scholarly journals Homogeneity of inverse semigroups

2018 ◽  
Vol 28 (05) ◽  
pp. 837-875 ◽  
Author(s):  
Thomas Quinn-Gregson
Keyword(s):  
Inverse Semigroup ◽  
Finite Groups ◽  
Abelian Groups ◽  
Semigroup Theory ◽  
Inverse Semigroups ◽  
Finitely Generated ◽  
Homogeneous Groups ◽  
Inverse Subsemigroups

An inverse semigroup [Formula: see text] is a semigroup in which every element has a unique inverse in the sense of semigroup theory, that is, if [Formula: see text] then there exists a unique [Formula: see text] such that [Formula: see text] and [Formula: see text]. We say that a countable inverse semigroup [Formula: see text] is a homogeneous (inverse) semigroup if any isomorphism between finitely generated (inverse) subsemigroups of [Formula: see text] extends to an automorphism of [Formula: see text]. In this paper, we consider both these concepts of homogeneity for inverse semigroups, and show when they are equivalent. We also obtain certain classifications of homogeneous inverse semigroups, in particular periodic commutative inverse semigroups. Our results may be seen as extending both the classification of homogeneous semilattices and the classification of certain classes of homogeneous groups, such as homogeneous abelian groups and homogeneous finite groups.

FINITE PRESENTABILITY OF HNN EXTENSIONS OF INVERSE SEMIGROUPS

2005 ◽  
Vol 15 (03) ◽  
pp. 423-436 ◽  
Author(s):  
E. R. DOMBI ◽  
N. D. GILBERT ◽  
N. RUŠKUC
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Finitely Generated ◽  
Semigroup Presentation ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Finite Presentability ◽  
Order Ideals ◽  
Inverse Subsemigroups ◽  
Finitely Presented

HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base, are defined by means of a construction based upon the isomorphism between the categories of inverse semigroups and inductive groupoids. The resulting HNN extension may conveniently be described by an inverse semigroup presentation, and we determine when an HNN extension with finitely generated or finitely presented base is again finitely generated or finitely presented. Our main results depend upon properties of the [Formula: see text]-preorder in the associated subsemigroups. Let S be a finitely generated inverse semigroup and let U, V be inverse subsemigroups of S, isomorphic via φ: U → V, that are order ideals in S. We prove that the HNN extension S*U,φ is finitely generated if and only if U is finitely [Formula: see text]-dominated. If S is finitely presented, we give a necessary and suffcient condition for S*U,φ to be finitely presented. Here, in contrast to the theory of HNN extensions of groups, it is not necessary that U be finitely generated.

scholarly journals ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

scholarly journals A NOTE ON THE HOWSON PROPERTY IN INVERSE SEMIGROUPS

2016 ◽  
Vol 94 (3) ◽  
pp. 457-463 ◽  
Author(s):  
PETER R. JONES
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Necessary And Sufficient

An algebra has the Howson property if the intersection of any two finitely generated subalgebras is again finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.

Conjugacy Separability of Certain Polygonal Products

1996 ◽  
Vol 39 (3) ◽  
pp. 294-307 ◽  
Author(s):  
Goansu Kim
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Cyclic Subgroups ◽  
Conjugacy Separability ◽  
Conjugacy Separable

AbstractWe show that polygonal products of polycyclic-by-finite groups amalgamating central cyclic subgroups, with trivial intersections, are conjugacy separable. Thus polygonal products of finitely generated abelian groups amalgamating cyclic subgroups, with trivial intersections, are conjugacy separable. As a corollary of this, we obtain that the group A1 *〈a1〉A2 *〈a2〉 • • • *〈am-1〉Am is conjugacy separable for the abelian groups Ai.

scholarly journals Inverse semigroups whose full inverse subsemigroups form a chain

1981 ◽  
Vol 22 (2) ◽  
pp. 159-165 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Analogous Problem ◽  
A Chain ◽  
Inverse Subsemigroups

The structure of semigroups whose subsemigroups form a chain under inclusion was determined by Tamura [9]. If we consider the analogous problem for inverse semigroups it is immediate that (since idempotents are singleton inverse subsemigroups) any inverse semigroup whose inverse subsemigroups form a chain is a group. We will therefore, continuing the approach of [5, 6], consider inverse semigroups whose full inverse subsemigroups form a chain: we call these inverse ▽-semigroups.

scholarly journals A note on shortly connected inverse semigroups

1991 ◽  
Vol 43 (3) ◽  
pp. 463-466 ◽  
Author(s):  
S.M. Goberstein
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
The Other ◽  
Inverse Subsemigroups ◽  
Completely Semisimple

Shortly connected and shortly linked inverse semigroups arise in the study of inverse semigroups determined by the lattices of inverse subsemigroups and by the partial automorphism semigroups. It has been shown that every shortly linked inverse semigroup is shortly connected, but the question of whether the converse is true has not been addressed. Here we construct two examples of combinatorial shortly connected inverse semigroups which are not shortly linked. One of them is completely semisimple while the other is not.

scholarly journals Inverse semigroups determined by their lattices of inverse subsemigroups

1981 ◽  
Vol 30 (3) ◽  
pp. 321-346 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Inverse Semigroups ◽  
Lattice Isomorphism ◽  
Free Inverse Semigroup ◽  
Inverse Subsemigroups ◽  
Completely Semisimple

AbstractIn a previous paper ([14]) the author showed that a free inverse semigroup is determined by its lattice of inverse subsemigroups, in the sense that for any inverse semigroup T, implies . (In fact, the lattice isomorphism is induced by an isomorphism of upon T.) In this paper the results leading up to that theorem are generalized (from completely semisimple to arbitrary inverse semigroups) and applied to various classes, including simple, fundamental and E-unitary inverse semigroups. In particular it is shown that the free product of two groups in the category of inverse semigroups is determined by its lattice of inverse subsemigroups.

scholarly journals Commensurability and QI classification of free products of finitely generated abelian groups

2008 ◽  
Vol 137 (03) ◽  
pp. 811-813 ◽  
Author(s):  
Jason A. Behrstock ◽  
Tadeusz Januszkiewicz ◽  
Walter D. Neumann
Keyword(s):  
Abelian Groups ◽  
Finitely Generated ◽  
Free Products

scholarly journals THE ÉTALE GROUPOID OF AN INVERSE SEMIGROUP AS A GROUPOID OF FILTERS

2013 ◽  
Vol 94 (2) ◽  
pp. 234-256 ◽  
Author(s):  
M. V. LAWSON ◽  
S. W. MARGOLIS ◽  
B. STEINBERG
Keyword(s):  
Functional Analysis ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Groupoid ◽  
Linear Representations ◽  
Inverse Subsemigroups ◽  
Étale Groupoid

AbstractPaterson showed how to construct an étale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz’s construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters. In addition, idempotent filters are closed inverse subsemigroups and so determine transitive representations by means of partial bijections. This connection between filters and representations by partial bijections is exploited to show how linear representations of inverse semigroups can be constructed from the groups occurring in the associated topological groupoid.

On growth, identities and free subsemigroups for inverse semigroups of deficiency one

2015 ◽  
Vol 25 (01n02) ◽  
pp. 233-258 ◽  
Author(s):  
L. M. Shneerson
Keyword(s):  
Positive Integer ◽  
Inverse Semigroup ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Inverse Semigroups ◽  
Defining Relations ◽  
Rank 2 ◽  
Inverse Subsemigroups

For any positive integer n > 1 we construct an example of inverse semigroup with n generators and n - 1 defining relations which has cubic growth and at least n generators in any presentation. This semigroup has the same set of identities as the free monogenic inverse semigroup. In particular, we give the first example of a one relation nonmonogenic inverse semigroup having polynomial growth. We also prove that for any positive integer n there exists an inverse semigroup ϒn of deficiency 1 and rank n + 1 such that ϒn has exponential growth and it does not contain nonmonogenic free inverse subsemigroups. Furthermore, ϒn satisfies the identity [[x, y], [z, t]]2 = [[x, y], [z, t]] of quasi-solvability and it contains a free subsemigroup of rank 2.

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