽  
pp. 95-108 ◽  
Author(s):  
Victoria Gould
Keyword(s):  
Cancellation Condition ◽  
Left Order ◽  
Right Order

We make the convention that if a is an element of a semigroup Q then by writing a–1 it is implicit that a lies in a subgroup of Q and has inverse a–1 in this subgroup; equivalently, a ℋ a2 and a–1 is the inverse of a in Ha.A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be written as a–1b where a, b ∈ S and, in addition, every element of S satisfying a weak cancellation condition which we call square-cancellable lies in a subgroup of Q. The notions of right order and semigroup of right quotients are defined dually; if S is both a left order and a right order in Q then S is an order in Q and Q is a semigroup of quotients of S.

scholarly journals No positive cone in a free product is regular

2017 ◽  
Vol 27 (08) ◽  
pp. 1113-1120
Author(s):  
Susan Hermiller ◽  
Zoran Šunić
Keyword(s):  
Free Product ◽  
Positive Cone ◽  
Finitely Generated ◽  
Free Products ◽  
Language Complexity ◽  
Chomsky Hierarchy ◽  
Positive Elements ◽  
Left Order ◽  
Context Free ◽  
Product Of Groups

We show that there exists no left order on the free product of two nontrivial, finitely generated, left-orderable groups such that the corresponding positive cone is represented by a regular language. Since there are orders on free groups of rank at least two with positive cone languages that are context-free (in fact, 1-counter languages), our result provides a bound on the language complexity of positive cones in free products that is the best possible within the Chomsky hierarchy. It also provides a strengthening of a result by Cristóbal Rivas which states that the positive cone in a free product of nontrivial, finitely generated, left-orderable groups cannot be finitely generated as a semigroup. As another illustration of our method, we show that the language of all geodesics (with respect to the natural generating set) that represent positive elements in a graph product of groups defined by a graph of diameter at least 3 cannot be regular.

Hebrew, Aramaic, and The Greek of The Gospels

1951 ◽  
Vol 20 (60) ◽  
pp. 115-122
Author(s):  
W. Leonard Grant
Keyword(s):  
Initial Position ◽  
Average Reader ◽  
Left Order ◽  
The Many

It is frequently remarked of the κοινή Greek of the Gospels that it bears distinct traces of Hebrew and Aramaic influence, conscious or unconscious. The average reader of the Gospels may often wonder just which of the many linguistic oddities he encounters are common to all or most forms of κοινή and which are the genuine Hebraisms or Aramaisms; he can, if of a heroic cast of mind, learn Hebrew for this one purpose (a plan which will commend itself to few), or he can read various weighty tomes, from those of Wellhausen and Burney, who see Aramaic influence everywhere, to those of Moulton and Milligan, who minimize that influence wherever possible, to that of Lagrange, who placidly pursues a via media. But he would probably prefer a brief essay which tries to sum up the main facts without a footnote, without an appendix, and without an axe to grind.First, influence on syntax. Every κοινή scholar would agree that one of the commonest Semitisms in the Gospels is the initial position of the verb in the sentence. In both Hebrew and Aramaic the verb almost invariably appears first: ‘the king saw the man’ would (though in the Hebrew right-to-left order in which the first becomes last and the last becomes first) be expressed as ra'ah (verb) hammelech (subject) 'eth (sign of the accusative) ha'ish (object). Obviously the average Aramaic-speaking Jew would write εἶεν ὁ βασιλεύς τòν ἄνλpα. Now it is perfectly true that there is nothing un-Greek in the position of ενεν here; but it is the frequency of the occurrence, not the fact, which is significant; the initially placed verb meets us on every page of the Gospels—examples are unnecessary.

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