scholarly journals Largest subsemigroups of the full transformation monoid

2008 ◽  
Vol 308 (20) ◽  
pp. 4801-4810 ◽  
Author(s):  
R. Gray ◽  
J.D. Mitchell
Keyword(s):  
Full Transformation ◽  
Transformation Monoid

scholarly journals An extension of properties of symmetric group to monoids and a pretorsion theory in a category of mappings

2019 ◽  
Vol 18 (12) ◽  
pp. 1950234 ◽  
Author(s):  
Alberto Facchini ◽  
Leila Heidari Zadeh
Keyword(s):  
Symmetric Group ◽  
Full Transformation ◽  
Transformation Monoid ◽  
Finite Set ◽  
Torsion Part

Several elementary properties of the symmetric group [Formula: see text] extend in a nice way to the full transformation monoid [Formula: see text] of all maps of the set [Formula: see text] into itself. The group [Formula: see text] turns out to be the torsion part of the monoid [Formula: see text]. That is, there is a pretorsion theory in the category of all maps [Formula: see text], [Formula: see text] an arbitrary finite set, in which bijections are exactly the torsion objects.

scholarly journals Centralising Monoids with Low-Arity Witnesses on a Four-Element Set

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1471
Author(s):  
Mike Behrisch ◽  
Edith Vargas-García
Keyword(s):  
The Other ◽  
Full Transformation ◽  
Group Operation ◽  
Binary Operations ◽  
Other Hand ◽  
Transformation Monoid ◽  
Element Set

As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Maľcev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal.

scholarly journals HALL'S CONDITION AND IDEMPOTENT RANK OF IDEALS OF ENDOMORPHISM MONOIDS

2008 ◽  
Vol 51 (1) ◽  
pp. 57-72 ◽  
Author(s):  
R. Gray
Keyword(s):  
Simple Semigroup ◽  
Analogous Result ◽  
Direct Consequence ◽  
Full Transformation ◽  
Generating Set ◽  
Endomorphism Monoids ◽  
Idempotent Rank ◽  
Transformation Monoid ◽  
Hall's Condition ◽  
Property Holding

AbstractIn 1990, Howie and McFadden showed that every proper two-sided ideal of the full transformation monoid $T_n$, the set of all maps from an $n$-set to itself under composition, has a generating set, consisting of idempotents, that is no larger than any other generating set. This fact is a direct consequence of the same property holding in an associated finite $0$-simple semigroup. We show a correspondence between finite $0$-simple semigroups that have this property and bipartite graphs that satisfy a condition that is similar to, but slightly stronger than, Hall's condition. The results are applied in order to recover the above result for the full transformation monoid and to prove the analogous result for the proper two-sided ideals of the monoid of endomorphisms of a finite vector space.

scholarly journals On the Monoid of Unital Endomorphisms of a Boolean Ring

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 305
Author(s):  
Bana Al Al Subaiei ◽  
Noômen Jarboui
Keyword(s):  
Automorphism Group ◽  
Explicit Description ◽  
Boolean Ring ◽  
Full Transformation ◽  
Unital Ring ◽  
Transformation Monoid ◽  
Power Set

Let X be a nonempty set and P(X) the power set of X. The aim of this paper is to provide an explicit description of the monoid End1P(X)(P(X)) of unital ring endomorphisms of the Boolean ring P(X) and the automorphism group Aut(P(X)) when X is finite. Among other facts, it is shown that if X has cardinality n≥1, then End1P(X)(P(X))≅Tnop, where Tn is the full transformation monoid on the set X and Aut(P(X))≅Sn.

scholarly journals Nonrandom cytogenetic changes accompany malignant progression in clonal lines abelson virus-infected lymphocytes

Blood ◽  
1994 ◽  
Vol 84 (12) ◽  
pp. 4301-4309 ◽  
Author(s):  
SS Clark ◽  
Y Liang ◽  
CK Reedstrom ◽  
SQ Wu
Keyword(s):  
Tumor Progression ◽  
Murine Leukemia Virus ◽  
Leukemia Virus ◽  
Lymphoid Cells ◽  
Full Transformation ◽  
Cytogenetic Abnormalities ◽  
Cytogenetic Changes ◽  
Abelson Murine Leukemia Virus ◽  
Murine Leukemia ◽  
Over Time

Initially, lymphoid cells transformed by v-abl or BCR/ABL oncogenes are poorly oncogenic but progress to full transformation over time. Although expression of the oncogene is necessary to initiate and maintain transformation, other molecular mechanisms are thought to be required for full transformation. To determine whether tumor progression in ABL oncogene-transformed lymphoid cells has a genetic basis, we examined whether progression of the malignant phenotype of transformed clones correlates with particular cytogenetic abnormalities. A modified in vitro bone marrow transformation model was used to obtain clonal Abelson murine leukemia virus-transformed B lymphoid cells that were poorly oncogenic. Multiple subclones were then derived from each clone and maintained over a marrow-derived stromal cell line for several weeks. Over time, clonally related Abelson murine leukemia virus-transformed subclones progressed asynchronously to full transformation. The data show that tumor progression can occur in the absence of detectable cytogenetic changes but, more importantly, that certain cytogenetic abnormalities appear reproducibly in highly malignant subclones. Therefore, three independent subclones showed deletion in a common region of chromosome 13. Other highly malignant cells carried a common breakpoint in the X chromosome, and, finally, two subclones carried an additional chromosome 5. These results are consistent with the hypothesis that ABL oncogenes are sufficient for the initial transformation of cells but that additional genetic events can drive oncogenic progression. These observations further suggest that diverse genetic mechanisms may be able to drive tumor progression in cells transformed with ABL oncogenes.

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