scholarly journals Maximum idempotents in naturally ordered regular semigroups

1983 ◽  
Vol 26 (2) ◽  
pp. 213-220 ◽  
Author(s):  
D. B. McAlister ◽  
R. McFadden
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Natural Partial Order ◽  
Ordered Semigroup ◽  
Regular Semigroups ◽  
Partially Ordered ◽  
Image Position ◽  
Reverse Implication

We shall denote by ω the natural partial order on the idempotents E = E(S) of a regular semigroup S, so that in E,A partially ordered semigroup S(≦) is called naturally partially ordered [9] if the imposed partial order ≦ extends ω in the sense thatNo assumption is made about the reverse implication.

scholarly journals Regular semigroups, fundamental semigroups and groups

1980 ◽  
Vol 29 (4) ◽  
pp. 475-503 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Direct Product ◽  
Homomorphic Image ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Natural Partial Order ◽  
Regular Semigroups ◽  
Necessary And Sufficient ◽  
Fundamental Semigroups

AbstractIn this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ: S → T between regular semigroups. This is a mapping such that (ab) θ ≦ aθ bθ in the natural partial order on T.

scholarly journals Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups

1981 ◽  
Vol 31 (3) ◽  
pp. 325-336 ◽  
Author(s):  
D. B. Mcalister
Keyword(s):  
Partial Order ◽  
Local Structure ◽  
Structure Theorem ◽  
Main Tool ◽  
Natural Partial Order ◽  
Ordered Semigroup ◽  
Ordered Group ◽  
Partially Ordered ◽  
Rees Matrix Semigroups ◽  
Matrix Semigroups

AbstractA partially ordered semigroup S is said to be a Dubreil-Jacotin semigroup if there is an isotone homomorphism θ of S onto a partially ordered group such that {} has a greatest member. In this paper we investigate the structure of regular Dubreil-Jacotin semigroups in which the imposed partial order extends the natural partial order on the idempotents. The main tool used is a local structure theorem which is introduced in Section 2. This local structure theorem applies to many other contexts as well.

Partial orders on linear transformation semigroups

2005 ◽  
Vol 135 (2) ◽  
pp. 413-437 ◽  
Author(s):  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Regular Semigroup ◽  
Partial Order ◽  
Linear Transformation ◽  
Natural Partial Order ◽  
Transformation Semigroups ◽  
Linear Transformations ◽  
Partially Ordered ◽  
Partial Linear ◽  
A Chain

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear α: A → B, where A, B are subspaces of V. Then P(V) is a semigroup under composition, which is partially ordered by ⊆ (that is, α ⊆ β if and only if dom α ⊆ dom β and α = β | dom α). We compare this order with the so-called 'natural partial order' ≤ on P(V) and we determine their meet and join. We also describe all elements of P(V) that are minimal (or maximal) with respect to each of these four orders, and we characterize all elements that are 'compatible' with them. In addition, we answer similar questions for the semigroup T(V) consisting of all α ∈ P(V) whose domain equals V. Other orders have been defined by Petrich on any regular semigroup: three of them form a chain below ≤, and we show that two of these are equal on the semigroup P(V) and on the ring T(V). We also consider questions for these orders that are similar to those already mentioned

Embedding Regular Semigroups into Idempotent Generated Ones

2010 ◽  
Vol 17 (02) ◽  
pp. 229-240 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Natural Partial Order ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Close Relationship ◽  
Completely Regular Semigroups ◽  
Relationship Of ◽  
And Inversion ◽  
The Relationship

Any semigroup S can be embedded into a semigroup, denoted by ΨS, having some remarkable properties. For general semigroups there is a close relationship between local submonoids of S and of ΨS. For a number of usual semigroup properties [Formula: see text], we prove that S and ΨS simultaneously satisfy [Formula: see text] or not. For a regular semigroup S, the relationship of S and ΨS is even closer, especially regarding the natural partial order and Green's relations; in addition, every element of ΨS is a product of at most four idempotents. For completely regular semigroups S, the relationship of S and ΨS is still closer. On the lattice [Formula: see text] of varieties of completely regular semigroups [Formula: see text] regarded as algebras with multiplication and inversion, by means of ΨS, we define an operator, denoted by Ψ. We compare Ψ with some of the standard operators on [Formula: see text] and evaluate it on a small sublattice of [Formula: see text].

scholarly journals The natural partial order on a regular semigroup

1980 ◽  
Vol 23 (3) ◽  
pp. 249-260 ◽  
Author(s):  
K. S. Subramonian Nambooripad
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Partial Order ◽  
Inverse Semigroups ◽  
Natural Partial Order ◽  
Simple Description ◽  
Regular Semigroups ◽  
Simple Congruence ◽  
Pseudo Inverse ◽  
Completely Simple

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.

scholarly journals Primitive idempotent measures on compact semitopological semigroups

1972 ◽  
Vol 13 (4) ◽  
pp. 451-455 ◽  
Author(s):  
Stephen T. L. Choy
Keyword(s):  
Partial Order ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Zero Element ◽  
Primitive Idempotent ◽  
Natural Partial Order ◽  
Partially Ordered

For a semigroup S let I(S) be the set of idempotents in S. A natural partial order of I(S) is defined by e ≦ f if ef = fe = e. An element e in I(S) is called a primitive idempotent if e is a minimal non-zero element of the partially ordered set (I(S), ≦). It is easy to see that an idempotent e in S is primitive if and only if, for any idempotent f in S, f = ef = fe implies f = e or f is the zero element of S. One may also easily verify that an idempotent e is primitive if and only if the only idempotents in eSe are e and the zero element. We let П(S) denote the set of primitive idempotent in S.

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.

THE NATURAL PARTIAL ORDER ON SOME TRANSFORMATION SEMIGROUPS

2013 ◽  
Vol 89 (2) ◽  
pp. 279-292 ◽  
Author(s):  
SUREEPORN CHAOPRAKNOI ◽  
TEERAPHONG PHONGPATTANACHAROEN ◽  
PATTANACHAI RAWIWAN
Keyword(s):  
Partial Order ◽  
Partially Ordered Sets ◽  
Natural Partial Order ◽  
Transformation Semigroups ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Partially Ordered

AbstractFor a semigroup $S$, let ${S}^{1} $ be the semigroup obtained from $S$ by adding a new symbol 1 as its identity if $S$ has no identity; otherwise let ${S}^{1} = S$. Mitsch defined the natural partial order $\leqslant $ on a semigroup $S$ as follows: for $a, b\in S$, $a\leqslant b$ if and only if $a= xb= by$ and $a= ay$ for some $x, y\in {S}^{1} $. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements.

scholarly journals Naturally ordered bands

1967 ◽  
Vol 8 (1) ◽  
pp. 55-58 ◽  
Author(s):  
J. M. Howie
Keyword(s):  
Partial Order ◽  
Order Relation ◽  
Natural Partial Order ◽  
Partial Order Relation ◽  
Image Position

In the terminology of Clifford and Preston [2], a band B is a semigroup in which every element is idempotent. On such a semigroup there is a natural (partial) order relation defined by the ruleIf the order relation ≧ is compatible with the multiplication in B, in the sense that e ≧ f and g ≧ h together imply that eg ≧ fh, we shall say that B is a naturally ordered band. The object of this note is to describe the structure of naturally ordered bands.

Certain Varieties and Quasivarieties of Completely Regular Semigroups

1977 ◽  
Vol 29 (6) ◽  
pp. 1171-1197 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Regular Semigroup ◽  
Unique Element ◽  
Completely Regular Semigroup ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups ◽  
Definition Of ◽  
Image Position

We adopt the following definition of a completely regular semigroup S: for every element a of S, there exists a unique element a-1 of S such that

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