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Journal articles on the topic 'Transformation semigroup'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Yusuf, Usman Mohammed, Moses Anayo Mbah, and Abimiku Alaku. "On Signed Full Transformation Semigroup of a Finite Set." FULafia Journal of Science and Technology 9, no. 1 (2025): 54–56. https://doi.org/10.62050/fjst2025.v9n1.510.

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If we define [n] = {1,2,3,...,n} and [n*] = {±1,±2,±3...,±n}. A map α: [n] → [n*] is called a signed transformation on [n]. The collection of all these maps together with composition forms a semigroup called a signed transformation semigroup. Given that dom(α) = [n], the signed transformation semigroup will be called a signed full transformation semigroup on [n]. In this paper, we obtain formulas that count the number of elements in the semigroups of order decreasing, order preserving and order decreasing signed transformations on [n]. We equally do same for the sub-semigroup of the signed tra

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2

Haynes, Tyler. "Thickness in topological transformation semigroups." International Journal of Mathematics and Mathematical Sciences 16, no. 3 (1993): 493–502. http://dx.doi.org/10.1155/s0161171293000602.

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This article deals with thickness in topological transformation semigroups (τ-semigroups). Thickness is used to establish conditions guaranteeing an invariant mean on a function space defined on aτ-semigroup if there exists an invariant mean on its functions restricted to a sub-τ-semigroup of the originalτ-semigroup. We sketch earlier results, then give many equivalent conditions for thickness onτ-semigroups, and finally present theorems giving conditions for an invariant mean to exist on a function space.

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3

Jude. A., Omelebele, Udoaka O. G., and Udoakpan I. U. "Ranks of Identity Difference Transformation Semigroup." International Journal of Pure Mathematics 9 (March 30, 2022): 49–54. http://dx.doi.org/10.46300/91019.2022.9.10.

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This study focuses on the ranks of identity difference transformation semigroup. The ideals of all the (sub) semigroups; identity difference full transformation semigroup (IDT_n), identity difference order preserving transformation semigroup, (IDO_n), identity difference symmetric inverse transformation semigroup( IDI_n), identity difference partial order preserving symmetric inverse transformation semigroup( IDPOI_n) and identity difference partial order preserving transformation semigroup ( IDPO_n) were investigated for rank and their combinatorial results obtained respectively.

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4

Emunefe, J. O., A. O. Atonuje, and J. Tsetimi,. "Conjugacy Classes in Order- Preserving Transformation Semi groups with Injective Contraction." Nigerian Journal of Science and Environment 22, no. 3 (2024): 22–43. https://doi.org/10.61448/njse223243.

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Enumerating the elements within transformation semigroups poses a significant challenge. Prior knowledge has been more on the injective order-preserving and order-decreasing transformation semigroup, a sub-semigroup of the injective transformation semigroup. This work categorized elements within the injective order-preserving sub-semigroup with contraction, arranging them into conjugacy classes using a path decomposition approach based on circuit and proper paths. Furthermore, these conjugacy classes were organized according to the number of images. A general expression was derived for the num

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5

Nenthein, S., та Y. Kemprasit. "On transformation semigroups which areℬ𝒬-semigroups". International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–10. http://dx.doi.org/10.1155/ijmms/2006/12757.

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A semigroup whose bi-ideals and quasi-ideals coincide is called aℬ𝒬-semigroup. The full transformation semigroup on a setXand the semigroup of all linear transformations of a vector spaceVover a fieldFinto itself are denoted, respectively, byT(X)andLF(V). It is known that every regular semigroup is aℬ𝒬-semigroup. Then bothT(X)andLF(V)areℬ𝒬-semigroups. In 1966, Magill introduced and studied the subsemigroupT¯(X,Y)ofT(X), where∅≠Y⊆XandT¯(X,Y)={α∈T(X,Y)|Yα⊆Y}. IfWis a subspace ofV, the subsemigroupL¯F(V,W)ofLF(V)will be defined analogously. In this paper, it is shown thatT¯(X,Y)is aℬ𝒬-semigroup i

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6

Imam, A.T, M. Balarabe, S. Kasim, and C. Eze. "Perfect Product of two Squares in Finite Full Transformation Semigroup." International Journal of Mathematical Sciences and Optimization: Theory and Applications 11, no. 1 (2025): 107–13. https://doi.org/10.5281/zenodo.15176079.

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 In this paper, we investigate the concept of the perfect product of two squares in the context  of finite full transformation semigroups. We provide a comprehensive analysis of the conditions  under which the product of two idempotent elements in a transformation semigroup forms a  perfect product of two squares. Specifically, we examine the relationship between the kernel and image of idempotents, as well as the interplay between the domain and image of these  transformations. The main result establishes that for two idempotent elements α and β in Tn,

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7

Shirazi, Zadeh, and Nasser Golestani. "On classifications of transformation semigroups: Indicator sequences and indicator topological spaces." Filomat 26, no. 2 (2012): 313–29. http://dx.doi.org/10.2298/fil1202313s.

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In this paper considering a transformation semigroup with finite height we define the notion of indicator sequence in such a way that any two transformation semigroups with the same indicator sequence have the same height. Also related to any transformation semigroup a topological space, called indicator topological space, is defined in such a way that transformation semigroups with homeomorphic indicator topological spaces have the same height. Moreover any two transformation semigroups with homeomorphic indicator topological spaces and finite height have the same indicator sequences.

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8

Dimitrova, I., and J. Koppitz. "ON THE MAXIMAL SUBSEMIGROUPS OF SOME TRANSFORMATION SEMIGROUPS." Asian-European Journal of Mathematics 01, no. 02 (2008): 189–202. http://dx.doi.org/10.1142/s1793557108000187.

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Let Singn be the semigroup of all singular transformations on an n-element set. We consider two subsemigroups of Singn: the semigroup On of all isotone singular transformations and the semigroup Mn of all monotone singular transformations. We describe the maximal subsemigroups of these two semigroups, and study the connections between them.

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9

Rakbud, Jittisak, та Malinee Chaiya. "Regularity of Semigroups of Transformations Whose Characters Form the Semigroup of a Δ -Structure". International Journal of Mathematics and Mathematical Sciences 2020 (27 грудня 2020): 1–7. http://dx.doi.org/10.1155/2020/8872391.

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In this paper, we make use of the notion of the character of a transformation on a fixed set X , provided by Purisang and Rakbud in 2016, and the notion of a Δ -structure on X , provided by Magill Jr. and Subbiah in 1974, to define a sub-semigroup of the full-transformation semigroup T X . We also define a sub-semigroup of that semigroup. The regularity of those two semigroups is also studied.

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10

SINGHA, BOORAPA, JINTANA SANWONG, and R. P. SULLIVAN. "PARTIAL ORDERS ON PARTIAL BAER–LEVI SEMIGROUPS." Bulletin of the Australian Mathematical Society 81, no. 2 (2010): 195–207. http://dx.doi.org/10.1017/s0004972709001038.

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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 t

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11

Srisawat, Jitsupa, and Yanisa Chaiya. "A Note on Coregular Elements in Certain Semigroups of Transformations." Progress in Applied Science and Technology 14, no. 1 (2024): 87–90. http://dx.doi.org/10.60101/past.2024.252662.

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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.

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12

Konieczny, Janusz. "A new definition of conjugacy for semigroups." Journal of Algebra and Its Applications 17, no. 02 (2018): 1850032. http://dx.doi.org/10.1142/s0219498818500329.

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The conjugacy relation plays an important role in group theory. If [Formula: see text] and [Formula: see text] are elements of a group [Formula: see text], [Formula: see text] is conjugate to [Formula: see text] if [Formula: see text] for some [Formula: see text]. The group conjugacy extends to inverse semigroups in a natural way: for [Formula: see text] and [Formula: see text] in an inverse semigroup [Formula: see text], [Formula: see text] is conjugate to [Formula: see text] if [Formula: see text] and [Formula: see text] for some [Formula: see text]. In this paper, we define a conjugacy for

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13

Luangchaisri, Panuwat, Thawhat Changphas, and Chalida Phanlert. "Left (right) magnifying elements of a partial transformation semigroup." Asian-European Journal of Mathematics 13, no. 01 (2018): 2050016. http://dx.doi.org/10.1142/s1793557120500163.

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An element [Formula: see text] of a semigroup [Formula: see text] is a called a left (respectively right) magnifying element of [Formula: see text] if [Formula: see text] (respectively [Formula: see text]) for some proper subset [Formula: see text] of [Formula: see text]. In this paper, left magnifying elements and right magnifying elements of a partial transformation semigroup will be characterized. The results obtained generalize the results of Magill [K. D. Magill, Magnifying elements of transformation semigroups, Semigroup Forum, 48 (1994) 119–126].

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14

Marques-Smith, M. Paula O., and R. P. Sullivan. "The ideal structure of nilpotent-generated transformation semigroups." Bulletin of the Australian Mathematical Society 60, no. 2 (1999): 303–18. http://dx.doi.org/10.1017/s0004972700036418.

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In 1987 Sullivan determined the elements of the semigroup N(X) generated by all nilpotent partial transformations of an infinite set X; and later in 1997 he studied subsemigroups of N(X) defined by restricting the index of the nilpotents and the cardinality of the set. Here, we describe the ideals and Green's relations on such semigroups, like Reynolds and Sullivan did in 1985 for the semigroup generated by all idempotent total transformations of X. We then use this information to describe the congruences on certain Rees factor semigroups and to construct families of congruence-free semigroups

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15

CHAOPRAKNOI, SUREEPORN, TEERAPHONG PHONGPATTANACHAROEN, and PATTANACHAI RAWIWAN. "THE NATURAL PARTIAL ORDER ON SOME TRANSFORMATION SEMIGROUPS." Bulletin of the Australian Mathematical Society 89, no. 2 (2013): 279–92. http://dx.doi.org/10.1017/s0004972713000580.

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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.

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16

Luangchaisri, Panuwat, and Thawhat Changphas. "Skew Pairs of Idempotents in Partial Transformation Semigroups." Mathematics 10, no. 1 (2021): 2. http://dx.doi.org/10.3390/math10010002.

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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.

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17

Levi, Inessa. "Automorphisms of normal partial transformation semigroups." Glasgow Mathematical Journal 29, no. 2 (1987): 149–57. http://dx.doi.org/10.1017/s0017089500006790.

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We let X be an arbitrary infinite set. A semigroup S of total or partial transformations of X is called -normal if hSh-1 = S, for all h in , the symmetric group on X. For example, the full transformation semigroup , the semigroup of all partial transformations , the semigroup of all 1–1 partial transformations and all ideals of and are -normal.

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18

Azeef Muhammed, P. A., and A. R. Rajan. "Cross-connections of the singular transformation semigroup." Journal of Algebra and Its Applications 17, no. 03 (2018): 1850047. http://dx.doi.org/10.1142/s0219498818500470.

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Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup [Formula: see text] of all non-invertible transformations on a set [Formula: see text]. The categories involved are characterized as the powerset category [Formula: see text] and the category of partitions [Formula: see text]. We describe these categories and show how a permutation on [Formula: see text] gives rise to a cross-connection.

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19

FITZGERALD, D. G., and KWOK WAI LAU. "ON THE PARTITION MONOID AND SOME RELATED SEMIGROUPS." Bulletin of the Australian Mathematical Society 83, no. 2 (2010): 273–88. http://dx.doi.org/10.1017/s0004972710001851.

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AbstractThe partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new in

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20

Sanwong, Jintana, and Worachead Sommanee. "Regularity and Green's Relations on a Semigroup of Transformations with Restricted Range." International Journal of Mathematics and Mathematical Sciences 2008 (2008): 1–11. http://dx.doi.org/10.1155/2008/794013.

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LetT(X)be the full transformation semigroup on the setXand letT(X,Y)={α∈T(X):Xα⊆Y}. ThenT(X,Y)is a sub-semigroup ofT(X)determined by a nonempty subsetYofX. In this paper, we give a necessary and sufficient condition forT(X,Y)to be regular. In the case thatT(X,Y)is not regular, the largest regular sub-semigroup is obtained and this sub-semigroup is shown to determine the Green's relations onT(X,Y). Also, a class of maximal inverse sub-semigroups ofT(X,Y)is obtained.

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21

Geem, Methaq Hamza, Ahmed Raad Hassan, and Hayder Ismael Neamah. "0-Semigroup of g-transformation." Journal of Interdisciplinary Mathematics 28, no. 1 (2025): 313–18. https://doi.org/10.47974/jim-1986.

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In this paper, we introduce a novel type of semi group which is called the g-C0-semigroup. We get the new relation between C0-semigroup and integral transformations by using g-transformation to induce certain type of c0-semigroup. We give some results and some theorems on this concept.

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22

EAST, JAMES. "PRESENTATIONS FOR SINGULAR SUBSEMIGROUPS OF THE PARTIAL TRANSFORMATION SEMIGROUP." International Journal of Algebra and Computation 20, no. 01 (2010): 1–25. http://dx.doi.org/10.1142/s0218196710005509.

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The partial transformation semigroup [Formula: see text] is the semigroup of all partial transformations on the finite set n = {1,…, n}. The transformation semigroup [Formula: see text] and the symmetric group [Formula: see text] consist of all (full) transformations on n and permutations on n, respectively. We obtain presentations, in terms of generators and relations, for the singular subsemigroups [Formula: see text] and [Formula: see text]. We also calculate the ranks of both subsemigroups.

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23

Bulatov, Andrei, Marcin Kozik, Peter Mayr, and Markus Steindl. "The subpower membership problem for semigroups." International Journal of Algebra and Computation 26, no. 07 (2016): 1435–51. http://dx.doi.org/10.1142/s0218196716500612.

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Fix a finite semigroup [Formula: see text] and let [Formula: see text] be tuples in a direct power [Formula: see text]. The subpower membership problem (SMP) asks whether [Formula: see text] can be generated by [Formula: see text]. If [Formula: see text] is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in [Formula: see text]. For semigroups this problem always lies in PSPACE. We show that the [Formula: see text] for a full transformation semigroup on [Formula: see text] or more letters is actually PSPACE-complete, while on [Formula: see text] l

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24

Kachalla, Aliyu, and Babagana A. Madu. "Expressing Partial Order-Preserving Transformations as Products of Nilpotents." International Journal of Development Mathematics (IJDM) 1, no. 3 (2024): 112–25. http://dx.doi.org/10.62054/ijdm/0103.08.

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Let S be a semigroup with zero, then an element a∈S is called nilpotent, if there exists a positive integer n such that . In the partial transformation semigroup on , where is a non-empty set, denoted by , the empty map is the zero of . Further, nilpotency is a structural property of fundamental importance which arises in many situations, for instance in linear algebra, we have nilpotent matrices and much more in other branches of mathematics. In this study, we will investigate, some elements of semigroups of partial one-to-one order-preserving transformations (denoted by ) and partial order-p

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25

Margolis, Stuart, and John Rhodes. "Degree 2 transformation semigroups as continuous maps on graphs: Foundations and structure." International Journal of Algebra and Computation 31, no. 06 (2021): 1065–91. http://dx.doi.org/10.1142/s0218196721400051.

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We develop the theory of transformation semigroups that have degree 2, that is, act by partial functions on a finite set such that the inverse image of points have at most two elements. We show that the graph of fibers of such an action gives a deep connection between semigroup theory and graph theory. It is known that the Krohn–Rhodes complexity of a degree 2 action is at most 2. We show that the monoid of continuous maps on a graph is the translational hull of an appropriate 0-simple semigroup. We show how group mapping semigroups can be considered as regular covers of their right letter map

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26

Jin, Jiulin. "On the rank of semigroup of transformations with restricted partial range." Filomat 35, no. 14 (2021): 4925–36. http://dx.doi.org/10.2298/fil2114925j.

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Let T(X) be the full transformation semigroup on a nonempty set X. For ? ? Z ? Y ? X, let T(X,Y,Z) = {? ? T(X):Y? ? Z}. It is not difficult to see that it is a generalized form of three well-known semigroups. This paper obtains an isomorphism theorem of T(X,Y,Z). In addition, when X is finite and Z ? Y ? X, the rank of the semigroup T(X,Y,Z) is calculated.

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Yang, Xiuliang, and Haobo Yang. "ISOMORPHISMS OF TRANSFORMATION SEMIGROUPS ASSOCIATED WITH SIMPLE DIGRAPHS." Asian-European Journal of Mathematics 02, no. 04 (2009): 727–37. http://dx.doi.org/10.1142/s1793557109000601.

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For each simple digraph D, we introduce its associated semigroup S(D) and its widened digraph W(D). We show that, for any two finite simple digraphs D1 and D2, S(D1) and S(D2) are isomorphic as semigroups if and only if W(D1) and W(D2) are isomorphic as digraphs.

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MASHEVITZKY, G. "A NEW METHOD IN THE FINITE BASIS PROBLEM WITH APPLICATIONS TO RANK 2 TRANSFORMATION SEMIGROUPS." International Journal of Algebra and Computation 17, no. 07 (2007): 1431–63. http://dx.doi.org/10.1142/s0218196707004232.

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We prove that the semigroup of all transformations of a 3-element set with rank at most 2 does not have a finite basis of identities. This gives a negative answer to a question of Shevrin and Volkov. It is worthwhile to notice that the semigroup of transformations with rank at most 2 of an n-element set, where n > 4, has a finite basis of identities. A new method of constructing finite non-finitely based semigroups is developed.

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Sabbaghan, Masoud, and Fatemah Ayatollah Zadeh Shirazi. "a-minimal sets and related topics in transformation semigroups (I)." International Journal of Mathematics and Mathematical Sciences 25, no. 10 (2001): 637–54. http://dx.doi.org/10.1155/s016117120100388x.

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We deal witha-minimal sets instead of minimal right ideals of the enveloping semigroup and obtain a partition of disjoint isomorphic subgroups of some of its subsets. We also give some generalizations of almost periodicity and distality in the transformation semigroups and obtain similar results.

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30

Wasanawichit, Amorn, and Yupaporn Kemprasit. "Dense subsemigroups of generalised transformation semigroups." Journal of the Australian Mathematical Society 73, no. 3 (2002): 433–46. http://dx.doi.org/10.1017/s1446788700009071.

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AbstractIn 1986, Higgins proved that T(X), the semigroup (under composition) of all total transformations of a set X, has a proper dense subsemigroup if and only if X is infinite, and he obtained similar results for partial and partial one-to-one transformations. We consider the generalised transformation semigroup T(X, Y) consisting of all total transformations from X into Y under the operation α * β = αθβ, where θ is any fixed element of T(Y, X). We show that this semigroup has a proper dense subsemigroup if and only if X and Y are infinite and | Yθ| = min{|X|,|Y|}, and we obtain similar res

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31

NICA, ALEXANDRU. "ON A GROUPOID CONSTRUCTION FOR ACTIONS OF CERTAIN INVERSE SEMIGROUPS." International Journal of Mathematics 05, no. 03 (1994): 349–72. http://dx.doi.org/10.1142/s0129167x94000206.

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We consider a version of the notion of F-inverse semigroup (studied in the algebraic theory of inverse semigroups). We point out that an action of such an inverse semigroup on a locally compact space has associated a natural groupoid construction, very similar to the one of a transformation group. We discuss examples related to Toeplitz algebras on subsemigroups of discrete groups, to Cuntz-Krieger algebras, and to crossed-products by partial automorphisms in the sense of Exel.

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32

Thomas, Rachel. "Products of two idempotent transformations over arbitrary sets and vector spaces." Bulletin of the Australian Mathematical Society 57, no. 1 (1998): 59–71. http://dx.doi.org/10.1017/s0004972700031427.

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In this paper we consider the characterisation of those elements of a transformation semigroup S which are a product of two proper idempotents. We give a characterisation where S is the endomorphism monoid of a strong independence algebra A, and apply this to the cases where A is an arbitrary set and where A is an arbitrary vector space. The results emphasise the analogy between the idempotent generated subsemigroups of the full transformation semigroup of a set and of the semigroup of linear transformations from a vector space to itself.

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33

Bakare, G. N., G. R. Ibrahim, I. F. Usamot, B. M. Ahmed, and Y. T. Oyebo. "Combinatorial Properties on Subsemigroup of Order-Decreasing Alternating Semigroup Using Three Distinct Combinatorial Functions." DIU Journal of Science & Technology 17, no. 1 (2024): 29–35. https://doi.org/10.5281/zenodo.13827986.

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Analysing problems of combinatorial nature arise naturally in the study of transformation semigroup. Combinatorial properties of many transformation semigroups and its subsemigroups have been studied with interesting and delightful results obtained. This paper investigated combinatorial properties on subsemigroup of order-decreasing alternating semigroups in which combinatorial functions and were used to derive some triangular arrays of numbers and some combinatorial results for each function were established. More so, the results on and were also obtained and generalized.

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STEINDL, MARKUS. "ON SEMIGROUPS WITH PSPACE-COMPLETE SUBPOWER MEMBERSHIP PROBLEM." Journal of the Australian Mathematical Society 106, no. 1 (2018): 127–42. http://dx.doi.org/10.1017/s1446788718000010.

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Fix a finite semigroup $S$ and let $a_{1},\ldots ,a_{k},b$ be tuples in a direct power $S^{n}$. The subpower membership problem (SMP) for $S$ asks whether $b$ can be generated by $a_{1},\ldots ,a_{k}$. For combinatorial Rees matrix semigroups we establish a dichotomy result: if the corresponding matrix is of a certain form, then the SMP is in P; otherwise it is NP-complete. For combinatorial Rees matrix semigroups with adjoined identity, we obtain a trichotomy: the SMP is either in P, NP-complete, or PSPACE-complete. This result yields various semigroups with PSPACE-complete SMP including the

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35

Fleischer, Lukas, and Trevor Jack. "The complexity of properties of transformation semigroups." International Journal of Algebra and Computation 30, no. 03 (2019): 585–606. http://dx.doi.org/10.1142/s0218196720500125.

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We investigate the computational complexity for determining various properties of a finite transformation semigroup given by generators. We introduce a simple framework to describe transformation semigroup properties that are decidable in [Formula: see text]. This framework is then used to show that the problems of deciding whether a transformation semigroup is a group, commutative or a semilattice are in [Formula: see text]. Deciding whether a semigroup has a left (respectively, right) zero is shown to be [Formula: see text]-complete, as are the problems of testing whether a transformation se

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36

Chinram, Ronnason. "REGULARITY AND GREEN'S RELATIONS OF GENERALIZED PARTIAL TRANSFORMATION SEMIGROUPS." Asian-European Journal of Mathematics 01, no. 03 (2008): 295–302. http://dx.doi.org/10.1142/s1793557108000266.

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Let X be any set and P(X) be the partial transformation semigroup on X. It is well-known that P(X) is regular. To generalize this, let X and Y be any sets and P(X, Y) be the set of all partial transformations from X to Y. For θ ∈ P(Y, X), let (P(X, Y), θ) be a semigroup (P(X, Y), *) where α * β = αθβ for all α, β ∈ P(X, Y). In this paper, we characterize the semigroup (P(X, Y), θ) to be regular, regular elements of the semigroup (P(X, Y), θ), [Formula: see text]-classes, [Formula: see text]-classes, [Formula: see text]-classes and [Formula: see text]-classes of the semigroup (P(X, Y), θ).

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37

Kozlowski, Wojciech M. "Convergence of Implicit Iterative Processes for Semigroups of Nonlinear Operators Acting in Regular Modular Spaces." Mathematics 12, no. 24 (2024): 4007. https://doi.org/10.3390/math12244007.

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This paper focuses on one-parameter semigroups of ρ-nonexpansive mappings Tt:C→C, where C is a subset of a modular space Xρ, the parameter t ranges over [0,+∞), and ρ is a convex modular with the Fatou property. The common fixed points of such semigroups can be interpreted as stationary points of a dynamic system defined by the semigroup, meaning they remain unchanged during the transformation Tt at any given time t. We demonstrate that, under specific conditions, the sequence {xk} generated by the implicit iterative process xk+1=ckTtk+1(xk+1)+(1−ck)xk is ρ-convergent to a common fixed point o

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38

Levi, Inessa. "Normal semigroups of one-to-one transformations." Proceedings of the Edinburgh Mathematical Society 34, no. 1 (1991): 65–76. http://dx.doi.org/10.1017/s0013091500005009.

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Let X be an infinite set and S be a transformation semigroup on X invariant under conjugations by permutations of X. Such S is termed x-normal. In the paper, we describe elements of a x-normal semigroup S of one-to-one transformations.

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39

Mendes-Gonçalves, Suzana, and R. P. Sullivan. "REGULAR ELEMENTS AND GREEN'S RELATIONS IN GENERALIZED TRANSFORMATION SEMIGROUPS." Asian-European Journal of Mathematics 06, no. 01 (2013): 1350006. http://dx.doi.org/10.1142/s179355711350006x.

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If X and Y are sets, we let P(X, Y) denote the set of all partial transformations from X into Y (that is, all mappings whose domain and range are subsets of X and Y, respectively). If θ ∈ P(Y, X), then P(X, Y) is a so-called "generalized semigroup" of transformations under the "sandwich operation": α * β = α ◦ θ ◦ β, for each α, β ∈ P(X, Y). We denote this semigroup by P(X, Y, θ) and, in this paper, we characterize Green's relations on it: that is, we study equivalence relations which determine when principal left (or right, or 2-sided) ideals in P(X, Y, θ) are equal. This solves a problem rai

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40

Kehinde, R., and O. H. Abdulazeez. "NUMERICAL AND GRAPHICAL RESULTS OF FINITE SYMMETRIC INVERSE (I_{n}) AND FULL (T_{n}) TRANSFORMATION SEMIGROUPS." FUDMA JOURNAL OF SCIENCES 4, no. 4 (2021): 443–53. http://dx.doi.org/10.33003/fjs-2020-0404-501.

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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.

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C., Eze, Olaiya O. O., and S. Kasim. "Graph-Theoretic Characterization of Quasi-Nilpotent Elements in Finite Semigroups of Full Order-Preserving Transformations." Mikailalsys Journal of Mathematics and Statistics 3, no. 2 (2025): 490–99. https://doi.org/10.58578/mjms.v3i2.5906.

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This paper investigates the structural behavior of quasi-nilpotent elements within the semigroup On of all full order-preserving transformations on a finite chain Xn = {1, 2, . . ., n}. While quasi-nilpotency has been extensively studied in full and partial transformation semigroups, its characterization in On remains largely unexplored. By employing a graph-theoretic approach, we associate to each transformation α ∈ On a digraph Γ(α) and establish necessary and sufficient conditions under which α is quasi-nilpotent. Specifically, we show that α is quasi-nilpotent if and only if Γ(α) has a uni

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42

C, Eze, A. T. Imam, M. Balarabe, and Olaiya O. O. "Graph-Theoretic Characterizations of Quasi-Idempotents in Full Order-Preserving Transformation Semigroup." Babylonian Journal of Mathematics 2025 (May 28, 2025): 88–91. https://doi.org/10.58496/bjm/2025/009.

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This paper presents a digraph-theoretic extension of the characterization of quasi-idempotent in the semigroup On of full order-preserving transformations on a finite chain. Building on earlier results . that describe quasi-idempotent as those transformations α ∈ On satisfying α≠α^2=α^4, we provide a novel interpretation using the functional digraphs of such maps. We show that a transformation is quasi-idempotent if and only if each vertex in its associated digraph is either fixed or maps directly into a fixed point, and every non-trivial strongly connected component forms a 2-cycle. Furthermo

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Kaewnoi, Thananya, Montakarn Petapirak, and Ronnason Chinram. "On Magnifying Elements in E-Preserving Partial Transformation Semigroups." Mathematics 6, no. 9 (2018): 160. http://dx.doi.org/10.3390/math6090160.

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Let S be a semigroup. An element a of S is called a right [left] magnifying element if there exists a proper subset M of S satisfying S = M a [ S = a M ] . Let E be an equivalence relation on a nonempty set X. In this paper, we consider the semigroup P ( X , E ) consisting of all E-preserving partial transformations, which is a subsemigroup of the partial transformation semigroup P ( X ) . The main propose of this paper is to show the necessary and sufficient conditions for elements in P ( X , E ) to be right or left magnifying.

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Livinsky, I. V., and T. G. Zhukovska. "On orders of two transformation semigroups of the boolean." Carpathian Mathematical Publications 6, no. 2 (2014): 317–19. http://dx.doi.org/10.15330/cmp.6.2.317-319.

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We consider the semigroup $\mathcal{O}(\mathcal{B}_n)$ of all order-preserving transformations $\varphi : \mathcal{B}_n \rightarrow \mathcal{B}_n$ of ordered by inclusion boolean $\mathcal{B}_n$ of $n$-element set (i.e. such transformations that $A \subseteq B$ implies $\varphi(A) \subseteq \varphi(B)$) and its subsemigroup $\mathcal{C}(\mathcal{B}_n)$ of those transformations for which $\varphi(A) \subseteq A$ for all $A \in \mathcal{B}_n$. Orders of these semigroups are calculated.

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45

Dolinka, Igor, and James East. "Variants of finite full transformation semigroups." International Journal of Algebra and Computation 25, no. 08 (2015): 1187–222. http://dx.doi.org/10.1142/s021819671550037x.

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The variant of a semigroup [Formula: see text] with respect to an element [Formula: see text], denoted [Formula: see text], is the semigroup with underlying set [Formula: see text] and operation ⋆ defined by [Formula: see text] for [Formula: see text]. In this paper, we study variants [Formula: see text] of the full transformation semigroup [Formula: see text] on a finite set [Formula: see text]. We explore the structure of [Formula: see text] as well as its subsemigroups [Formula: see text] (consisting of all regular elements) and [Formula: see text] (consisting of all products of idempotents

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46

Brown, B., and P. M. Higgins. "Finite full transformation semigroups as collections of random functions." Glasgow Mathematical Journal 30, no. 2 (1988): 203–11. http://dx.doi.org/10.1017/s0017089500007230.

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The collection of all self-maps on a non-empty set X under composition is known in algebraic semigroup theory as the full transformation semigroup on X and is written x. Its importance lies in the fact that any semigroup S can be embedded in the full transformation semigroup (where S1 is the semigroup S with identity 1 adjoined, if S does not already possess one). The proof is similar to Cayley's Theorem that a group G can be embedded in SG, the group of all bijections of G to itself. In this paper X will be a finite set of order n, which we take to be and so we shall write Tn for X.

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Chinram, Ronnason, Pattarawan Petchkaew, and Samruam Baupradist. "Left and Right Magnifying Elements in Generalized Semigroups of Transformations by Using Partitions of a Set." European Journal of Pure and Applied Mathematics 11, no. 3 (2018): 580–88. http://dx.doi.org/10.29020/nybg.ejpam.v11i3.3260.

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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 X be a nonempty set and T(X) be the semigroup of all transformation from X into itself under the composition of functions. For a partition P = {X_α | α ∈ I} of the set X, let T(X,P) = {f ∈ T(X) | (X_α)f ⊆ X_α for all α ∈ I}. Then T(X,P) is a subsemigroup of T(X) and if P = {X}, T(X,P) = T(X). Our aim in this paper is to give necessary and suﬃcient conditions for elements in T(X,P) to be left or right magnifying. Moreover, we apply those conditions to give necess

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LIN, XIAOGANG, DONGKUI MA, and YUPAN WANG. "On the measure-theoretic entropy and topological pressure of free semigroup actions." Ergodic Theory and Dynamical Systems 38, no. 2 (2016): 686–716. http://dx.doi.org/10.1017/etds.2016.41.

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In this paper we introduce the notions of topological pressure and measure-theoretic entropy for a free semigroup action. Suppose that a free semigroup acts on a compact metric space by continuous self-maps. To this action we assign a skew-product transformation whose fiber topological pressure is taken to be the topological pressure of the initial action. Some properties of these two notions are given, followed by two main results. One is the relationship between the topological pressure of the skew-product transformation and the topological pressure of the free semigroup action, the other is

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49

SUN, LEI, and LIMIN WANG. "NATURAL PARTIAL ORDER IN SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET." Bulletin of the Australian Mathematical Society 87, no. 1 (2012): 94–107. http://dx.doi.org/10.1017/s0004972712000287.

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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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50

O Francis, M., L. F. Joseph, A. T. Cole, and B. Oshatuyi. "Roots of Tropical Polynomial from Clopen and Non-Clopen Discrete m-Topological Transformation Semigroup." International Journal of Advanced Mathematical Sciences 10, no. 2 (2024): 37–47. https://doi.org/10.14419/4h3jjx97.

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This paper introduces a specific subclass of m-topological transformation semigroup spaces, referred to as closed m-topological full transformation semigroup spaces. It defines both the clopen and non-clopen elements within these spaces and explores the nature of their roots. The study presents formulas for clopen and non-clopen elements, as well as for the closed m-topological full transformation semigroup spaces. Furthermore, numerical and graphical results for degrees 2 and 3 are provided to illustrate the findings.

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