scholarly journals Congruences induced by transitive representations of inverse semigroups

1987 ◽  
Vol 29 (1) ◽  
pp. 21-40 ◽  
Author(s):  
Mario Petrich ◽  
Stuart Rankin
Keyword(s):  
General Theory ◽  
Inverse Semigroup ◽  
Symmetric Group ◽  
Transitive Group ◽  
Inverse Semigroups ◽  
Group Representations ◽  
Symmetric Inverse Semigroup ◽  
The Right ◽  
Special Case

Transitive group representations have their analogue for inverse semigroups as discovered by Schein [7]. The right cosets in the group case find their counterpart in the right ω-cosets and the symmetric inverse semigroup plays the role of the symmetric group. The general theory developed by Schein admits a special case discovered independently by Ponizovskiǐ [4] and Reilly [5]. For a discussion of this topic, see [1, §7.3] and [2, Chapter IV].

scholarly journals EMBEDDINGS IN COSET MONOIDS

2008 ◽  
Vol 85 (1) ◽  
pp. 75-80
Author(s):  
JAMES EAST
Keyword(s):  
Semigroup Forum ◽  
Inverse Semigroup ◽  
Simple Proof ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Simple Description ◽  
Group Of Units ◽  
Finite Monoids ◽  
Finite Set ◽  
Symmetric Inverse Semigroup

AbstractA submonoid S of a monoid M is said to be cofull if it contains the group of units of M. We extract from the work of Easdown, East and FitzGerald (2002) a sufficient condition for a monoid to embed as a cofull submonoid of the coset monoid of its group of units, and show further that this condition is necessary. This yields a simple description of the class of finite monoids which embed in the coset monoids of their group of units. We apply our results to give a simple proof of the result of McAlister [D. B. McAlister, ‘Embedding inverse semigroups in coset semigroups’, Semigroup Forum20 (1980), 255–267] which states that the symmetric inverse semigroup on a finite set X does not embed in the coset monoid of the symmetric group on X. We also explore examples, which are necessarily infinite, of embeddings whose images are not cofull.

The characters of the symmetric inverse semigroup

1957 ◽  
Vol 53 (1) ◽  
pp. 13-18 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Symmetric Group ◽  
Characteristic Zero ◽  
Semisimple Algebra ◽  
Matrix Representations ◽  
Natural Analogues ◽  
Completely Reducible ◽  
Symmetric Inverse Semigroup ◽  
Algebra Over A Field

There are two natural analogues of the symmetric group on n symbols in the theory of semigroups, namely, the set of all mappings of a set of n symbols into itself, and the set of all partial transformations of such a set, with the obvious definitions of multiplication. We are concerned here with the latter system. This is an inverse semigroup, and accordingly we call it the ‘symmetric inverse semigroup’. It gives rise to a semisimple algebra over a field of characteristic zero or a prime greater than n, and its matrix representations over such a field are thus completely reducible.

scholarly journals JOINS AND COVERS IN INVERSE SEMIGROUPS AND TIGHT -ALGEBRAS

2014 ◽  
Vol 90 (1) ◽  
pp. 121-133 ◽  
Author(s):  
ALLAN P. DONSIG ◽  
DAVID MILAN
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Inverse Semigroups ◽  
Natural Partial Order ◽  
Higher Rank ◽  
Special Case

AbstractWe show Exel’s tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the ${C}^{\ast } $-algebra of a finitely aligned category of paths, developed by Spielberg, is the tight ${C}^{\ast } $-algebra of a natural inverse semigroup. This includes as a special case finitely aligned higher-rank graphs: that is, for such a higher-rank graph $\Lambda $, the tight ${C}^{\ast } $-algebra of the inverse semigroup associated to $\Lambda $ is the same as the ${C}^{\ast } $-algebra of $\Lambda $.

scholarly journals Accounting for Context in Markup: Which Situation, Whose Semantics?

2015 ◽  
Author(s):  
Karen M. Wickett
Keyword(s):  
General Theory ◽  
Natural Language ◽  
Critical Issue ◽  
Situation Semantics ◽  
Theory Of Meaning ◽  
Open Questions ◽  
Full Analysis ◽  
The Right ◽  
Distinct Features

Situation semantics - as developed by Barwise and Perry - is a general theory of meaning for natural language, and can be used to understand the role of context in markup semantics. While the notion of a discourse situation provides many of the right hooks for accounting for contextual assignment of meaning to markup structures, there are still many open questions. One critical issue is that situation semantics itself is open enough to allow many different approaches to identifying the relevant discourse situation. Three core types of discourse situations for descriptive markup - documentary, transport, and discovery - lead to distinct features in the discourse situations connected to those scenarios. Beyond developing a fuller picture of the discourse situations that shape the meaning of markup, this exercise lays groundwork for the full analysis of the assignment of meaning to metadata records.

scholarly journals Inverse semigroups generated by nilpotent transformations

1984 ◽  
Vol 99 (1-2) ◽  
pp. 153-162 ◽  
Author(s):  
John M. Howie ◽  
M. Paula O. Marques-Smith
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Nilpotent Elements ◽  
Inverse Subsemigroup ◽  
Symmetric Inverse Semigroup ◽  
Infinite Cardinality ◽  
Index 2 ◽  
Rees Quotient ◽  
The Ideal

SynopsisLet X be a set with infinite cardinality m and let B be the Baer-Levi semigroup, consisting of all one-one mappings a:X→X for which ∣X/Xα∣ = m. Let Km=<B 1B>, the inverse subsemigroup of the symmetric inverse semigroup ℐ(X) generated by all products β−γ, with β,γ∈B. Then Km = <N2>, where N2 is the subset of ℐ(X) consisting of all nilpotent elements of index 2. Moreover, Km has 2-nilpotent-depth 3, in the sense that Let Pm be the ideal {α∈Km: ∣dom α∣<m} in Km and let Lm be the Rees quotient Km/Pm. Then Lm is a 0-bisimple, 2-nilpotent-generated inverse semigroup with 2-nilpotent-depth 3. The minimum non-trivial homomorphic image of Lm also has these properties and is congruence-free.

scholarly journals COUNTABLE VERSUS UNCOUNTABLE RANKS IN INFINITE SEMIGROUPS OF TRANSFORMATIONS AND RELATIONS

2003 ◽  
Vol 46 (3) ◽  
pp. 531-544 ◽  
Author(s):  
P. M. Higgins ◽  
J. M. Howie ◽  
J. D. Mitchell ◽  
N. Ruškuc
Keyword(s):  
Inverse Semigroup ◽  
Symmetric Group ◽  
Transformation Semigroup ◽  
Countable Subset ◽  
Binary Relations ◽  
Minimum Cardinality ◽  
Full Transformation ◽  
Relative Rank ◽  
Primary 20M20 ◽  
Symmetric Inverse Semigroup

AbstractThe relative rank $\rank(S:A)$ of a subset $A$ of a semigroup $S$ is the minimum cardinality of a set $B$ such that $\langle A\cup B\rangle=S$. It follows from a result of Sierpiński that, if $X$ is infinite, the relative rank of a subset of the full transformation semigroup $\mathcal{T}_{X}$ is either uncountable or at most $2$. A similar result holds for the semigroup $\mathcal{B}_{X}$ of binary relations on $X$.A subset $S$ of $\mathcal{T}_{\mathbb{N}}$ is dominated (by $U$) if there exists a countable subset $U$ of $\mathcal{T}_{\mathbb{N}}$ with the property that for each $\sigma$ in $S$ there exists $\mu$ in $U$ such that $i\sigma\le i\mu$ for all $i$ in $\mathbb{N}$. It is shown that every dominated subset of $\mathcal{T}_{\mathbb{N}}$ is of uncountable relative rank. As a consequence, the monoid of all contractions in $\mathcal{T}_{\mathbb{N}}$ (mappings $\alpha$ with the property that $|i\alpha-j\alpha|\le|i-j|$ for all $i$ and $j$) is of uncountable relative rank.It is shown (among other results) that $\rank(\mathcal{B}_{X}:\mathcal{T}_{X})=1$ and that $\rank(\mathcal{B}_{X}:\mathcal{I}_{X})=1$ (where $\mathcal{I}_{X}$ is the symmetric inverse semigroup on $X$). By contrast, if $\mathcal{S}_{X}$ is the symmetric group, $\rank(\mathcal{B}_{X}:\mathcal{S}_{X})=2$.AMS 2000 Mathematics subject classification: Primary 20M20

scholarly journals Inverse semigroups generated by linear transformations

2005 ◽  
Vol 71 (2) ◽  
pp. 205-213 ◽  
Author(s):  
Suzana Mendes-Gonçalves ◽  
R. P. Sullivan
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Vector Spaces ◽  
Natural Question ◽  
Linear Transformations ◽  
Symmetric Inverse Semigroup

Suppose X is a set with |X| = p ≥ q ≥ ℵ0 and let B = BL(p, q) denote the Baer-Levi semigroup defined on X. In 1984, Howie and Marques-Smith showed that, if p = q, then BB−1 = I(X), the symmetric inverse semigroup on X, and they described the subsemigroup of I(X) generated by B−1B. In 1994, Lima extended that work to ‘independence algebras’, and thus also to vector spaces. In this paper, we answer the natural question: what happens when p > q? We also show that, in this case, the analogues BB−1 for sets and GG−1 for vector spaces are never isomorphic, despite their apparent similarities.

scholarly journals Epidemic, Endemic, or Stewart–Bluefarb? When Several Forms of Kaposi Seem to Dispute Paternity

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Hugues Adegbidi ◽  
Bérénice Dégboé ◽  
Fabrice Akpadjan ◽  
Nadège Agbessi-Mekoun ◽  
Christiane Koudoukpo ◽  
...  
Keyword(s):  
Central Africa ◽  
Road Accident ◽  
Biological Assessment ◽  
Human Herpes Virus 8 ◽  
Noticeable Improvement ◽  
Local Trauma ◽  
History Of ◽  
The Right ◽  
Special Case

The role of human herpes virus 8 (HHV8) is demonstrated in the occurrence of Kaposi’s disease, but the role of cofactors is still hardly known. We report a case of Kaposi’s disease which occurred 10 years after a local trauma in an HIV-positive patient from Central Africa. A 38-year-old female, from and living in Central Africa, consulted for angiomatous papulo-nodules associated with purple-colored macules and painful lymphoedema of the right leg and foot that had been developing for 6 months. She reported a history of posttraumatic lymphoedema of the affected limb as a result of a road accident that occurred ten years earlier. The mucous were healthy. There was no sign of systemic lesions. The diagnosis of Kaposi’s disease was evoked with, in differential, a Stewart–Bluefarb syndrome-type of pseudo-Kaposi and an epidemic Kaposi disease. Retroviral serology was positive to HIV1 with a CD4 count of 600 cells/mm3. Histopathology of the lesions and duplex ultrasonography could not be performed. The rest of the biological assessment was without particularity. The diagnosis of epidemic Kaposi’s disease associated with cofactors involved in endemic Kaposi’s disease and Stewart–Bluefarb syndrome was retained. An antiretroviral treatment (emtricitabine, tenofovir, and efavirenz) allowed to obtain after 6 months a noticeable improvement of the lesions and a disappearance of the pain with however the persistence of a residual lymphoedema. This is a special case of Kaposi’s disease that seems to involve several factors. The role of cofactors in Kaposi’s disease remains to be elucidated.

scholarly journals QUANTIZATION OF A LOCALLY SUPERSYMMETRIC FRIEDMANN MODEL IN THE PRESENCE OF SUPERMATTER

1995 ◽  
Vol 04 (02) ◽  
pp. 189-206 ◽  
Author(s):  
A.D.Y. CHENG ◽  
P.R.L.V. MONIZ
Keyword(s):  
General Theory ◽  
Simple Form ◽  
Quantum Cosmology ◽  
Scalar Fields ◽  
Quantum States ◽  
Spherically Symmetric ◽  
Two Dimensional ◽  
Complex Scalar ◽  
Special Case

The general theory of N=1 supergravity with supermatter is restricted to the special case of a k=+1 Friedmann minisuperspace model with complex scalar fields and fermionic partners. The different supermatter models are given by specifying a Kähler metric for the scalar fields. For the two-dimensional spherically symmetric and flat Kähler geometries new solutions for the quantum states are found. These are shown to have simple form. In particular, although they permit a Hartle-Hawking solution, they do not allow a wormhole state. The role of these solutions with regard to more general issues in supersymmetric quantum cosmology is also briefly discussed.

Combinatorially Factorizable Cryptic Inverse Semigroups

2014 ◽  
Vol 57 (3) ◽  
pp. 621-630
Author(s):  
Mario Petrich
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Inverse Monoid ◽  
Direct Products ◽  
Inverse Subsemigroup ◽  
Equality Relation ◽  
Special Case

AbstractAn inverse semigroup S is combinatorially factorizable if S = TG where T is a combinatorial (i.e., 𝓗 is the equality relation) inverse subsemigroup of S and G is a subgroup of S. This concept was introduced and studied byMills, especially in the case when S is cryptic (i.e., 𝓗 is a congruence on S). Her approach is mainly analytical considering subsemigroups of a cryptic inverse semigroup.We start with a combinatorial inverse monoid and a factorizable Clifford monoid and from an action of the former on the latter construct the semigroups in the title. As a special case, we consider semigroups that are direct products of a combinatorial inverse monoid and a group.

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