scholarly journals The $O$-minimal ideal of the global of a combinatorial completely $O$-simple semigroup

1987 ◽  
Vol 63 (9) ◽  
pp. 364-366
Author(s):  
Takayuki Tamura ◽  
Kenya Yamaoka
Keyword(s):  
Simple Semigroup ◽  
Minimal Ideal

PROFINITE SEMIGROUPS, VARIETIES, EXPANSIONS AND THE STRUCTURE OF RELATIVELY FREE PROFINITE SEMIGROUPS

2001 ◽  
Vol 11 (06) ◽  
pp. 627-672 ◽  
Author(s):  
JOHN RHODES ◽  
BENJAMIN STEINBERG
Keyword(s):  
Fixed Points ◽  
Structural Properties ◽  
Simple Semigroup ◽  
Cayley Graphs ◽  
Minimal Ideal ◽  
Completely Simple Semigroup ◽  
Finite Semigroups ◽  
Finite Set ◽  
Profinite Semigroup ◽  
Completely Simple

Building on the now generally accepted thesis that profinite semigroups are important to the study of finite semigroups, this paper proposes to apply various of the techniques, already used in studying algebraic semigroups, to profinite semigroups. The goal in mind is to understand free profinite semigroups on a finite set. To do this we define profinite varieties. We then introduce expansions of profinite semigroups, giving examples of several classes of such expansions. These expansions will then be useful in studying various structural properties of relatively free profinite semigroups, since these semigroups will be fixed points of certain expansions. This study also requires a look at profinite categories, semigroupoids, and Cayley graphs, all of which we handle in turn. We also study the structure of the minimal ideal of relatively free profinite semigroups showing, in particular, that the minimal ideal of the free profinite semigroup on a finite set with more than two generators is not a relatively free profinite completely simple semigroup, as well as some generalizations to related pseudovarieties.

scholarly journals Skew compact semigroups

2003 ◽  
Vol 4 (1) ◽  
pp. 133
Author(s):  
Ralph D. Kopperman ◽  
Desmond Robbie
Keyword(s):  
Continuous Semigroup ◽  
Minimal Ideal ◽  
Compact Semigroup ◽  
Closed Ideal ◽  
Hausdorff Spaces ◽  
Compact Spaces ◽  
Compact Semigroups ◽  
Compact Topology ◽  
Hausdorff Group ◽  
Non Hausdorff Spaces

<p>Skew compact spaces are the best behaving generalization of compact Hausdorff spaces to non-Hausdorff spaces. They are those (X ; τ ) such that there is another topology τ* on X for which τ V τ* is compact and (X; τ ; τ*) is pairwise Hausdorff; under these conditions, τ uniquely determines τ *, and (X; τ*) is also skew compact. Much of the theory of compact T<sub>2</sub> semigroups extends to this wider class. We show:</p> <p>A continuous skew compact semigroup is a semigroup with skew compact topology τ, such that the semigroup operation is continuous τ<sup>2</sup>→ τ. Each of these contains a unique minimal ideal which is an upper set with respect to the specialization order.</p> <p>A skew compact semigroup which is a continuous semigroup with respect to both topologies is called a de Groot semigroup. Given one of these, we show:</p> <p>It is a compact Hausdorff group if either the operation is cancellative, or there is a unique idempotent and S<sup>2</sup> = S.</p> <p>Its topology arises from its subinvariant quasimetrics.</p> <p>Each *-closed ideal ≠ S is contained in a proper open ideal.</p>

scholarly journals On when a topologically simple semigroup is simple

1978 ◽  
Vol 26 (1) ◽  
pp. 126-128
Author(s):  
Kermit Sigmon
Keyword(s):  
Open Problem ◽  
Simple Semigroup ◽  
Infinite Order ◽  
Compact Semigroup ◽  
Subject Classification ◽  
Locally Compact ◽  
M 10 ◽  
Compact Semigroups ◽  
20 M 10

AbstractThe compact semigroups in which each topologically simple subsemigroup is simple are characterized as those in which no subgroup sontains an element of infinite order. It is also shown that a locally compact toplogically simple subsemigroup of a compact semigroup must be simple. The note closes with an open problem.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 22 A 15; secondary 20 M 10.

scholarly journals Semimodularity and bisimple ω-semigroups

1970 ◽  
Vol 17 (1) ◽  
pp. 79-81 ◽  
Author(s):  
H. E. Scheiblich
Keyword(s):  
Simple Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Primitive Idempotent ◽  
Necessary And Sufficient ◽  
Lattice Of Congruences

Let S be a completely 0-simple semigroup and let Λ(S) be the lattice of congruences on S. G. Lallement (2) has described necessary and sufficient conditions on S for Λ(S) to be modular, and has shown that Λ(S) is always semimodular . This result may be stated: If S is 0-bisimple and contains a primitive idempotent, then Λ(S) is semimodular.

scholarly journals On products of all elements of a finite semigroup

1999 ◽  
Vol 42 (3) ◽  
pp. 551-557 ◽  
Author(s):  
P. Z. Hermann ◽  
E. F. Robertson ◽  
N. Ruškuc
Keyword(s):  
Maximal Subgroup ◽  
Finite Semigroup ◽  
Minimal Ideal

Let S be a finite semigroup. Consider the set p(S) of all elements of S which can be represented as a product of all the elements of S in some order. It is shown that p(S) is contained in the minimal ideal M of S and intersects each maximal subgroup H of M in essentially the same way. The main result shows that p(S) intersects H in a union of cosets of H′.

The Arithmetic of the Quasi-Uniserial Semigroups without Zero

1971 ◽  
Vol 23 (3) ◽  
pp. 507-516 ◽  
Author(s):  
Ernst August Behrens
Keyword(s):  
Simple Semigroup ◽  
Disjoint Union ◽  
Maximal Subgroups ◽  
Ordered Semigroup ◽  
Partially Ordered ◽  
Completely Simple Semigroup ◽  
Image Position ◽  
Previous Paragraph ◽  
Completely Simple ◽  
Identity Elements

An element a in a partially ordered semigroup T is called integral ifis valid. The integral elements form a subsemigroup S of T if they exist. Two different integral idempotents e and f in T generate different one-sided ideals, because eT = fT, say, implies e = fe ⊆ f and f = ef ⊆ e.Let M be a completely simple semigroup. M is the disjoint union of its maximal subgroups [4]. Their identity elements generate the minimal one-sided ideals in M. The previous paragraph suggests the introduction of the following hypothesis on M.Hypothesis 1. Every minimal one-sided ideal in M is generated by an integral idempotent.

scholarly journals On minimal ideal triangulations of cusped hyperbolic 3‐manifolds

2020 ◽  
Vol 13 (1) ◽  
pp. 308-342
Author(s):  
William Jaco ◽  
Hyam Rubinstein ◽  
Jonathan Spreer ◽  
Stephan Tillmann
Keyword(s):  
Minimal Ideal ◽  
Ideal Triangulations

scholarly journals Soluble Lie algebras having finite-dimensional maximal subalgebras

1974 ◽  
Vol 11 (1) ◽  
pp. 145-156 ◽  
Author(s):  
Ian N. Stewart
Keyword(s):  
Lie Algebras ◽  
Minimal Ideal ◽  
Prime Characteristic ◽  
Maximal Subalgebra ◽  
Maximal Subalgebras ◽  
Infinite Dimensional ◽  
The Core ◽  
Zero Characteristic ◽  
Finite Dimensional ◽  
Split Extensions

Infinite-dimensional soluble Lie algebras can possess maximal subalgebras which are finite-dimensional. We give a fairly complete description of such algebras: over a field of prime characteristic they do not exist; over a field of zero characteristic then, modulo the core of the aforesaid maximal subalgebra, they are split extensions of an abelian minimal ideal by the maximal subalgebra. If the field is algebraically closed, or if the maximal subalgebra is supersoluble, then all finite-dimensional maximal subalgebras are conjugate under the group of automorphisms generated by exponentials of inner derivations by elements of the Fitting radical. An example is given to indicate the differences encountered in the insoluble case, and the nonexistence of group-theoretic analogues is briefly discussed.

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