Semigroup Of Partial Transformations Research Articles

A Note on Coregular Elements in Certain Semigroups of Transformations

For a non-empty set X, let P(X) represent the partial transformation semigroup on X. For a non-empty subset Y of X, define as the semigroup . Then is a generalization of P(X) , encompassing all partial transformations on that maintain Y as an invariant set. This paper delves into exploring the characterization of coregular elements within and applies the results to obtain a similar characterization in relevant semigroups.

A note on intra regularity on semigroups of partial transformations with invariant set

A note on intra regularity on semigroups of partial transformations with invariant set

Green’s relations and natural partial order on Baer-Levi semigroups of partial transformations with restricted range

Green’s relations and natural partial order on Baer-Levi semigroups of partial transformations with restricted range

Regularity and abundance on semigroups of partial transformations with invariant set

Abstract Let P ( X ) P\left(X) be a partial transformation semigroup on a non-empty set X X . For a fixed non-empty subset Y Y of X X , let P T ¯ ( X , Y ) = { α ∈ P ( X ) ∣ ( dom α ∩ Y ) α ⊆ Y } . \overline{PT}\left(X,Y)=\left\{\alpha \in P\left(X)| \left({\rm{dom}}\hspace{0.33em}\alpha \cap Y)\alpha \subseteq Y\right\}. Then, P T ¯ ( X , Y ) \overline{PT}\left(X,Y) consists of all the mapping in P ( X ) P\left(X) that leave Y ⊆ X Y\subseteq X as an invariant. It is a generalization of P ( X ) P\left(X) since P T ¯ ( X , X ) = P ( X ) \overline{PT}\left(X,X)=P\left(X) . In this article, we present the necessary and sufficient conditions for elements of P T ¯ ( X , Y ) \overline{PT}\left(X,Y) to be regular, left regular, and right regular. The results are used to describe the relationships between these elements and determine their number when X X is a finite set. Moreover, we show that P T ¯ ( X , Y ) \overline{PT}\left(X,Y) is always abundant.

Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chain

Let [n]={1,2,…,n} be a finite chain and let Pn be the semigroup of partial transformations on [n]. Let CPn={α∈Pn:(for allx,y∈Domα)|xα−yα|≤|x−y|} be the semigroup of partial contractions on [n]. Then CPn is a subsemigroup of Pn. In this paper, in the second and third sections we give a necessary and sufficient condition for an element in CPn to be regular and characterize all the Green’s relations in the semigroup CPn, respectively. In the subsequent fourth and fifth sections we obtain analogous results for the subsemigroups of order preserving and/or order reversing contractions, and that of full contractions, respectively. Finally, in the last section, we characterize all the maximal subgroups of these semigroups.

A Note About the Abundance of Partial Transformation Semigroups with Fixed Point Sets

A Note About the Abundance of Partial Transformation Semigroups with Fixed Point Sets

Green’s relations and regularity on semigroups of partial transformations with fixed sets

Given a non-empty set X and let P(X) be a partial transformation semigroup on X. For a fixed subset Y of X, define a generalization of P(X) by In this article, we give a complete description of Green’s relations on PFix(X, Y) and apply this to obtain the characterizations of left regular, right regular, intra-regular and completely regular elements of PFix(X, Y). The number and the relationship between these elements are also investigated by using finiteness of X and

Counting of Conjugacy classes in partial transformation semigroup.

J.Koneiczny in [8] introduced the new notion $sim_{n}$ notion of conjugacy in semigroups. In this paper, we count the number of conjugacy classes in Partial Transformation semigroup $mathcal{P}(A)$ for an infinite set $A$ with respect to $sim_{n}$ notion of conjugacy.

On the extended zero-divisor graph of strictly partial transformation semigroup

Given a commutative ring $R$, the zero-divisor graph of $R$ is an undirected simple graph with vertices the nonzero zero-divisors of $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. In [8], Redmond presented different versions of zero-divisor graphs of noncommutative rings. The main aim of this paper is to analyse these graphs for the semigroup $\mathcal{SP}_{n}$ of all strictly partial transformations on the set $X_{n}=\{1,2,\dots,n\}$.

Skew Pairs of Idempotents in Partial Transformation Semigroups

Let S be a regular semigroup. A pair (e,f) of idempotents of S is said to be a skew pair of idempotents if fe is idempotent, but ef is not. T. S. Blyth and M. H. Almeida (T. S. Blyth and M. H. Almeida, skew pair of idempotents in transformation semigroups, Acta Math. Sin. (English Series), 22 (2006), 1705–1714) gave a characterization of four types of skew pairs—those that are strong, left regular, right regular, and discrete—existing in a full transformation semigroup T(X). In this paper, we do in this line for partial transformation semigroups.

Zero-divisor graphs of partial transformation semigroups

Let $\mathcal P_{n}$ be the partial transformation semigroup on $X_{n}=\{1,2,\ldots ,n\}$. In this paper, we find the left zero-divisors, right zero-divisors and two sided zero-divisors of $\mathcal P_{n}$, and their numbers. For $n \geq 3$, we define an undirected graph $\Gamma(\mathcal P_{n})$ associated with $\mathcal P_{n}$ whose vertices are the two sided zero-divisors of $\mathcal P_{n}$ excluding the zero element $\theta$ of $\mathcal P_{n}$ with distinct two vertices $\alpha$ and $\beta$ joined by an edge in case $\alpha\beta=\theta =\beta\alpha$. First, we prove that $\Gamma(\mathcal P_{n})$ is a connected graph, and find the diameter, girth, domination number and the degrees of the all vertices of $\Gamma(\mathcal P_{n})$. Furthermore, we give lower bounds for clique number and chromatic number of $\Gamma(\mathcal P_{n})$.

NUMERICAL AND GRAPHICAL RESULTS OF FINITE SYMMETRIC INVERSE (I_{n}) AND FULL (T_{n}) TRANSFORMATION SEMIGROUPS

Supposed is a finite set, then a function is called a finite partial transformation semigroup , which moves elements of from its domain to its co-domain by a distance of where . The total work done by the function is therefore the sum of these distances. It is a known fact that and . In this this research paper, we have mainly presented the numerical solutions of the total work done, the average work done by functions on the finite symmetric inverse semigroup of degree , and the finite full transformation semigroup of degree , as well as their respective powers for a given fixed time in space. We used an effective methodology and valid combinatorial results to generalize the total work done, the average work done and powers of each of the transformation semigroups. The generalized results were tested by substituting small values of and a specified fixed times in space. Graphs were plotted in each case to illustrate the nature of the total work done and the average work done. The results obtained in this research article have an important application in some branch of physics and theoretical computer science

On the large rank of certain semigroups of transformations on a finite chain

The large rank of a finite semigroup [Formula: see text] is the least number [Formula: see text] such that every subset of [Formula: see text] with [Formula: see text] elements generates [Formula: see text]. This paper obtains the large ranks of [Formula: see text] and [Formula: see text], the semigroups of singular transformations, injective partial and partial transformations on a finite chain [Formula: see text], which preserve or reverse the order, respectively. As a consequence, we obtain the large ranks of [Formula: see text] and [Formula: see text], the semigroups of injective order-preserving partial transformations and order-preserving partial transformations on [Formula: see text], respectively.

Left and right magnifying elements in some generalized partial transformation semigroups

An element a of a semigroup S is called left [right] magnifying if there exists a proper subset M of S such that S = aM [S = Ma]. Let P(X) be a semigroup of all partial transformations on a set X under the composition of maps. A number of results concerning the necessary and sufficient conditions for elements in some interesting generalized semigroups of partial transformations to be left or right magnifiers are presented.

Fixed points of order-preserving transformations

Let [Formula: see text] be the semigroup of order-preserving partial transformations on an [Formula: see text]-chain. We establish several results for the number [Formula: see text] of elements of [Formula: see text] with [Formula: see text] fixed points, including recurrence relations and generating functions. We also prove that the probability that, for large [Formula: see text], a randomly chosen map in [Formula: see text] has exactly [Formula: see text] fixed points is asymptotically [Formula: see text].

Natural partial order on semigroup partial linear transformation with restricted range

The set with assosiative binary operations is semigroup. Of all set partial linear transformations with a composition function is semigroup which is called a partial linear transformation semigroup that is denoted by (PT (X), o). If Y is a subspace of X, with all images of PT (X) a subset of Y, so the semigroup which is formed is a semigroup of partial linear transformation with restricted range that is denoted by (PT(X,Y),o). A natural partial order is a relation which defined by a ≤ b if only if a = xb = by, xa = a for some x,y ∈PT(X,Y) in a semigroup partial linear transformation with restricted range. In this article using the literature study method, it discusses the characterization of natural partial order on partial linear transformations semigroups with restricted range resulting neccesary and sufficient condition for a natural partial order in semigroup partial linear transformation with restricted range.

A transversal property for permutation groups motivated by partial transformations

In this paper we introduce the definition of the (k,l)-universal transversal property for permutation groups, which is a refinement of the definition of k-universal transversal property, which in turn is a refinement of the classical definition of k-homogeneity for permutation groups. In particular, a group possesses the (2,n)-universal transversal property if and only if it is primitive; it possesses the (2,2)-universal transversal property if and only if it is 2-homogeneous. Up to a few undecided cases, we give a classification of groups satisfying the (k,l)-universal transversal property, for k≥3. Then we apply this result for studying regular semigroups of partial transformations.

On certain semigroups of contraction mappings of a finite chain

Let[n] ={1,2, . . . , n} be a finite chain and let Pn (resp.,Tn) be the semigroup of partial transformations on[n] (resp., full transformations on[n]). Let CPn={α∈ Pn: (for allx, y ∈ Dom α)|xα−yα|⩽|x−y|}(resp., CTn={α∈ Tn: (for allx, y∈[n])|xα−yα|⩽|x−y|}) be the subsemigroup of partial contractionmappings on[n](resp., subsemigroup of full contraction mappingson[n]). We characterize all the starred Green’s relations on C Pn and it subsemigroup of order preserving and/or order reversingand subsemigroup of order preserving partial contractions on[n], respectively. We show that the semigroups CPn and CTn, and some of their subsemigroups are left abundant semigroups for all n but not right abundant forn⩾4. We further show that the set ofregular elements of the semigroup CTn and its subsemigroup of order preserving or order reversing full contractions on[n], each formsa regular subsemigroup and an orthodox semigroup, respectively.

Multiplicative invertibility characterization on star-like cyclicpoid CyPω*n finite partial transformation semigroups

Multiplicative invertibility characterization on star-like cyclicpoid CyPω*n finite partial transformation semigroups

REGULARITAS DAN RELASI GREEN PADA SEMIGRUP TRANSFORMASI LINEAR PARSIAL INJEKTIF DENGAN RESTRIKSI RANGE

Injective partial linear transformations in the vector space V and the function composition operations form semigroups called semigroups of injective partial linear transformations denoted by I (V). Suppose Z is a subspace in a vector space V, then the set of all members of I (V) whose function results are contained in Z to the function composition operation is also a semigroup denoted by I (V, Z) which is called a semigroup of injectable partial linear transformations with range restrictions. . In this article, we will examine how the characteristics of regular elements and the characteristics of two set members in I (V, Z) are related to Green using the literature study method. The results of the research in this article are the necessary and sufficient conditions needed so that any element is called a regular element and the characteristics of the two members of the set I (V, Z) are related.