ON A NEW APPROACH TO THE DUAL SYMMETRIC INVERSE MONOID $\mathcal{I}_{X}^{\ast}$

2007 ◽  
Vol 17 (03) ◽  
pp. 567-591 ◽  
Author(s):  
VICTOR MALTCEV
Keyword(s):  
Inverse Semigroup ◽  
Cross Section ◽  
Inverse Monoid ◽  
Finite Sets ◽  
New Approach ◽  
Generating Set ◽  
Maximal Subsemigroups ◽  
Symmetric Inverse Semigroup ◽  
Transitive Representation ◽  
Symmetric Inverse Monoid

We construct the inverse partition semigroup[Formula: see text], isomorphic to the dual symmetric inverse monoid[Formula: see text], introduced in [6]. We give a convenient geometric illustration for elements of [Formula: see text]. We describe all maximal subsemigroups of [Formula: see text] and find a generating set for [Formula: see text] when X is finite. We prove that all the automorphisms of [Formula: see text] are inner. We show how to embed the symmetric inverse semigroup into the inverse partition one. For finite sets X, we establish that, up to equivalence, there is a unique faithful effective transitive representation of [Formula: see text], namely to [Formula: see text]. Finally, we construct an interesting [Formula: see text]-cross-section of [Formula: see text], which is reminiscent of [Formula: see text], the [Formula: see text]-cross-section of [Formula: see text], constructed in [4].

scholarly journals LARGEST 2-GENERATED SUBSEMIGROUPS OF THE SYMMETRIC INVERSE SEMIGROUP

2007 ◽  
Vol 50 (3) ◽  
pp. 551-561 ◽  
Author(s):  
J. M. André ◽  
V. H. Fernandes ◽  
J. D. Mitchell
Keyword(s):  
Inverse Semigroup ◽  
Inverse Monoid ◽  
Symmetric Inverse Semigroup ◽  
Symmetric Inverse Monoid ◽  
Element Set

AbstractThe symmetric inverse monoid $\mathcal{I}_{n}$ is the set of all partial permutations of an $n$-element set. The largest possible size of a $2$-generated subsemigroup of $\mathcal{I}_{n}$ is determined. Examples of semigroups with these sizes are given. Consequently, if $M(n)$ denotes this maximum, it is shown that $M(n)/|\mathcal{I}_{n}|\rightarrow1$ as $n\rightarrow\infty$. Furthermore, we deduce the known fact that $\mathcal{I}_{n}$ embeds as a local submonoid of an inverse $2$-generated subsemigroup of $\mathcal{I}_{n+1}$.

scholarly journals On the semigroup of all partial fence-preserving injections on a finite set

2017 ◽  
Vol 16 (12) ◽  
pp. 1750223 ◽  
Author(s):  
Ilinka Dimitrova ◽  
Jörg Koppitz
Keyword(s):  
Inverse Semigroup ◽  
Transformation Formula ◽  
Green’S Relations ◽  
Generating Set ◽  
Regular Elements ◽  
Green's Relations ◽  
Finite Set ◽  
Symmetric Inverse Semigroup ◽  
Text Element ◽  
Element Set

For [Formula: see text], let [Formula: see text] be an [Formula: see text]-element set and let [Formula: see text] be a fence, also called a zigzag poset. As usual, we denote by [Formula: see text] the symmetric inverse semigroup on [Formula: see text]. We say that a transformation [Formula: see text] is fence-preserving if [Formula: see text] implies that [Formula: see text], for all [Formula: see text] in the domain of [Formula: see text]. In this paper, we study the semigroup [Formula: see text] of all partial fence-preserving injections of [Formula: see text] and its subsemigroup [Formula: see text]. Clearly, [Formula: see text] is an inverse semigroup and contains all regular elements of [Formula: see text] We characterize the Green’s relations for the semigroup [Formula: see text]. Further, we prove that the semigroup [Formula: see text] is generated by its elements with rank [Formula: see text]. Moreover, for [Formula: see text], we find the least generating set and calculate the rank of [Formula: see text].

scholarly journals INJECTIVE TRANSFORMATIONS WITH EQUAL GAP AND DEFECT

2009 ◽  
Vol 79 (2) ◽  
pp. 327-336 ◽  
Author(s):  
JINTANA SANWONG ◽  
R. P. SULLIVAN
Keyword(s):  
Inverse Semigroup ◽  
Green’S Relations ◽  
Maximal Subsemigroups ◽  
Green's Relations ◽  
Algebraic Properties ◽  
Symmetric Inverse Semigroup ◽  
Infinite Set

AbstractSuppose that X is an infinite set and I(X) is the symmetric inverse semigroup defined on X. If α∈I(X), we let dom α and ran α denote the domain and range of α, respectively, and we say that g(α)=|X/dom α| and d(α)=|X/ran α| is the ‘gap’ and the ‘defect’ of α, respectively. In this paper, we study algebraic properties of the semigroup $A(X)=\{\alpha \in I(X)\mid g(\alpha )=d(\alpha )\}$. For example, we describe Green’s relations and ideals in A(X), and determine all maximal subsemigroups of A(X) when X is uncountable.

scholarly journals Dual symmetric inverse monoids and representation theory

1998 ◽  
Vol 64 (3) ◽  
pp. 345-367 ◽  
Author(s):  
D. G. Fitzgerald ◽  
Jonathan Leech
Keyword(s):  
Representation Theory ◽  
Inverse Semigroup ◽  
The Other ◽  
Natural Order ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Dual Pairs ◽  
Representations Of Groups ◽  
Dual Symmetric Inverse Monoids ◽  
Symmetric Inverse Monoid

AbstractThere is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid Ix, that is, a monoid of partial one-to-one selfmaps of a set X. The present paper describes the structure of a categorical dual Ix* to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symmetric inverse monoid. It is shown how a representation of S by (full) selfmaps of a set X leads to dual pairs of representations in Ix and Ix*, and how a number of known representations arise as one or the other of these pairs. Conditions on S are described which ensure that representations of S preserve such infima or suprema as exist in the natural order of S. The categorical treatment allows the construction, from standard functors, of representations of S in certain other inverse algebras (that is, inverse monoids in which all finite infima exist). The paper concludes by distinguishing two subclasses of inverse algebras on the basis of their embedding properties.

scholarly journals On Maximal Subsemigroups of Partial Baer-Levi Semigroups

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Boorapa Singha ◽  
Jintana Sanwong
Keyword(s):  
Inverse Semigroup ◽  
Maximal Subsemigroups ◽  
Symmetric Inverse Semigroup ◽  
Infinite Set

Suppose thatXis an infinite set with|X|≥q≥ℵ0andI(X)is the symmetric inverse semigroup defined onX. In 1984, Levi and Wood determined a class of maximal subsemigroupsMA(using certain subsetsAofX) of the Baer-Levi semigroupBL(q)={α∈I(X):domα=Xand|X∖Xα|=q}. Later, in 1995, Hotzel showed that there are many other classes of maximal subsemigroups ofBL(q), but these are far more complicated to describe. It is known thatBL(q)is a subsemigroup of the partial Baer-Levi semigroupPS(q)={α∈I(X):|X∖Xα|=q}. In this paper, we characterize all maximal subsemigroups ofPS(q)when|X|>q, and we extendMAto obtain maximal subsemigroups ofPS(q)when|X|=q.

A New Approach to the Demonstration of Oxidase-Localization Using Tannic Acid Or Catechin

1973 ◽  
Vol 31 ◽  
pp. 698-699
Author(s):  
V. Mizuhira ◽  
Y. Futaesaku
Keyword(s):  
Cross Section ◽  
Tannic Acid ◽  
Elastic Fiber ◽  
The Other ◽  
New Approach ◽  
Tissue Block ◽  
Active Reaction ◽  
The Cross

Previously we reported that tannic acid is a very effective fixative for proteins including polypeptides. Especially, in the cross section of microtubules, thirteen submits in A-tubule and eleven in B-tubule could be observed very clearly. An elastic fiber could be demonstrated very clearly, as an electron opaque, homogeneous fiber. However, tannic acid did not penetrate into the deep portion of the tissue-block. So we tried Catechin. This shows almost the same chemical natures as that of proteins, as tannic acid. Moreover, we thought that catechin should have two active-reaction sites, one is phenol,and the other is catechole. Catechole site should react with osmium, to make Os- black. Phenol-site should react with peroxidase existing perhydroxide.

scholarly journals New Method for Analysis of the Temporomandibular Joint Using Cone Beam Computed Tomography

Sensors ◽  
2021 ◽  
Vol 21 (9) ◽  
pp. 3070
Author(s):  
Sebastian Iwaszenko ◽  
Jakub Munk ◽  
Stefan Baron ◽  
Adam Smoliński
Keyword(s):  
Computed Tomography ◽  
Temporomandibular Joint ◽  
Cross Section ◽  
Cone Beam Computed Tomography ◽  
Cone Beam ◽  
Section Plane ◽  
New Approach ◽  
Volume Of Interest ◽  
Gap Width

Modern dentistry commonly uses a variety of imaging methods to support diagnosis and treatment. Among them, cone beam computed tomography (CBCT) is particularly useful in presenting head structures, such as the temporomandibular joint (TMJ). The determination of the morphology of the joint is an important part of the diagnosis as well as the monitoring of the treatment results. It can be accomplished by measurement of the TMJ gap width at three selected places, taken at a specific cross-section. This study presents a new approach to these measurements. First, the CBCT images are denoised using curvilinear methods, and the volume of interest is determined. Then, the orientation of the vertical cross-section plane is computed based on segmented axial sections of the TMJ head. Finally, the cross-section plane is used to determine the standardized locations, at which the width of the gap between condyle and fossa is measured. The elaborated method was tested on selected TMJ CBCT scans with satisfactory results. The proposed solution lays the basis for the development of an autonomous method of TMJ index identification.

scholarly journals Object-Oriented Type Inference

1991 ◽  
Vol 20 (345) ◽  
Author(s):  
Jens Palsberg ◽  
Michael I. Schwartzbach
Keyword(s):  
Fixed Point ◽  
Object Oriented ◽  
Type Inference ◽  
Finite Sets ◽  
New Approach ◽  
Optimizing Compiler ◽  
Point Derivation ◽  
Set Inclusion ◽  
Least Solution ◽  
Type Information

We present a new approach to inferring types in untyped object-oriented programs with inheritance, assignments, and late binding. It guarantees that all messages are understood, annotates the program with type information, allows polymorphic methods, and can be used as the basis of an optimizing compiler. Types are finite sets of classes and subtyping is set inclusion. Using a trace graph, our algorithm constructs a set of conditional type constraints and computes the least solution by least fixed-point derivation.

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