scholarly journals Natural Partial Orders on Transformation Semigroups with Fixed Sets

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Yanisa Chaiya ◽  
Preeyanuch Honyam ◽  
Jintana Sanwong
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Natural Partial Order ◽  
Transformation Semigroups ◽  
Maximal Elements ◽  
Fixed Subset ◽  
Regular Monoid

LetXbe a nonempty set. For a fixed subsetYofX, letFixX,Ybe the set of all self-maps onXwhich fix all elements inY. ThenFixX,Yis a regular monoid under the composition of maps. In this paper, we characterize the natural partial order onFix(X,Y)and this result extends the result due to Kowol and Mitsch. Further, we find elements which are compatible and describe minimal and maximal elements.

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 PARTIAL ORDERS ON PARTIAL BAER–LEVI SEMIGROUPS

2010 ◽  
Vol 81 (2) ◽  
pp. 195-207 ◽  
Author(s):  
BOORAPA SINGHA ◽  
JINTANA SANWONG ◽  
R. P. SULLIVAN
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Partial Orders ◽  
The Other ◽  
Natural Order ◽  
Natural Partial Order ◽  
Transformation Semigroups ◽  
Minimal Elements ◽  
Symmetric Inverse Semigroup ◽  
Containment Order

AbstractMarques-Smith and Sullivan [‘Partial orders on transformation semigroups’, Monatsh. Math.140 (2003), 103–118] studied various properties of two partial orders on P(X), the semigroup (under composition) consisting of all partial transformations of an arbitrary set X. One partial order was the ‘containment order’: namely, if α,β∈P(X) then α⊆β means xα=xβ for all x∈dom α, the domain of α. The other order was the so-called ‘natural order’ defined by Mitsch [‘A natural partial order for semigroups’, Proc. Amer. Math. Soc.97(3) (1986), 384–388] for any semigroup. In this paper, we consider these and other orders defined on the symmetric inverse semigroup I(X) and the partial Baer–Levi semigroup PS(q). We show that there are surprising differences between the orders on these semigroups, concerned with their compatibility with respect to composition and the existence of maximal and minimal elements.

scholarly journals PARTIAL ORDERS ON SEMIGROUPS OF PARTIAL TRANSFORMATIONS WITH RESTRICTED RANGE

2012 ◽  
Vol 86 (1) ◽  
pp. 100-118 ◽  
Author(s):  
KRITSADA SANGKHANAN ◽  
JINTANA SANWONG
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Natural Partial Order ◽  
Restricted Range ◽  
Maximal Elements

AbstractLet X be any set and P(X) the set of all partial transformations defined on X, that is, all functions α:A→B where A,B are subsets of X. Then P(X) is a semigroup under composition. Let Y be a subset of X. Recently, Fernandes and Sanwong defined PT(X,Y )={α∈P(X):Xα⊆Y } and defined I(X,Y ) to be the set of all injective transformations in PT(X,Y ) . Hence PT(X,Y ) and I(X,Y ) are subsemigroups of P(X) . In this paper, we study properties of the so-called natural partial order ≤ on PT(X,Y ) and I(X,Y ) in terms of domains, images and kernels, compare ≤ with the subset order, characterise the meet and join of these two orders, then find elements of PT(X,Y ) and I(X,Y ) which are compatible. Also, the minimal and maximal elements are described.

THE NATURAL PARTIAL ORDER ON LINEAR SEMIGROUPS WITH NULLITY AND CO-RANK BOUNDED BELOW

2014 ◽  
Vol 91 (1) ◽  
pp. 104-115 ◽  
Author(s):  
SUREEPORN CHAOPRAKNOI ◽  
TEERAPHONG PHONGPATTANACHAROEN ◽  
PONGSAN PRAKITSRI
Keyword(s):  
Semigroup Forum ◽  
Partial Order ◽  
Lower Bounds ◽  
Partial Orders ◽  
Natural Partial Order ◽  
Linear Transformations ◽  
Bounded Below

AbstractHiggins [‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.

A PARTIAL ORDER ON TRANSFORMATION SEMIGROUPS THAT PRESERVE DOUBLE DIRECTION EQUIVALENCE RELATION

2013 ◽  
Vol 12 (08) ◽  
pp. 1350041 ◽  
Author(s):  
LEI SUN ◽  
JUNLING SUN
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Natural Partial Order ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Minimal Elements ◽  
Full Transformation Semigroup

Let [Formula: see text] be the full transformation semigroup on a nonempty set X and E be an equivalence relation on X. Then [Formula: see text] is a subsemigroup of [Formula: see text]. In this paper, we endow it with the natural partial order. With respect to this partial order, we determine when two elements are related, find the elements which are compatible and describe the maximal (minimal) elements. Also, we investigate the greatest lower bound of two elements.

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

scholarly journals THE NATURAL PARTIAL ORDER ON THE SEMIGROUP OF ALL TRANSFORMATIONS OF A SET THAT REFLECT AN EQUIVALENCE RELATION

2013 ◽  
Vol 88 (3) ◽  
pp. 359-368
Author(s):  
LEI SUN ◽  
XIANGJUN XIN
Keyword(s):  
Partial Order ◽  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Natural Partial Order ◽  
Maximal Elements ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Image Position

AbstractLet ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote $$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$ so that ${T}_{\exists } (X)$ is a subsemigroup of ${ \mathcal{T} }_{X} $. In this paper, we endow ${T}_{\exists } (X)$ with the natural partial order and investigate when two elements are related, then find elements which are compatible. Also, we characterise the minimal and maximal elements.

scholarly journals NATURAL PARTIAL ORDER IN SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET

2012 ◽  
Vol 87 (1) ◽  
pp. 94-107 ◽  
Author(s):  
LEI SUN ◽  
LIMIN WANG
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Nonempty Subset ◽  
Transformation Semigroup ◽  
Invariant Set ◽  
Natural Partial Order ◽  
Maximal Elements ◽  
Full Transformation ◽  
Minimal Elements ◽  
Image Position

AbstractLet 𝒯X be the full transformation semigroup on the nonempty set X. We fix a nonempty subset Y of X and consider the semigroup of transformations that leave Y invariant, and endow it with the so-called natural partial order. Under this partial order, we determine when two elements of S(X,Y ) are related, find the elements which are compatible and describe the maximal elements, the minimal elements and the greatest lower bound of two elements. Also, we show that the semigroup S(X,Y ) is abundant.

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