Journal articles: 'Regular semigroup'

1

Luo, Xiao Qiang. "Π^*-Regular Semigroups." Bulletin of Mathematical Sciences and Applications 1 (August 2012): 46–51. http://dx.doi.org/10.18052/www.scipress.com/bmsa.1.46.

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El-Qallali, Abdulsalam. "Left regular bands of groups of left quotients." Glasgow Mathematical Journal 33, no. 1 (January 1991): 29–40. http://dx.doi.org/10.1017/s0017089500008004.

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In this paper we characterize semigroups S which have a semigroup Q of left quotients, where Q is an ℛ-unipotent semigroup which is a band of groups. Recall that an ℛ-unipotent (or left inverse) semigroup S is one in which every ℛ-class contains a unique idempotent. It is well-known that any ℛ-unipotent semigroup 5 is a regular semigroup in which the set of idempotents is a left regular band in that efe = ef for any idempotents e, fin S. ℛ-unipotent semigroups were studied by several authors, see for example [1] and [13].Bailes [1]characterized ℛ-unipotent semigroups which are bands of groups. This characterization extended the structure of inverse semigroups which are semilattices of groups. Recently, Gould studied in [7]the semigroup S which has a semigroup Q of left quotients where Q is an inverse semigroup which is a semilattice of groups.

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Et. al., Dr D. Mrudula Devi. "A characterization of Commutative Semigroups." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 3 (April 11, 2021): 5150–55. http://dx.doi.org/10.17762/turcomat.v12i3.2065.

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This paper deals with some results on commutative semigroups. We consider (s,.) is externally commutative right zero semigroup is regular if it is intra regular and (s,.) is externally commutative semigroup then every inverse semigroup is u – inverse semigroup. We will also prove that if (S,.) is a H - semigroup then weakly cancellative laws hold in H - semigroup. In one case we will take (S,.) is commutative left regular semi group and we will prove that (S,.) is ∏ - inverse semigroup. We will also consider (S,.) is commutative weakly balanced semigroup and then prove every left (right) regular semigroup is weakly separate, quasi separate and separate. Additionally, if (S,.) is completely regular semigroup we will prove that (S,.) is permutable and weakly separtive. One a conclusing note we will show and prove some theorems related to permutable semigroups and GC commutative Semigroups.

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KANTOROVITZ, SHMUEL. "GENERATORS OF REGULAR SEMIGROUPS." Glasgow Mathematical Journal 50, no. 1 (January 2008): 47–53. http://dx.doi.org/10.1017/s0017089507003916.

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AbstractA regular semigroup (cf. [4, p. 38]) is a C0-semigroup T(⋅) that has an extension as a holomorphic semigroup W(⋅) in the right halfplane $\Bbb C^+$, such that ||W(⋅)|| is bounded in the ‘unit rectangle’ Q:=(0, 1]× [−1, 1]. The important basic facts about a regular semigroup T(⋅) are: (i) it possesses a boundary groupU(⋅), defined as the limit lims → 0+W(s+i⋅) in the strong operator topology; (ii) U(⋅) is a C0-group, whose generator is iA, where A denotes the generator of T(⋅); and (iii) W(s+it)=T(s)U(t) for all s+it ∈$\Bbb C^+$ (cf. Theorems 17.9.1 and 17.9.2 in [3]). The following converse theorem is proved here. Let A be the generator of a C0-semigroup T(⋅). If iA generates a C0-group, U(⋅), then T(⋅) is a regular semigroup, and its holomorphic extension is given by (iii). This result is related to (but not included in) known results of Engel (cf. Theorem II.4.6 in [2]), Liu [7] and the author [6] for holomorphic extensions into arbitrary sectors, of C0-semigroups that are bounded in every proper subsector. The method of proof is also different from the method used in these references. Criteria for generators of regular semigroups follow as easy corollaries.

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Kelarev, A. V. "The regular radical of semigroup rings of commutative semigroups." Glasgow Mathematical Journal 34, no. 2 (May 1992): 133–41. http://dx.doi.org/10.1017/s001708950000865x.

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A description of regular group rings is well known (see [12]). Various authors have considered regular semigroup rings (see [17], [8], [10], [11], [4]). These rings have been characterized for many important classes of semigroups, although the general problem turns out to be rather difficult and still has not got a complete solution. It seems natural to describe the regular radical in semigroup rings for semigroups of the classes mentioned. In [10], the regular semigroup rings of commutative semigroups were described. The aim of the present paper is to characterize the regular radical ρ(R[S]) for each associative ring R and commutative semigroup S.

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6

Shoji, Kunitaka. "Regular Semigroups Which Are Amalgamation Bases for Finite Semigroups." Algebra Colloquium 14, no. 02 (June 2007): 245–54. http://dx.doi.org/10.1142/s1005386707000247.

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In this paper, we prove that a completely 0-simple (or completely simple) semigroup is an amalgamation base for finite semigroups if and only if it is an amalgamation base for semigroups. By adopting the same method as used in a previous paper, we prove that a finite regular semigroup is an amalgamation base for finite semigroups if its [Formula: see text]-classes are linearly ordered and all of its principal factor semigroups are amalgamation bases for finite semigroups. Finally, we give an example of a finite semigroup U which is an amalgamation base for semigroups, but not all of its principal factor semigroups are amalgamation bases either for semigroups or for finite semigroups.

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NI, XIANGFEI, and HAIZHOU CHAO. "REGULAR SEMIGROUPS WITH NORMAL IDEMPOTENTS." Journal of the Australian Mathematical Society 103, no. 1 (March 29, 2017): 116–25. http://dx.doi.org/10.1017/s1446788717000088.

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In this paper, we investigate regular semigroups that possess a normal idempotent. First, we construct a nonorthodox nonidempotent-generated regular semigroup which has a normal idempotent. Furthermore, normal idempotents are described in several different ways and their properties are discussed. These results enable us to provide conditions under which a regular semigroup having a normal idempotent must be orthodox. Finally, we obtain a simple method for constructing all regular semigroups that contain a normal idempotent.

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8

Wang, Shoufeng. "On E-semiabundant semigroups with a multiplicative restriction transversal." Studia Scientiarum Mathematicarum Hungarica 55, no. 2 (June 2018): 153–73. http://dx.doi.org/10.1556/012.2018.55.2.1374.

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Multiplicative inverse transversals of regular semigroups were introduced by Blyth and McFadden in 1982. Since then, regular semigroups with an inverse transversal and their generalizations, such as regular semigroups with an orthodox transversal and abundant semigroups with an ample transversal, are investigated extensively in literature. On the other hand, restriction semigroups are generalizations of inverse semigroups in the class of non-regular semigroups. In this paper we initiate the investigations of E-semiabundant semigroups by using the ideal of "transversals". More precisely, we first introduce multiplicative restriction transversals for E-semiabundant semigroups and obtain some basic properties of E-semiabundant semigroups containing a multiplicative restriction transver- sal. Then we provide a construction method for E-semiabundant semigroups containing a multiplicative restriction transversal by using the Munn semigroup of an admissible quadruple and a restriction semigroup under some natural conditions. Our construction is similar to Hall's spined product construction of an orthodox semigroup. As a corollary, we obtain a new construction of a regular semigroup with a multiplicative inverse transversal and an abundant semigroup having a multiplicative ample transversal, which enriches the corresponding results obtained by Blyth-McFadden and El-Qallali, respectively.

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9

GIGOŃ, ROMAN S. "REGULAR CONGRUENCES ON AN IDEMPOTENT-REGULAR-SURJECTIVE SEMIGROUP." Bulletin of the Australian Mathematical Society 88, no. 2 (July 30, 2013): 190–96. http://dx.doi.org/10.1017/s0004972713000270.

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AbstractA semigroup $S$ is called idempotent-surjective (respectively, regular-surjective) if whenever $\rho $ is a congruence on $S$ and $a\rho $ is idempotent (respectively, regular) in $S/ \rho $, then there is $e\in {E}_{S} \cap a\rho $ (respectively, $r\in \mathrm{Reg} (S)\cap a\rho $), where ${E}_{S} $ (respectively, $\mathrm{Reg} (S)$) denotes the set of all idempotents (respectively, regular elements) of $S$. Moreover, a semigroup $S$ is said to be idempotent-regular-surjective if it is both idempotent-surjective and regular-surjective. We show that any regular congruence on an idempotent-regular-surjective (respectively, regular-surjective) semigroup is uniquely determined by its kernel and trace (respectively, the set of equivalence classes containing idempotents). Finally, we prove that all structurally regular semigroups are idempotent-regular-surjective.

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10

Lawson, M. V. "Enlargements of regular semigroups." Proceedings of the Edinburgh Mathematical Society 39, no. 3 (October 1996): 425–60. http://dx.doi.org/10.1017/s001309150002321x.

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We introduce a class of regular extensions of regular semigroups, called enlargements; a regular semigroup T is said to be an enlargement of a regular subsemigroup S if S = STS and T = TST. We show that S and T have many properties in common, and that enlargements may be used to analyse a number of questions in regular semigroup theory.

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11

Albayrak, Barış, Didem Yeşil, and Didem Karalarlioğlu Camci. "The Source of Semiprimeness of Semigroups." Journal of Mathematics 2021 (June 1, 2021): 1–8. http://dx.doi.org/10.1155/2021/4659756.

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In this study, we define new semigroup structures using the set S S = a ∈ S | a S a = 0 which is called the source of semiprimeness for a semigroup S with zero element. S S − idempotent semigroup, S S − regular semigroup, S S − reduced semigroup, and S S − nonzero divisor semigroup which are generalizations of idempotent, regular, reduced, and nonzero divisor semigroups in semigroup theory are investigated, and their basic properties are determined. In addition, we adapt some well-known results in semigroup theory to these new semigroups.

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12

O'Meara, K. C. "Products of idempotents in regular rings." Glasgow Mathematical Journal 28, no. 2 (July 1986): 143–52. http://dx.doi.org/10.1017/s0017089500006467.

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The problem of describing the subsemigroup generated by the idempotents in various natural semigroups has received the attention of several semigroup theorists ([1], [2], [3], [5], [7]). However, in those cases where the parent semigroup is in fact the multiplicative semigroup of a natural ring, the known ring structure has not been exploited. When this ring structure is taken into account, proofs can often be streamlined and can lead to more general arguments (such as not requiring that the elements of the semigroup be already transformations of some known structure).

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13

Petrich, Mario. "Embedding Regular Semigroups into Idempotent Generated Ones." Algebra Colloquium 17, no. 02 (June 2010): 229–40. http://dx.doi.org/10.1142/s1005386710000246.

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Any semigroup S can be embedded into a semigroup, denoted by ΨS, having some remarkable properties. For general semigroups there is a close relationship between local submonoids of S and of ΨS. For a number of usual semigroup properties [Formula: see text], we prove that S and ΨS simultaneously satisfy [Formula: see text] or not. For a regular semigroup S, the relationship of S and ΨS is even closer, especially regarding the natural partial order and Green's relations; in addition, every element of ΨS is a product of at most four idempotents. For completely regular semigroups S, the relationship of S and ΨS is still closer. On the lattice [Formula: see text] of varieties of completely regular semigroups [Formula: see text] regarded as algebras with multiplication and inversion, by means of ΨS, we define an operator, denoted by Ψ. We compare Ψ with some of the standard operators on [Formula: see text] and evaluate it on a small sublattice of [Formula: see text].

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14

Guo, Xiaojiang, K. P. Shum, and Yongqian Zhu. "REES MATRIX COVERS FOR TIGHT ABUNDANT SEMIGROUPS." Asian-European Journal of Mathematics 03, no. 03 (September 2010): 409–25. http://dx.doi.org/10.1142/s1793557110000398.

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Rees matrix covers for regular semigroups were first studied by McAlister in 1984. Lawson extended McAlister's results to abundant semigroups in 1987. We consider here a semigroup whose set of regular elements forms a subsemigroup, named tight semigroups. In this paper, it is proved that an abundant semigroup is tight and locally E-solid if and only if it is an F-local isomorphic image of an abundant Rees matrix semigroup [Formula: see text] over a tight E-solid abundant semigroup T, where the entries of the sandwich matrix P of [Formula: see text] are regular elements of T. Our results enrich the result of Lawson on Rees matrix covers for a class of abundant semigroups and extend the results of McAlister on Rees matrix covers for regular semigroups.

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15

Szendrei, Mária B. "Extensions of regular orthogroups by groups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 59, no. 1 (August 1995): 28–60. http://dx.doi.org/10.1017/s1446788700038465.

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AbstractA common generalization of the author's embedding theorem concerning the E-unitary regular semigroups with regular band of idempotents, and Billhardt's and Ismaeel's embedding theorem on the inverse semigroups, the closure of whose set of idempotents is a Clifford semigroup, is presented. We prove that each orthodox semigroup with a regular band of idempotents, which is an extension of an orthogroup K by a group, can be embedded into a semidirect product of an orthogroup K′ by a group, where K′ belongs to the variety of orthogroups generated by K. The proof is based on a criterion of embeddability into a semidirect product of an orthodox semigroup by a group and uses bilocality of orthogroup bivarieties.

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16

Yuan, Zhiling, and K. P. Shum. "$\widetilde{\cal H}$-Supercryptogroups Having Regular Band Congruence." Algebra Colloquium 16, no. 04 (December 2009): 709–20. http://dx.doi.org/10.1142/s1005386709000674.

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We consider a generalized superabundant semigroup within the class of semiabundant semigroups, called a supercryptogroup since it is an analogy of a cryptogroup in the class of regular semigroups. We prove that a semigroup S is an [Formula: see text]-regular supercryptogroup if and only if S can be expressed as a refined semilattice of completely [Formula: see text]-simple semigroups. Some results on regular cryptogroups are extended to [Formula: see text]-regular supercryptogroups. Some results on superabundant semigroups are also generalized.

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17

Guo, Junying, and Xiaojiang Guo. "Self-injectivity of semigroup algebras." Open Mathematics 18, no. 1 (May 26, 2020): 333–52. http://dx.doi.org/10.1515/math-2020-0023.

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Abstract It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K, if K 0[S] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K[S] is a right (respectively, left) self-injective imply that S is finite? (Semigroup Algebras, Marcel Dekker, 1990), for IC abundant semigroups (primitively semisimple semigroups; primitive abundant semigroups). Moreover, we determine the structure of K 0[S] being right (left) self-injective when K 0[S] has a unity. As their applications, we determine some sufficient and necessary conditions for the algebra of an IC abundant semigroup (a primitively semisimple semigroup; a primitive abundant semigroup) over a field to be semisimple.

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18

Pinto, G. A. "Eventually Pointed Principally Ordered Regular Semigroups." Sultan Qaboos University Journal for Science [SQUJS] 24, no. 2 (January 19, 2020): 139. http://dx.doi.org/10.24200/squjs.vol24iss2pp139-146.

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An ordered regular semigroup, , is said to be principally ordered if for every there exists . A principally ordered regular semigroup is pointed if for every element, we have . Here we investigate those principally ordered regular semigroups that are eventually pointed in the sense that for all there exists a positive integer, , such that . Necessary and sufficient conditions for an eventually pointed principally ordered regular semigroup to be naturally ordered and to be completely simple are obtained. We describe the subalgebra of generated by a pair of comparable idempotents and such that .

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Pastijn, Francis. "The lattice of completely regular semigroup varieties." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 49, no. 1 (August 1990): 24–42. http://dx.doi.org/10.1017/s1446788700030214.

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AbstractA completely regular semigroup is a semigroup which is a union of groups. The class CR of completely regular semigroups forms a variety. On the lattice L (CR) of completely regular semigroup varieties we define two closure operations which induce complete congruences. The consideration of a third complete congruence on L (CR) yields a subdirect decomposition of L (CR). Using these results we show that L (CR) is arguesian. This confirms the (tacit) conjecture that L (CR) is modular.

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20

Tamilarasi, A. "Idempotent-separating extensions of regular semigroups." International Journal of Mathematics and Mathematical Sciences 2005, no. 18 (2005): 2945–75. http://dx.doi.org/10.1155/ijmms.2005.2945.

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For a regular biordered setE, the notion ofE-diagram and the associated regular semigroup was introduced in our previous paper (1995). Given a regular biordered setE, anE-diagram in a categoryCis a collection of objects, indexed by the elements ofEand morphisms ofCsatisfying certain compatibility conditions. With such anE-diagramAwe associate a regular semigroupRegE(A)havingEas its biordered set of idempotents. This regular semigroup is analogous to automorphism group of a group. This paper provides an application ofRegE(A)to the idempotent-separating extensions of regular semigroups. We introduced the concept of crossed pair and used it to describe all extensions of a regular semigroup S by a groupE-diagramA. In this paper, the necessary and sufficient condition for the existence of an extension ofSbyAis provided. Also we study cohomology and obstruction theories and find a relationship with extension theory for regular semigroups.

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21

Çullhaj, Fabiana, and Anjeza Krakulli. "On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups." Open Mathematics 18, no. 1 (December 15, 2020): 1501–9. http://dx.doi.org/10.1515/math-2020-0107.

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22

SALIOLA, FRANCO V. "THE QUIVER OF THE SEMIGROUP ALGEBRA OF A LEFT REGULAR BAND." International Journal of Algebra and Computation 17, no. 08 (December 2007): 1593–610. http://dx.doi.org/10.1142/s0218196707004219.

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Recently it has been noticed that many interesting combinatorial objects belong to a class of semigroups called left regular bands, and that random walks on these semigroups encode several well-known random walks. For example, the set of faces of a hyperplane arrangement is endowed with a left regular band structure. This paper studies the module structure of the semigroup algebra of an arbitrary left regular band, extending results for the semigroup algebra of the faces of a hyperplane arrangement. In particular, a description of the quiver of the semigroup algebra is given and the Cartan invariants are computed. These are used to compute the quiver of the face semigroup algebra of a hyperplane arrangement and to show that the semigroup algebra of the free left regular band is isomorphic to the path algebra of its quiver.

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23

Guo, Xiaojiang, and Lin Chen. "Semigroup algebras of finite ample semigroups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 2 (March 21, 2012): 371–89. http://dx.doi.org/10.1017/s0308210510000715.

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An adequate semigroup S is called ample if ea = a(ea)* and ae = (ae)†a for all a ∈ S and e ∈ E(S). Inverse semigroups are exactly those ample semigroups that are regular. After obtaining some characterizations of finite ample semigroups, it is proved that semigroup algebras of finite ample semigroups have generalized triangular matrix representations. As applications, the structure of the radicals of semigroup algebras of finite ample semigroups is obtained. In particular, it is determined when semigroup algebras of finite ample semigroup are semiprimitive.

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24

Lawson, Mark V. "Rees matrix covers for a class of abundant semigroups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 107, no. 1-2 (1987): 109–20. http://dx.doi.org/10.1017/s0308210500029383.

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SynopsisRecently considerable attention has been paid to the study of locally inverse regular semigroups. McAlister [14] obtained a description of such semigroups as locally isomorphic images of regular Rees matrix semigroups over an inverse semigroup. The class of abundant semigroups originally arose from ‘homological’ considerations in the theory of S-systems: they are the semigroup theoretic analogue of PP-rings. Cancellative monoids, full subsemigroups of regular semigroups as well as the multiplicative semigroups of PP rings are abundant. The aim of this paper is to show how the structure theory described above for regular semigroups may be generalised to a class of abundant semigroups.

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Munn, W. D. "Congruence-free regular semigroups." Proceedings of the Edinburgh Mathematical Society 28, no. 1 (February 1985): 113–19. http://dx.doi.org/10.1017/s0013091500003254.

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A semigroup is said to be congruence-free if and only if its only congruences are the universal relation and the identical relation. Congruence-free inverse semigroups were studied by Baird [2], Trotter [19], Munn [15,16] and Reilly [18]. In addition, results on congruence-free regular semigroups have been obtained by Trotter [20], Hall [4] and Howie [7].

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Wang, Yu, and Zhixiang Yin. "TWO PARTICULAR EVENTUALLY REGULAR SEMIGROUPS WITH 0-MODULAR OR 0-DISTRIBUTIVE SUBSEMIGROUP LATTICES." Asian-European Journal of Mathematics 06, no. 04 (December 2013): 1350046. http://dx.doi.org/10.1142/s1793557113500460.

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The structure of a completely π-regular semigroup with 0-modular or 0-distributive subsemigroup lattice is given. Furthermore, it is shown that an eventually inverse semigroup to have 0-modular or 0-distributive subsemigroup lattice is a completely π-regular semigroup which is a semilattice of completely archimedean semigroups. Thus the structure of an eventually inverse semigroup whose subsemigroup lattice is 0-modular or 0-distributive is characterized as well.

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Pastijn, F. J., and Mario Petrich. "Rees matrix semigroups over inverse semigroups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 102, no. 1-2 (1986): 61–90. http://dx.doi.org/10.1017/s0308210500014499.

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SynopsisA Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

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Ćirić, Miroslav, and Stojan Bogdanović. "Strong bands of groups of left quotients." Glasgow Mathematical Journal 38, no. 2 (May 1996): 237–42. http://dx.doi.org/10.1017/s0017089500031499.

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An interesting concept of semigroups (and also rings) of (left) quotients, based on the notion of group inverse in a semigroup, was developed by J. B. Fountain, V. Gould and M. Petrich, in a series of papers (see [5]-[12]). Among the most interesting are semigroups having a semigroup of (left) quotients that is a union of groups. Such semigroups have been widely studied. Recall from [3] that a semigroup has a group of left quotients if and only if it is right reversible and cancellative. A more general result was obtained by V. Gould [10]. She proved that a semigroup has a semilattice of groups as its semigroup of left quotients if and only if it is a semilattice of right reversible, cancellative semigroups. This result has been since generalized by A. El-Qallali [4]. He proved that a semigroup has a left regular band of groups as its semigroup of left quotients if and only if it is a left regular band of right reversible, cancellative semigroups. Moreover, he proved that such semigroups can be also characterised as punched spined products of a left regular band and a semilattice of right reversible, cancellative semigroups. If we consider the proofs of their theorems, we will observe that the principal problem treated there can be formulated in the following way: Given a semigroup S that is a band B of right reversible, cancellative semigroups Si, i ε B, to each Si, we can associate its group of left quotients Gi. When is it possible to define a multiplication of such that Q becomes a semigroup having S as its left order, and especially, that Q becomes a band B of groups Gi, i E B?Applying the methods developed in [1] (see also [2]), in the present paper we show how this problem can be solved for Qto become a strong band of groups (that is in fact a band of groups whose idempotents form a subsemigroup, by [16, Theorem 2]. Moreover, we show how Gould's and El-Quallali's constructions of semigroups of left quotients of a semilattice and a left regular band of right reversible, cancellative semigroups, can be simplified.

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Guo, Junying, and Xiaojiang Guo. "Semiprimeness of semigroup algebras." Open Mathematics 19, no. 1 (January 1, 2021): 803–32. http://dx.doi.org/10.1515/math-2021-0026.

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Abstract Abundant semigroups originate from p.p. rings and are generalizations of regular semigroups. The main aim of this paper is to study the primeness and the primitivity of abundant semigroup algebras. We introduce and study D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices of a semigroup. Based on D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices, we determine when a contracted semigroup algebra of a primitive abundant semigroup is prime (respectively, semiprime, semiprimitive, or primitive). It is well known that for any algebra A {\mathcal{A}} with unity, A {\mathcal{A}} is primitive (prime) if and only if so is M n ( A ) {M}_{n}\left({\mathcal{A}}) . Our results can be viewed as some kind of generalizations of such a known result. In addition, it is proved that any contracted semigroup algebra of a locally ample semigroup whose set of idempotents is locally finite (respectively, locally pseudofinite and satisfying the regularity condition) is isomorphic to some contracted semigroup algebra of primitive abundant semigroups. Moreover, we obtain sufficient and necessary conditions for these classes of contracted semigroup algebras to be prime (respectively, semiprime, semiprimitive, or primitive). Finally, the structure of simple contracted semigroup algebras of idempotent-connected abundant semigroups is established. Our results enrich and extend the related results on regular semigroup algebras.

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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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Azeef Muhammed, P. A., P. G. Romeo, and K. S. S. Nambooripad. "Cross-connection structure of concordant semigroups." International Journal of Algebra and Computation 30, no. 01 (October 11, 2019): 181–216. http://dx.doi.org/10.1142/s021819671950070x.

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Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalized Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalization of the cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories.

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Khan, Tanveer A., and Mark V. Lawson. "REES MATRIX COVERS FOR A CLASS OF SEMIGROUPS WITH LOCALLY COMMUTING IDEMPOTENTS." Proceedings of the Edinburgh Mathematical Society 44, no. 1 (February 2001): 173–86. http://dx.doi.org/10.1017/s0013091599001066.

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AbstractMcAlister proved that every regular locally inverse semigroup can be covered by a regular Rees matrix semigroup over an inverse semigroup by means of a homomorphism which is locally an isomorphism. We generalize this result to the class of semigroups with local units whose local submonoids have commuting idempotents and possessing what we term a ‘McAlister sandwich function’.AMS 2000 Mathematics subject classification: Primary 20M10. Secondary 20M17

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Thomas, Julie, K. Indhira, and V. M. Chandrasekaran. "A Study on Regular Semigroups and its Idempotents." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 511. http://dx.doi.org/10.14419/ijet.v7i4.10.21214.

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An idempotent of a semigroup T is an element e in T such that In many semigroups, idempotents can be recognized easily. Thus it plays an important role in the structure of semigroups especially on regular semigroups. This article reviews about some research work done about the structure of regular semigroups with a special emphasis on its idempotents.

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REN, XUEMING, DANDAN YANG, and K. P. SHUM. "ON LOCALLY EHRESMANN SEMIGROUPS." Journal of Algebra and Its Applications 10, no. 06 (December 2011): 1165–86. http://dx.doi.org/10.1142/s0219498811005129.

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It was first proved by McAlister in 1983 that every locally inverse semigroup is a locally isomorphic image of a regular Rees matrix semigroup over an inverse semigroup and Lawson in 2000 further generalized this result to some special locally adequate semigroups. In this paper, we show how these results can be extended to a class of locally Ehresmann semigroups.

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Byleen, Karl. "Embedding any countable semigroup without idempotents in a 2-generated simple semigroup without idempotents." Glasgow Mathematical Journal 30, no. 2 (May 1988): 121–28. http://dx.doi.org/10.1017/s0017089500007126.

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Although the classes of regular simple semigroups and simple semigroups without idempotents are evidently at opposite ends of the spectrum of simple semigroups, their theories involve some interesting connections. Jones [5] has obtained analogues of the bicyclic semigroup for simple semigroups without idempotents. Megyesi and Pollák [7] have classified all combinatorial simple principal ideal semigroups on two generators, showing that all are homomorphic images of one such semigroup Po which has no idempotents.

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TALWAR, SUNIL. "STRONG MORITA EQUIVALENCE AND THE SYNTHESIS THEOREM." International Journal of Algebra and Computation 06, no. 02 (April 1996): 123–41. http://dx.doi.org/10.1142/s0218196796000064.

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In recent work we associated a natural category to a semigroup and developed Morita theory for semigroups. In particular we gave a generalisation of Rees’ Theorem which led us to define what we call a Morita semigroup, this is our analogue of a structure matrix semigroup. In this article we formulate a method for extending Morita semigroups by groups. We say that a semigroup is an iterative Morita semigroup if it is obtained by successive applications of pasting families of Morita semigroups which have been extended by groups. By relying on Morita theory we show that every regular unambiguous semigroup is isomorphic to an iterative Morita semigroup of a special form. Our result can be viewed as a co-ordinate free version of the Synthesis Theorem.

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ALMEIDA, JORGE, and ALFREDO COSTA. "ON THE TRANSITION SEMIGROUPS OF CENTRALLY LABELED RAUZY GRAPHS." International Journal of Algebra and Computation 22, no. 02 (March 2012): 1250018. http://dx.doi.org/10.1142/s021819671250018x.

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Rauzy graphs of subshifts are endowed with an automaton structure. For Sturmian subshifts, it is shown that its transition semigroup is the syntactic semigroup of the language recognized by the automaton. An inverse limit of the partial semigroups of nonzero regular elements of their transition semigroups is described. If the subshift is minimal, then this inverse limit is isomorphic, as a partial semigroup, to the [Formula: see text]-class associated to it in the free pro-aperiodic semigroup.

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PETRICH, MARIO. "CHARACTERIZING SOME COMPLETELY REGULAR SEMIGROUPS BY THEIR SUBSEMIGROUPS." Journal of the Australian Mathematical Society 94, no. 3 (June 2013): 397–416. http://dx.doi.org/10.1017/s1446788713000049.

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AbstractWe consider several familiar varieties of completely regular semigroups such as groups and completely simple semigroups. For each of them, we characterize their members in terms of absence of certain kinds of subsemigroups, as well as absence of certain divisors, and in terms of a homomorphism of a concrete semigroup into the semigroup itself. For each of these varieties $ \mathcal{V} $ we determine minimal non-$ \mathcal{V} $ varieties, provide a basis for their identities, determine their join and give a basis for its identities. Most of this is complete; one of the items missing is a basis for identities for minimal nonlocal orthogroups. Three tables and a figure illustrate the results obtained.

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BILLHARDT, BERND. "EXTENSIONS OF REGULAR ORTHOGROUPS BY INVERSE SEMIGROUPS." International Journal of Algebra and Computation 05, no. 03 (June 1995): 317–42. http://dx.doi.org/10.1142/s0218196795000197.

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Let V be a variety of regular orthogroups, i.e. completely regular orthodox semigroups whose band of idempotents is regular. Let S be an orthodox semigroup which is a (normal) extension of an orthogroup K from V by an inverse semigroup G, that is, there is a congruence ρ on S such that the semigroup ker ρ of all idempotent related elements of S is isomorphic to K and S/ρ≅G. It is shown that S can be embedded into an orthodox subsemigroup T of a double semidirect product A**G where A belongs to V. Moreover T itself can be chosen to be an extension of a member from V by G. If in addition ρ is a group congruence we obtain a recent result due to M.B. Szendrei [16] which says that each orthodox semigroup which is an extension of a regular orthogroup K by a group G can be embedded into a semidirect product of an orthogroup K′ by G where K′ belongs to the variety of orthogroups generated by K.

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Kumar Bhuniya, Anjan, and Kalyan Hansda. "On Radicals of Green’s Relations in Ordered Semigroups." Canadian Mathematical Bulletin 60, no. 2 (June 1, 2017): 246–52. http://dx.doi.org/10.4153/cmb-2016-093-7.

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AbstractIn this paper, we give a new definition of radicals of Green’s relations in an ordered semigroup and characterize left regular (right regular), intra regular ordered semigroups by radicals of Green’s relations. We also characterize the ordered semigroups that are unions and complete semilattices of t-simple ordered semigroups.

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PETRICH, MARIO. "COMPLETELY REGULAR MONOIDS WITH TWO GENERATORS." Journal of the Australian Mathematical Society 90, no. 2 (April 2011): 271–87. http://dx.doi.org/10.1017/s1446788711001108.

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AbstractWe classify semigroups in the title according to whether they have a finite or an infinite number ofℒ-classes or ℛ-classes. For each case, we provide a concrete construction using Rees matrix semigroups and their translational hulls. An appropriate relatively free semigroup is used to complete the classification. All this is achieved by first treating the special case in which one of the generators is idempotent. We conclude by a discussion of a possible classification of 2-generator completely regular semigroups.

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Bonzini, C., A. Cherubini, and B. Piochi. "The Least Commutative Congruence on a simple regular ω-semigroup." Glasgow Mathematical Journal 32, no. 1 (January 1990): 13–23. http://dx.doi.org/10.1017/s0017089500009022.

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Piochi in [10] gives a description of the least commutative congruence γ of an inverse semigroup in terms of congruence pairs and generalizes to inverse semigroups the notion of solvability. The object of this paper is to give an explicit construction of λ for simple regular ω-semigroups exploiting the work of Baird on congruences on such semigroups. Moreover the connection between the solvability classes of simple regular ω-semigroups and those of their subgroups is studied.

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Xie, Xiang-yun. "On Strongly Ordered Congruences and Decompositions of Ordered Semigroups." Algebra Colloquium 15, no. 04 (December 2008): 589–98. http://dx.doi.org/10.1142/s1005386708000564.

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In this paper, we introduce the concept of a strongly ordered congruence on a directed ordered semigroup S. We prove that any strongly ordered congruence on S is a strongly regular congruence. We characterize the finite direct product, subdirect product and full subdirect product of ordered semigroups by using the concepts of strongly ordered congruence and regular congruence on an ordered semigroup S.

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Wang, Qiumei, Jianming Zhan, and R. A. Borzooei. "A study on soft rough semigroups and corresponding decision making applications." Open Mathematics 15, no. 1 (December 5, 2017): 1400–1413. http://dx.doi.org/10.1515/math-2017-0119.

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Abstract In this paper, we study a kind of soft rough semigroups according to Shabir’s idea. We define the upper and lower approximations of a subset of a semigroup. According to Zhan’s idea over hemirings, we also define a kind of new C-soft sets and CC-soft sets over semigroups. In view of this theory, we investigate the soft rough ideals (prime ideals, bi-ideals, interior ideals, quasi-ideals, regular semigroups). Finally, we give two decision making methods: one is for looking a best a parameter which is to the nearest semigroup, the other is to choose a parameter which keeps the maximum regularity of regular semigroups.

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Habib, Sana, Harish Garg, Yufeng Nie, and Faiz Muhammad Khan. "An Innovative Approach towards Possibility Fuzzy Soft Ordered Semigroups for Ideals and Its Application." Mathematics 7, no. 12 (December 3, 2019): 1183. http://dx.doi.org/10.3390/math7121183.

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The objective of this paper is put forward the novel concept of possibility fuzzy soft ideals and the possibility of fuzzy soft interior ideals. The various results in the form of the theorems with these notions are presented and further validated by suitable examples. In modern life decision-making problems, there is a wide applicability of the possibility fuzzy soft ordered semigroup which has also been constructed in the paper to solve the decision-making process. Elementary and fundamental concepts including regular, intra-regular and simple ordered semigroups in terms of possibility fuzzy soft ordered semigroup are presented. Later, the concept of left (resp. right) regular and left (resp. right) simple in terms of possibility fuzzy soft ordered semigroups are delivered. Finally, the notion of possibility fuzzy soft semiprime ideals in an ordered semigroup is defined and illustrated by theorems and example.

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Guo, Xiaojiang, Ming Zhao, and K. P. Shum. "Wreath Product Structure of Left C-rpp Semigroups." Algebra Colloquium 15, no. 01 (March 2008): 101–8. http://dx.doi.org/10.1142/s1005386708000102.

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The concept of wreath product of semigroups was first introduced by Neumann in 1960, and later on, this concept was used by Preston to investigate the structure of some inverse semigroups. In this paper, we modify the wreath product given by Neumann and Preston to study the structure of some generalized Clifford semigroups. In particular, we prove that a semigroup is a left C-rpp semigroup if and only if it is the wreath product of a left regular band and a C-rpp semigroup. Our result provides a new insight to the structure of left C-rpp semigroups.

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Auinger, K., J. Doyle, and P. R. Jones. "On existence varieties of locally inverse semigroups." Mathematical Proceedings of the Cambridge Philosophical Society 115, no. 2 (March 1994): 197–217. http://dx.doi.org/10.1017/s0305004100072042.

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AbstractA locally inverse semigroup is a regular semigroup S with the property that eSe is inverse for each idempotent e of S. Motivated by natural examples such as inverse semigroups and completely simple semigroups, these semigroups have been the subject of deep structure-theoretic investigations. The class ℒ ℐ of locally inverse semigroups forms an existence variety (or e-variety): a class of regular semigroups closed under direct products, homomorphic images and regular subsemigroups. We consider the lattice ℒ(ℒℐ) of e-varieties of such semigroups. In particular we investigate the operations of taking meet and join with the e-variety CS of completely simple semigroups. An important consequence of our results is a determination of the join of CS with the e-variety of inverse semigroups – it comprises the E-solid locally inverse semigroups. It is shown, however, that not every e-variety of E-solid locally inverse semigroups is the join of completely simple and inverse e-varieties.

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Blyth, T. S., and G. A. Pinto. "Residuated regular semigroups." Proceedings of the Edinburgh Mathematical Society 35, no. 3 (October 1992): 501–9. http://dx.doi.org/10.1017/s0013091500005770.

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We prove that in a residuated regular semigroup the elements of the form and are idempotents, and derive some consequences of this fact. In particular, we show how the maximality of such idempotents is related to the semigroup being naturally ordered, and obtain from this a characterisation of the boot-lace semigroup of [2].

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AUINGER, K., and T. E. HALL. "REPRESENTATIONS OF SEMIGROUPS BY TRANSFORMATIONS AND THE CONGRUENCE LATTICE OF AN EVENTUALLY REGULAR SEMIGROUP." International Journal of Algebra and Computation 06, no. 06 (December 1996): 655–85. http://dx.doi.org/10.1142/s0218196796000386.

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On any eventually regular semigroup S, congruences ν, μL, μR, μ, K, KL, KR, ζ are introduced which are the greatest congruences over: nil-extensions (n.e.) of completely simple semigroups, n.e. of left groups, n.e. of right groups, n.e. of groups, n.e. of rectangular bands, n.e. of left zero semigroups, n.e. of right zero semigroups, nil-semigroups, respectively. Each of these congruences is induced by a certain representation of S which is defined on an arbitrary semigroup. These congruences play an important role in the study of lattices of varieties, pseudovarieties and existence varieties. The investigation also leads to eight complete congruences U, Tt, Tr, T, K, Kl, Kr, Z on the congruence lattice Con (S) of S.

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CHAOPRAKNOI, SUREEPORN, TEERAPHONG PHONGPATTANACHAROEN, and PONGSAN PRAKITSRI. "THE NATURAL PARTIAL ORDER ON LINEAR SEMIGROUPS WITH NULLITY AND CO-RANK BOUNDED BELOW." Bulletin of the Australian Mathematical Society 91, no. 1 (October 14, 2014): 104–15. http://dx.doi.org/10.1017/s0004972714000793.

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

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Journal articles on the topic 'Regular semigroup'

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