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Journal articles on the topic 'Transformation semigroup'

Author: Grafiati

Published: 4 June 2021

Last updated: 20 February 2023

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1

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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2

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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3

Nenthein, S., and 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 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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4

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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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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5

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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6

Rakbud, Jittisak, and Malinee Chaiya. "Regularity of Semigroups of Transformations Whose Characters Form the Semigroup of a Δ -Structure." International Journal of Mathematics and Mathematical Sciences 2020 (December 27, 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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7

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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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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8

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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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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9

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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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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10

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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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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11

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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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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12

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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13

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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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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14

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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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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15

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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16

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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17

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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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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18

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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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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19

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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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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20

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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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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21

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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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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22

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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23

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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24

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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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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25

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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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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26

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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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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27

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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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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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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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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29

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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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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30

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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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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31

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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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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32

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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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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33

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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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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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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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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35

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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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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36

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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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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37

Sabbaghan, M., F. Ayatollah Zadeh Shirazi, and A. Hosseini. "Codecomposition of a Transformation Semigroup." Ukrainian Mathematical Journal 65, no. 11 (April 2014): 1670–80. http://dx.doi.org/10.1007/s11253-014-0888-9.

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Abolghasemi, M., A. Rejali, and H. R. E. Vishki. "On the transformation semitopological semigroup." International Mathematical Forum 2 (2007): 2245–54. http://dx.doi.org/10.12988/imf.2007.07198.

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39

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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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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40

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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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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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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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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Konieczny, Janusz. "The Semigroup of Surjective Transformations on an Infinite Set." Algebra Colloquium 26, no. 01 (March 2019): 9–22. http://dx.doi.org/10.1142/s1005386719000038.

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For an infinite set X, denote by Ω(X) the semigroup of all surjective mappings from X to X. We determine Green’s relations in Ω(X), show that the kernel (unique minimum ideal) of Ω(X) exists and determine its elements and cardinality. For a countably infinite set X, we describe the elements of Ω(X) for which the 𝒟-class and 𝒥-class coincide. We compare the results for Ω(X) with the corresponding results for other transformation semigroups on X.

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Levi, Inessa. "Group closures of one-to-one transformations." Bulletin of the Australian Mathematical Society 64, no. 2 (October 2001): 177–88. http://dx.doi.org/10.1017/s000497270003985x.

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For a semigroup S of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve S under conjugation. Fix a permutation group H on X and a transformation f of X, and let 〈f: H〉 = 〈{hfh−1: h ∈ H}〉 be the H-closure of f. We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G〈f:H〉 = H. We also show that if S is a semigroup of one-to-one transformations of X and GS contains the alternating group on X then Aut(S) = Inn(S) ≅ GS.

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Sullivan, R. P. "Partial orders on linear transformation semigroups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 135, no. 2 (April 2005): 413–37. http://dx.doi.org/10.1017/s0308210500003942.

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

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45

Levi, Inessa. "On the inner automorphisms of finite transformation semigroups." Proceedings of the Edinburgh Mathematical Society 39, no. 1 (February 1996): 27–30. http://dx.doi.org/10.1017/s0013091500022732.

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If the group of inner automorphisms of a semigroup S of transformations of a finite n-element set contains an isomorphic copy of the alternating group Altn, then S is an Sn-normal semigroup and all the automorphisms of S are inner.

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46

Kumduang, Thodsaporn, and Sorasak Leeratanavalee. "Semigroups of Terms, Tree Languages, Menger Algebra of n-Ary Functions and Their Embedding Theorems." Symmetry 13, no. 4 (March 27, 2021): 558. http://dx.doi.org/10.3390/sym13040558.

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The concepts of terms and tree languages are significant tools for the development of research works in both universal algebra and theoretical computer science. In this paper, we establish a strong connection between semigroups of terms and tree languages, which provides the tools for studying monomorphisms between terms and generalized hypersubstitutions. A novel concept of a seminearring of non-deterministic generalized hypersubstitutions is introduced and some interesting properties among subsets of its are provided. Furthermore, we prove that there are monomorphisms from the power diagonal semigroup of tree languages and the monoid of generalized hypersubstitutions to the power diagonal semigroup of non-deterministic generalized hypersubstitutions and the monoid of non-deterministic generalized hypersubstitutions, respectively. Finally, the representation of terms using the theory of n-ary functions is defined. We then present the Cayley’s theorem for Menger algebra of terms, which allows us to provide a concrete example via full transformation semigroups.

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Marques-Smith, M. Paula O., and R. P. Sullivan. "Congruences on Nilpotent-generated Partial Transformation Semigroups." Algebra Colloquium 16, no. 02 (June 2009): 229–42. http://dx.doi.org/10.1142/s1005386709000236.

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In 1987, Sullivan characterised the elements of the semigroup NP(X) generated by the nilpotents in P(X), the semigroup (under composition) consisting of all partial transformations of a set X; and in 1999, Marques-Smith and Sullivan determined all the ideals of NP(X) for arbitrary X. In this paper, we use that work to describe all the congruences on NP(X).

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48

Jude. A., Omelebele, Udoaka Otobong. G., and Udoakpan Itoro. U. "Idempotent Rank Identity Difference Transformation Semigroup." International Journal of Pure Mathematics 9 (September 22, 2022): 99–102. http://dx.doi.org/10.46300/91019.2022.9.14.

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49

Tantong, Piyaporn, and Nares Sawatraksa. "Regularity on Variants of Transformation Semigroups that Preserve an Equivalence Relation." European Journal of Pure and Applied Mathematics 15, no. 4 (October 31, 2022): 2116–26. http://dx.doi.org/10.29020/nybg.ejpam.v15i4.4596.

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The variant of a semigroup $ S $ with respect to an element $ a\in S $, is the semigroup with underlying set $ S $ and a new binary operation $ \ast $ defined by $ x\ast y=xay $ for $ x, y\in S $. Let $ T(X) $ be the full transformation semigroup on a nonempty set $ X $. For an arbitrary equivalence $ E $ on $ X $, let \[T_{E}(X)=\{\alpha\in T(X) : \forall a, b\in X, (a, b)\in E \Rightarrow (a\alpha, b\alpha\in E)\}.\] Then $ T_{E}(X) $ is a subsemigroup of $ T(X) $. In this paper, we investigate regular, left regular and right regular elements for the variant of some subsemigroups of the semigroup $ T_{E}(X) $.

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Zhao, Ping, Taijie You, and Huabi Hu. "The (p,q)-potent ranks of certain semigroups of transformations." Journal of Algebra and Its Applications 16, no. 07 (July 11, 2016): 1750138. http://dx.doi.org/10.1142/s0219498817501389.

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Let [Formula: see text] and [Formula: see text] be the partial transformation and the strictly partial transformation semigroups on the finite set [Formula: see text]. It is well known that the ranks of the semigroups [Formula: see text] and [Formula: see text] are [Formula: see text], for [Formula: see text], and [Formula: see text], for [Formula: see text], respectively. The idempotent rank, defined as the smallest number of idempotents generating set, of the semigroup [Formula: see text] has the same value as the rank. Idempotent can be seen as a special case (with [Formula: see text]) of [Formula: see text]-potent. In this paper, we determine the [Formula: see text]-potent ranks, defined as the smallest number of [Formula: see text]-potents generating set, of the semigroups [Formula: see text], for [Formula: see text], and [Formula: see text], for [Formula: see text].

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