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Статті в журналах з теми "Transformation semigroup"

Автор: Grafiati

Опубліковано: 4 червня 2021

Оновлено: 9 червня 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 (April 10, 2025): 54–56. https://doi.org/10.62050/fjst2025.v9n1.510.

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Анотація:

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 transformation semigroup consisting only of idempotents.

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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 (December 20, 2024): 22–43. https://doi.org/10.61448/njse223243.

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Анотація:

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 number of conjugacy classes in the injective order-preserving contraction transformation semigroup. We found that the number of conjugacy classes in this transformation is precisely given by the sequence A000070 (OEIS). This alignment suggests a profound relationship between the structure of these transformations and the theory of partitions (number theory), opening up new avenues for research and analysis.

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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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Анотація:

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 if and only ifY=X,|Y|=1, or|X|≤3, andL¯F(V,W)is aℬ𝒬-semigroup if and only if (i)W=V, (ii)W={0}, or (iii)F=ℤ2,dimFV=2, anddimFW=1.

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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 (March 30, 2025): 107–13. https://doi.org/10.5281/zenodo.15176079.

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Анотація:

 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,  if the domain and image of α and β satisfy certain equivalence conditions, then their product is  a perfect product of two squares. We also explore related properties of disjoint cycles and how these contribute to the structural characteristics of the semigroup. Our findings extend the  existing theory of transformation semigroups and offer valuable insights into the decomposition  of semigroup elements into squares, contributing to the broader field of semigroup theory.

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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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Анотація:

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 (June 2008): 189–202. http://dx.doi.org/10.1142/s1793557108000187.

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Анотація:

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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Анотація:

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 (January 26, 2010): 195–207. http://dx.doi.org/10.1017/s0004972709001038.

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Анотація:

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.

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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 (April 29, 2024): 87–90. http://dx.doi.org/10.60101/past.2024.252662.

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Анотація:

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 (January 23, 2018): 1850032. http://dx.doi.org/10.1142/s0219498818500329.

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Анотація:

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 an arbitrary semigroup [Formula: see text] that reduces to the inverse semigroup conjugacy if [Formula: see text] is an inverse semigroup. (None of the existing notions of conjugacy for semigroups has this property.) We compare our new notion of conjugacy with existing definitions, characterize the conjugacy in basic transformation semigroups and their ideals using the representation of transformations as directed graphs, and determine the number of conjugacy classes in these semigroups.

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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 (August 3, 2018): 2050016. http://dx.doi.org/10.1142/s1793557120500163.

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Анотація:

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 (October 1999): 303–18. http://dx.doi.org/10.1017/s0004972700036418.

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Анотація:

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 with interesting algebraic properties. We also study analogous questions for X finite and for one-to-one partial transformations.

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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 (June 28, 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 (December 21, 2021): 2. http://dx.doi.org/10.3390/math10010002.

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Анотація:

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 (July 1987): 149–57. http://dx.doi.org/10.1017/s0017089500006790.

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Анотація:

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 (February 5, 2018): 1850047. http://dx.doi.org/10.1142/s0219498818500470.

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Анотація:

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. Further, we prove that every cross-connection between them is induced by a permutation and construct the regular semigroups that arise from the cross-connections. We show that each of the cross-connection semigroups arising this way is isomorphic to [Formula: see text]. We also describe the right reductive subsemigroups of [Formula: see text] with the category of principal left ideals isomorphic to [Formula: see text]. This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of [Formula: see text].

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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 (December 6, 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 interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

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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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Анотація:

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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Анотація:

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 (February 2010): 1–25. http://dx.doi.org/10.1142/s0218196710005509.

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Анотація:

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 (November 2016): 1435–51. http://dx.doi.org/10.1142/s0218196716500612.

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Анотація:

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] letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup [Formula: see text] embeds into a direct product of a Clifford semigroup and a nilpotent semigroup, then [Formula: see text] is in P; otherwise it is NP-complete.

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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 (September 9, 2024): 112–25. http://dx.doi.org/10.62054/ijdm/0103.08.

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Анотація:

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-preserving transformations (denoted by ), and some basic properties of nilpotent transformations. We have studied certain properties of nilpotent elements in , symmetric inverse semigroup, where is finite and infinite, certain properties of nilpotent elements in , descriptions of the elements of 〈N〉, where N is the set of nilpotents of and we used the same letter N to denote that of , and used them to establish the following results, in this paper (that is, we have used the knowledge that we have acquired to establish the following results): - a partial ono-to-one order-preserving transformation , has from 1 to fixed, and there is an upper jump of length two(2) between and , and lower jumps of length one(1) and one(1), immediately after , can be expressed as a product of three nilpotents. – a partial one-to-one order-preserving transformation , has from 1 to fixed, and there is an upper jump of length three (3) between and and lower jumps of length two (2) and one (1), immediately after , can be expressed as a product of fewer than three nilpotents. – a partial order-preserving transformation , has from 1 to fixed, but belongs to , and there is an upper jump of length two(2) between and , and lower jumps of length two(2) and one(1), immediately after , cannot be expressed as a product of fewer than three nilpotents. – a partial order-preserving transformation , has from 1 to fixed, but belongs to , and there is an upper jump of length three (3) between and , and lower jumps of length three (3) and one (1), immediately after n-5, can be expressed as a product of fewer than three nilpotents. – a transformation , partial transformation semigroup, is not nilpotent if and only if the graph of the transformation has a loop, and - a transformation , partial one-to-one transformation semigroup, which has a nilpotent path and a permutation part is not nilpotent

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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 (June 3, 2021): 1065–91. http://dx.doi.org/10.1142/s0218196721400051.

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Анотація:

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 mapping image and relate this to their graph of fibers.

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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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Анотація:

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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27

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

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Анотація:

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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28

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 (November 2007): 1431–63. http://dx.doi.org/10.1142/s0218196707004232.

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Анотація:

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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29

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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Анотація:

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 (December 2002): 433–46. http://dx.doi.org/10.1017/s1446788700009071.

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Анотація:

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 results for partial and partial one-to-one transformations. The results of Higgins then become special cases.

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31

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

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Анотація:

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 (February 1998): 59–71. http://dx.doi.org/10.1017/s0004972700031427.

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Анотація:

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 (September 23, 2024): 29–35. https://doi.org/10.5281/zenodo.13827986.

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Анотація:

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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34

STEINDL, MARKUS. "ON SEMIGROUPS WITH PSPACE-COMPLETE SUBPOWER MEMBERSHIP PROBLEM." Journal of the Australian Mathematical Society 106, no. 1 (May 30, 2018): 127–42. http://dx.doi.org/10.1017/s1446788718000010.

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Анотація:

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 six-element Brandt monoid, the full transformation semigroup on three or more letters, and semigroups of all $n$ by $n$ matrices over a field for $n\geq 2$.

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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 (December 6, 2019): 585–606. http://dx.doi.org/10.1142/s0218196720500125.

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Анотація:

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 semigroup is nilpotent, [Formula: see text]-trivial or has central idempotents. We also give [Formula: see text] algorithms for testing whether a transformation semigroup is idempotent, orthodox, completely regular, Clifford or has commuting idempotents. Some of these algorithms are direct consequences of the more general result that arbitrary fixed semigroup equations can be tested in [Formula: see text]. Moreover, we show how to compute left and right identities of a transformation semigroup in polynomial time. Finally, we show that checking whether an element is regular is [Formula: see text]-complete.

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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 (September 2008): 295–302. http://dx.doi.org/10.1142/s1793557108000266.

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Анотація:

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 (December 20, 2024): 4007. https://doi.org/10.3390/math12244007.

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Анотація:

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 of the semigroup. Our findings extend existing convergence results for semigroups of operators, from Banach spaces to a broader class of regular modular spaces.

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38

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

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Анотація:

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 (March 2013): 1350006. http://dx.doi.org/10.1142/s179355711350006x.

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Анотація:

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 raised by Magill and Subbiah in 1975. We also discuss the same idea for important subsemigroups of P(X, Y, θ) and characterize when these semigroups satisfy certain regularity conditions.

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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 (June 14, 2021): 443–53. http://dx.doi.org/10.33003/fjs-2020-0404-501.

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Анотація:

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

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41

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 (May 27, 2025): 490–99. https://doi.org/10.58578/mjms.v3i2.5906.

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Анотація:

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 unique sink and all vertices are connected to it via directed paths. This char- acterization is further refined by relating the height of Γ(α) to the number of convex blocks in the domain partition of α. Illustrative examples and explicit constructions are provided to validate the theoretical findings. The results offer new insights into the interplay between algebraic properties of transformation semigroups and their combi- natorial representations.

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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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Анотація:

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. Furthermore, we prove that no directed path of the form v1 → v2 → v3 exists where all vertices are non-stationary. These findings offer a new perspective on the structure of On, bridging algebraic properties with graphical structure, and set the stage for visual and computational analysis of quasi-idempotent generation in transformation semigroups.

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43

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

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Анотація:

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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44

Livinsky, I. V., and T. G. Zhukovska. "On orders of two transformation semigroups of the boolean." Carpathian Mathematical Publications 6, no. 2 (December 27, 2014): 317–19. http://dx.doi.org/10.15330/cmp.6.2.317-319.

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Анотація:

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 (December 2015): 1187–222. http://dx.doi.org/10.1142/s021819671550037x.

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Анотація:

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), and the ideals of [Formula: see text]. Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.

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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 (May 1988): 203–11. http://dx.doi.org/10.1017/s0017089500007230.

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Анотація:

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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47

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 (July 31, 2018): 580–88. http://dx.doi.org/10.29020/nybg.ejpam.v11i3.3260.

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Анотація:

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 necessary and suﬃcient conditions for elements in some generalized linear transformation semigroups.

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48

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 (July 26, 2016): 686–716. http://dx.doi.org/10.1017/etds.2016.41.

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Анотація:

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 the partial variational principle about the topological pressure. Moreover, we apply this partial variational principle to study the measure-theoretic entropy and the topological entropy of finite affine transformations on a metrizable group.

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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 (May 22, 2012): 94–107. http://dx.doi.org/10.1017/s0004972712000287.

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Анотація:

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 (December 13, 2024): 37–47. https://doi.org/10.14419/4h3jjx97.

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Анотація:

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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