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Journal articles on the topic 'Idempotence'

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Author: Grafiati

Published: 25 May 2024

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1

Sheng, Yuqiu, and Hanyu Zhang. "Maps Preserving Idempotence on Matrix Spaces." Journal of Mathematics 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/428203.

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SupposeFis an arbitrary field. Let|F|be the number of the elements ofF. LetMn(F)be the space of alln×nmatrices overF, letSn(F)be the subset ofMn(F)consisting of all symmetric matrices, and letTn(F)be the subset ofMn(F)consisting of all upper-triangular matrices. LetV∈{Sn(F),Mn(F),Tn(F)}; a mapΦ:V→Vis said to preserve idempotence ifA-λBis idempotent if and only ifΦ(A)-λΦ(B)is idempotent for anyA,B∈Vandλ∈F. In this paper, the maps preserving idempotence onSn(F),Mn(F), andTn(F)were characterized in case|F|=3.

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2

Surbatovich, Milijana, Naomi Spargo, Limin Jia, and Brandon Lucia. "A Type System for Safe Intermittent Computing." Proceedings of the ACM on Programming Languages 7, PLDI (June 6, 2023): 736–60. http://dx.doi.org/10.1145/3591250.

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Batteryless energy-harvesting devices enable computing in inaccessible environments, at a cost to programmability and correctness. These devices operate intermittently as energy is available, using a recovery system to save and restore state. Some program tasks must execute atomically w.r.t. power failures, re-executing if power fails before completion. Any re-execution should typically be idempotent —its behavior should match the behavior of a single execution. Thus, a key aspect of correct intermittent execution is identifying and recovering state causing undesired non-idempotence. Unfortunately, past intermittent systems take an ad-hoc approach, using unsound dataflow analyses or conservatively recovering all written state. Moreover, no prior work allows the programmer to directly specify idempotence requirements (including allowable non-idempotence). We present curricle, the first type system approach to safe intermittence, for Rust. Type level reasoning allows programmers to express requirements and retains alias information crucial for sound analyses. Curricle uses information flow and type qualifiers to reject programs causing undesired non-idempotence. We implement Curricle’s type system on top of Rust’s compiler, evaluating the prototype on benchmarks from prior work. We find that Curricle benefits application programmers by allowing them to express idempotence requirements that are checked to be satisfied, and that targeting programs checked with Curricle allows intermittent system designers to write simpler recovery systems that perform better.

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3

Lee, Sang-Gu, and Jin-Woo Park. "Sign idempotent sign pattern matrices that allow idempotence." Linear Algebra and its Applications 487 (December 2015): 232–41. http://dx.doi.org/10.1016/j.laa.2015.09.020.

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4

Kuzma, B. "Additive idempotence preservers." Linear Algebra and its Applications 355, no. 1-3 (November 2002): 103–17. http://dx.doi.org/10.1016/s0024-3795(02)00340-3.

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5

HUMBERSTONE, LLOYD. "AGGREGATION AND IDEMPOTENCE." Review of Symbolic Logic 6, no. 4 (August 7, 2013): 680–708. http://dx.doi.org/10.1017/s175502031300021x.

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AbstractA 1-ary sentential context is aggregative (according to a consequence relation) if the result of putting the conjunction of two formulas into the context is a consequence (by that relation) of the results of putting first the one formula and then the other into that context. All 1-ary contexts are aggregative according to the consequence relation of classical propositional logic (though not, for example, according to the consequence relation of intuitionistic propositional logic), and here we explore the extent of this phenomenon, generalized to having arbitrary connectives playing the role of conjunction; among intermediate logics, LC, shows itself to occupy a crucial position in this regard, and to suggest a characterization, applicable to a broader range of consequence relations, in terms of a variant of the notion of idempotence we shall call componentiality. This is an analogue, for the consequence relations of propositional logic, of the notion of a conservative operation in universal algebra.

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6

Ramalingam, Ganesan, and Kapil Vaswani. "Fault tolerance via idempotence." ACM SIGPLAN Notices 48, no. 1 (January 23, 2013): 249–62. http://dx.doi.org/10.1145/2480359.2429100.

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7

Teply, Mark L. "On the idempotence and stability of kernel functors." Glasgow Mathematical Journal 37, no. 1 (January 1995): 37–43. http://dx.doi.org/10.1017/s0017089500030366.

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A kernel functor (equivalently, a left exact torsion preradical) is a left exact subfunctor of the identity on the category R-mod of left R-modules over a ring R with identity. A kernel functor is said to be idempotent if, in addition, σ satisfies σ(M / σ(M)) = 0 for every M ∊ R-mod. To every kernel functor / there corresponds a unique topologizing filter ℒσ = {I Ⅰ σ (R/I) = R/I} of left ideals and a unique class ℱσ = {M ∊ R-mod Ⅰ σ(M) = M} that is closed under homomorphic images, submodules, and direct sums. The idempotence of σ is characterized by either of the following additional conditions:(1) if I ∊ ℒσ, K ⊆ I, and (K:x) = {r ∊ R ∣ rx ∊ K} ∊ ℒσ for each x ∊ I, then K ∊ ℒ or(2) ℱσ is closed under extensions of one member of ℱσ by another member of ℱσ Idempotent kernel functors are important since they are the tool used to construct localization functors. For M∊ R-mod, let E(M) denote the injective hull of M. A kernel functor σ is called stable if Mℱ implies that E(M) ∊ ℱσ For more information about kernel functors, see [6], [7], [14], and [15].

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8

Eschenbach, Carolyn. "Idempotence for sign-pattern matrices." Linear Algebra and its Applications 180 (February 1993): 153–65. http://dx.doi.org/10.1016/0024-3795(93)90529-w.

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9

Kala, Vítězslav, and Miroslav Korbelář. "Idempotence of finitely generated commutative semifields." Forum Mathematicum 30, no. 6 (November 1, 2018): 1461–74. http://dx.doi.org/10.1515/forum-2017-0098.

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Abstract We prove that a commutative parasemifield S is additively idempotent, provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.

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10

Helland, Pat. "Idempotence is not a medical condition." Communications of the ACM 55, no. 5 (May 2012): 56–65. http://dx.doi.org/10.1145/2160718.2160734.

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11

Helland, Pat. "Idempotence Is Not a Medical Condition." Queue 10, no. 4 (April 2012): 30–46. http://dx.doi.org/10.1145/2181796.2187821.

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12

Aceto, Luca, Arnar Birgisson, Anna Ingolfsdottir, MohammadReza Mousavi, and Michel A. Reniers. "Rule formats for determinism and idempotence." Science of Computer Programming 77, no. 7-8 (July 2012): 889–907. http://dx.doi.org/10.1016/j.scico.2010.04.002.

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13

Tang, Xiao-Min, Jin-Li Xu, and Chong-Guang Cao. "A note on idempotence-preserving maps." Linear and Multilinear Algebra 56, no. 4 (July 2008): 399–414. http://dx.doi.org/10.1080/03081080701403840.

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14

Zhang, Yang, Jun Cao, and Bao-Dong Zheng. "Idempotence preserving maps between matrix spaces." Linear and Multilinear Algebra 60, no. 3 (March 2012): 349–75. http://dx.doi.org/10.1080/03081087.2011.599066.

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15

Osoba, Benard, and Temitope Gbolahan Jaiyeola. "Algebraic connections between right and middle Bol loops and their cores." Quasigroups and Related Systems 30, no. 1(47) (May 2022): 149–60. http://dx.doi.org/10.56415/qrs.v30.13.

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To every right or left Bol loop corresponds a middle Bol loop. In this paper, the cores of right Bol loops (RBL) and its corresponding middle Bol loops (MBL) were studied. Their algebraic connections were considered. It was shown that the core of a RBL is elastic and right idempotent. The core of a RBL was found to be alternative (or left idempotent) if and only if its corresponding MBL is right symmetric. If a MBL is right (left) symmetric, then, the core of its corresponding RBL is a medial (semimedial). The core of a middle Bol loop has the left inverse property (automorphic inverse property, right idempotence resp.) if and only if its corresponding RBL has the super anti-automorphic inverse property (automorphic inverse property, exponent 2 resp.). If a RBL is of exponent 2, then, the core of its corresponding MBL is left idempotent. If a RBL is of exponent 2 then: the core of a MBL has the left alternative property (right alternative property) if and only if its corresponding RBL has the cross inverse property (middle symmetry). Some other similar results were derived for RBL of exponent 3.

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16

HILL, PATRICIA M., ROBERTO BAGNARA, and ENEA ZAFFANELLA. "Soundness, idempotence and commutativity of set-sharing." Theory and Practice of Logic Programming 2, no. 2 (March 2002): 155–201. http://dx.doi.org/10.1017/s1471068401001338.

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It is important that practical data-flow analyzers are backed by reliably proven theoretical results. Abstract interpretation provides a sound mathematical framework and necessary generic properties for an abstract domain to be well-defined and sound with respect to the concrete semantics. In logic programming, the abstract domain Sharing is a standard choice for sharing analysis for both practical work and further theoretical study. In spite of this, we found that there were no satisfactory proofs for the key properties of commutativity and idempotence that are essential for Sharing to be well-defined and that published statements of the soundness of Sharing assume the occurs-check. This paper provides a generalization of the abstraction function for Sharing that can be applied to any language, with or without the occurs-check. Results for soundness, idempotence and commutativity for abstract unification using this abstraction function are proven.

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17

Cui, Jian Lian, and Jin Chuan Hou. "Linear Maps Preserving Idempotence on Nest Algebras." Acta Mathematica Sinica, English Series 20, no. 5 (June 21, 2004): 807–20. http://dx.doi.org/10.1007/s10114-004-0314-6.

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18

Liu, Sizhe. "Study for Identity Losses in Image-to-Image Domain Translation with Cycle-Consistent Generative Adversarial Network." Journal of Physics: Conference Series 2400, no. 1 (December 1, 2022): 012030. http://dx.doi.org/10.1088/1742-6596/2400/1/012030.

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Abstract The effect of identity loss in image-to-image domain translation using generative adversarial networks (GAN) remains largely uncertain given its wide application in many successful GAN architectures, e.g., CycleGAN. In this study, negative identity regularization and idempotence regularization are identified from several related studies, and a novel loss, positive identity loss which limits the generative model’s output on the input of positive samples from the source domain, is defined. Results from identity regularized GANs and those without are compared and analyzed. This study concludes that: 1) negative identity loss is helpful in regularizing CycleGAN’s tendency to apply inversible but non-identity transform on non-features such as coloration and background landscapes; 2) positive identity loss significantly increases resolution, details, and affinity with the original image on top of negative identity loss; 3) idempotence loss has positive but limited impacts on output image quality.

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19

J. D. H., Smith. "Quantum idempotence, distributivity, and the Yang-Baxter equation." Commentationes Mathematicae Universitatis Carolinae 57, no. 4 (February 20, 2017): 567–83. http://dx.doi.org/10.14712/1213-7243.2015.186.

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20

Ronse, C. "On idempotence and related requirements in edge detection." IEEE Transactions on Pattern Analysis and Machine Intelligence 15, no. 5 (May 1993): 484–91. http://dx.doi.org/10.1109/34.211468.

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21

Surbatovich, Milijana, Limin Jia, and Brandon Lucia. "I/O dependent idempotence bugs in intermittent systems." Proceedings of the ACM on Programming Languages 3, OOPSLA (October 10, 2019): 1–31. http://dx.doi.org/10.1145/3360609.

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22

Sheng, Yu Qiu, Bao Dong Zheng, and Xian Zhang. "Maps on spaces of symmetric matrices preserving idempotence." Linear Algebra and its Applications 420, no. 2-3 (January 2007): 576–85. http://dx.doi.org/10.1016/j.laa.2006.08.012.

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23

Sheng, Yuqiu, Baodong Zheng, and Xian Zhang. "Idempotence preserving maps on spaces of triangular matrices." Journal of Applied Mathematics and Computing 25, no. 1-2 (September 2007): 17–33. http://dx.doi.org/10.1007/bf02832336.

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24

Wu, Joz. "Idempotence and Orthogonality in Relation to Mixed Model Adjustments." Journal of Surveying Engineering 129, no. 4 (November 2003): 141–45. http://dx.doi.org/10.1061/(asce)0733-9453(2003)129:4(141).

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25

Zhang, Xian. "Idempotence-preserving maps without the linearity and surjectivity assumptions." Linear Algebra and its Applications 387 (August 2004): 167–82. http://dx.doi.org/10.1016/j.laa.2004.02.011.

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26

Schmidt-Schauss, Manfred. "Unification under associativity and idempotence is of type nullary." Journal of Automated Reasoning 2, no. 3 (September 1986): 277–81. http://dx.doi.org/10.1007/bf02328450.

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27

Torres, Pablo, and Mario Valencia-Pabon. "Shifts of the stable Kneser graphs and hom-idempotence." European Journal of Combinatorics 62 (May 2017): 50–57. http://dx.doi.org/10.1016/j.ejc.2016.11.012.

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28

Du, Xiankun. "LEFT T-IDEMPOTENCE AND LEFT T-STABILITY OF SEMIGROUP RINGS." Communications in Algebra 29, no. 12 (January 1, 2001): 5477–97. http://dx.doi.org/10.1081/agb-100107940.

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29

Liu, Shaowu, and Guifang Yuan. "Linear operators preserving idempotence on matrices spaces over skew-fields." Chinese Science Bulletin 42, no. 15 (August 1997): 1244–47. http://dx.doi.org/10.1007/bf02882749.

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30

BENVENUTI, PIETRO, DORETTA VIVONA, and MARIA DIVARI. "AGGREGATION OPERATORS AND ASSOCIATED FUZZY MEASURES." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 09, no. 02 (April 2001): 197–204. http://dx.doi.org/10.1142/s0218488501000739.

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The natural properties of the aggregation operators and the most elementary ones are the idempotence, the monotonicity and the continuity from below. We assume only these properties for the aggregation operators with infinitely many inputs, defined by functionals on the family of measurable functions. A family of fuzzy measures is associated with each aggregation operator. The properties of horizontal or vertical pseudo-additivity are recognizable by means of this family of fuzzy measures.

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31

Shaowu, Liu. "Linear maps preserving idempotence on matrix modules over principal ideal domains." Linear Algebra and its Applications 258 (June 1997): 219–31. http://dx.doi.org/10.1016/s0024-3795(96)00203-0.

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32

Xu, Jin-Li, Chong-Guang Cao, and Xiao-Min Tang. "Linear preservers of idempotence on triangular matrix spaces over any field." International Mathematical Forum 2 (2007): 2305–19. http://dx.doi.org/10.12988/imf.2007.07205.

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33

Ong, Kai Lin, and Miin Huey Ang. "Full Identification of Idempotens in Binary Abelian Group Rings." Journal of the Indonesian Mathematical Society 23, no. 2 (December 24, 2017): 67–75. http://dx.doi.org/10.22342/jims.23.2.288.67-75.

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Every code in the latest study of group ring codes is a submodule thathas a generator. Study reveals that each of these binary group ring codes can havemultiple generators that have diverse algebraic properties. However, idempotentgenerators get the most attention as codes with an idempotent generator are easierto determine its minimal distance. We have fully identify all idempotents in everybinary cyclic group ring algebraically using basis idempotents. However, the conceptof basis idempotent constrained the exibilities of extending our work into the studyof identication of idempotents in non-cyclic groups. In this paper, we extend theconcept of basis idempotent into idempotent that has a generator, called a generatedidempotent. We show that every idempotent in an abelian group ring is either agenerated idempotent or a nite sum of generated idempotents. Lastly, we show away to identify all idempotents in every binary abelian group ring algebraically by fully obtain the support of each generated idempotent.

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34

Minati, Ludovico, Julia Winkel, Angelo Bifone, Paweł Oświęcimka, and Jorge Jovicich. "Self-similarity and quasi-idempotence in neural networks and related dynamical systems." Chaos: An Interdisciplinary Journal of Nonlinear Science 27, no. 4 (April 2017): 043115. http://dx.doi.org/10.1063/1.4981908.

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35

BAETS, BERNARD DE, ETIENNE KERRE, and MADAN GUPTA. "THE FUNDAMENTALS OF FUZZY MATHEMATICAL MORPHOLOGY PART 2: IDEMPOTENCE, CONVEXITY AND DECOMPOSITION." International Journal of General Systems 23, no. 4 (March 1995): 307–22. http://dx.doi.org/10.1080/03081079508908045.

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36

Yao, Hong Mei, Chong Guang Cao, and Xian Zhang. "Additive preservers of idempotence and Jordan homorphisms between rings of square matrices." Acta Mathematica Sinica, English Series 25, no. 4 (March 25, 2009): 639–48. http://dx.doi.org/10.1007/s10114-009-7165-0.

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37

Cockett, J. Robin B., Jalel Zrida, and J. Douglas Birdwell. "Stochastic Decision Theory." Probability in the Engineering and Informational Sciences 3, no. 1 (January 1989): 13–54. http://dx.doi.org/10.1017/s0269964800000966.

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The manipulations and basic results of stochastic decision theory are introduced. The manipulations of idempotence, transposition, and repetition, introduced for deterministic decision trees, can be used to manipulate stochastic trees. However, there are two major differences. First, in order to obtain a complete set of manipulations it is necessary to introduce an additional rule called indifference. Second, these identities must be treated as rules of inference. Not all the rules can be soundly applied in both directions; in particular, idempotence is a one-way rule.A manipulation of a stochastic decision tree not only alters the structure of the tree, but also the probability distributions associated with the tree. This allows probability calculation to be viewed as structural manipulation. In particular, a retrieval corresponds to a conditional probability calculation. The algorithm for doing this calculation has, therefore, many applications. For example, the solution to the classical state-estimation problem and the retrieval of information from probabilistic or uncertain knowledge bases may both be viewed as an application of this algorithm.The main result of this paper is that these manipulations are complete and sound. In order to prove this result, it is necessary to have a semantic setting for these theories. The setting chosen is the category of description spaces which is a generalization of the category of bounded measure spaces with maps which do not increase measure. The proof of this result exploits the retrieval properties of stochastic terms and its relationship to conditional probability calculations in the models.

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38

RICCI, ROBERTO GHISELLI. "ASYMPTOTICALLY IDEMPOTENT AGGREGATION OPERATORS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17, no. 05 (October 2009): 611–31. http://dx.doi.org/10.1142/s0218488509006170.

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This paper deals with aggregation operators. A new form of idempotency, called asymptotic idempotency, is introduced. A critical discussion of the basic notion of aggregation operator, strictly connected with asymptotic idempotency, is provided. Some general construction methods of symmetric, asymptotically idempotent aggregation operators admitting a neutral element are illustrated.

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39

Chao, Haizhou, and Xiangfei Ni. "On abundant semigroups withweak normal idempotents." Filomat 34, no. 4 (2020): 1241–49. http://dx.doi.org/10.2298/fil2004241c.

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A weak normal idempotent of an abundant semigroup was introduced by Guo [7]. In this paper, weak normal idempotents and normal idempotents of abundant semigroups are respectively characterized in many different ways. These results enable us to obtain an example which shows that the class of normal idempotents of abundant semigroups is a proper subclass of normal idempotents of abundant semigroups. Furthermore, this example tell us that there exists a non-regular abundant semigroup containing a weak normal idempotent. At last, we investigate the relationships between weak normal idempotents and normal idempotents and deduce that the main result of [2] can not be generalized into the class of abundant semigroups.

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40

Chao, Haizhou, and Xiangfei Ni. "On abundant semigroups withweak normal idempotents." Filomat 34, no. 4 (2020): 1241–49. http://dx.doi.org/10.2298/fil2004241c.

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A weak normal idempotent of an abundant semigroup was introduced by Guo [7]. In this paper, weak normal idempotents and normal idempotents of abundant semigroups are respectively characterized in many different ways. These results enable us to obtain an example which shows that the class of normal idempotents of abundant semigroups is a proper subclass of normal idempotents of abundant semigroups. Furthermore, this example tell us that there exists a non-regular abundant semigroup containing a weak normal idempotent. At last, we investigate the relationships between weak normal idempotents and normal idempotents and deduce that the main result of [2] can not be generalized into the class of abundant semigroups.

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41

MinTang, Xiao, Chong GuangCao, and Xian Zhang. "Modular automorphisms preserving idempotence and Jordan isomorphisms of triangular matrices over commutative rings." Linear Algebra and its Applications 338, no. 1-3 (November 2001): 145–52. http://dx.doi.org/10.1016/s0024-3795(01)00379-2.

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42

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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43

GILBERT, N. D., and R. NOONAN HEALE. "THE IDEMPOTENT PROBLEM FOR AN INVERSE MONOID." International Journal of Algebra and Computation 21, no. 07 (November 2011): 1179–94. http://dx.doi.org/10.1142/s0218196711006893.

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We generalize the word problem for groups, the formal language of all words in the generators that represent the identity, to inverse monoids. In particular, we introduce the idempotent problem, the formal language of all words representing idempotents, and investigate how the properties of an inverse monoid are related to the formal language properties of its idempotent problem. We show that if an inverse monoid is either E-unitary or has a finite set of idempotents, then its idempotent problem is regular if and only if the inverse monoid is finite. We also give examples of inverse monoids with context-free idempotent problems, including all Bruck–Reilly extensions of finite groups.

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44

Liu, Xing Xiang, and Wen Ya Cheng. "The Generalized Idempotence of Linear Combination on Generalized Three-Idemfactor and an Arbitrary Matrix." Advanced Materials Research 765-767 (September 2013): 678–82. http://dx.doi.org/10.4028/www.scientific.net/amr.765-767.678.

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Generalized emfactor is an important sort of matrix, if we can ulteriorly study its characters and application, there will optimize the procedure of finding its answer. So the relevant properties discussed more and more.

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45

Sullivan, R. P., and Rachel Thomas. "Products of three idempotent transformations." Bulletin of the Australian Mathematical Society 68, no. 1 (August 2003): 57–71. http://dx.doi.org/10.1017/s0004972700037412.

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In 1988 Howie, Robertson and Schein characterised the transformations of a finite set X that can be written as a product of two or of three idempotent transformations of X; and in 1989 Saito did the same for products of four idempotents. In 1998 Thomas extended the characterisation of two idempotents to arbitrary sets, and here we characterise products of three idempotents in general. We also define a notion of complexity for transformations of any set and use it to provide a different solution to the three-idempotent problem.

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46

D’Osualdo, Emanuele, Azadeh Farzan, and Derek Dreyer. "Proving hypersafety compositionally." Proceedings of the ACM on Programming Languages 6, OOPSLA2 (October 31, 2022): 289–314. http://dx.doi.org/10.1145/3563298.

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Hypersafety properties of arity n are program properties that relate n traces of a program (or, more generally, traces of n programs). Classic examples include determinism, idempotence, and associativity. A number of relational program logics have been introduced to target this class of properties. Their aim is to construct simpler proofs by capitalizing on structural similarities between the n related programs. We propose an unexplored, complementary proof principle that establishes hyper-triples (i.e. hypersafety judgments) as a unifying compositional building block for proofs, and we use it to develop a Logic for Hyper-triple Composition (LHC), which supports forms of proof compositionality that were not achievable in previous logics. We prove LHC sound and apply it to a number of challenging examples.

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47

Cristea, Irina, Juš Kocijan, and Michal Novák. "Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures." Mathematics 7, no. 10 (September 23, 2019): 885. http://dx.doi.org/10.3390/math7100885.

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The aim of this paper is to study, from an algebraic point of view, the properties of interdependencies between sets of elements (i.e., pieces of secrets, atmospheric variables, etc.) that appear in various natural models, by using the algebraic hyperstructure theory. Starting from specific examples, we first define the relation of dependence and study its properties, and then, we construct various hyperoperations based on this relation. We prove that two of the associated hypergroupoids are H v -groups, while the other two are, in some particular cases, only partial hypergroupoids. Besides, the extensivity and idempotence property are studied and related to the cyclicity. The second goal of our paper is to provide a new interpretation of the dependence relation by using elements of the theory of algebraic hyperstructures.

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48

Zhu, Haiyang, Jianlong Chen, and Yukun Zhou. "On elements whose (b,c)-inverse is idempotent in a monoid." Filomat 36, no. 14 (2022): 4645–53. http://dx.doi.org/10.2298/fil2214645z.

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In this paper, we investigate the elements whose (b, c)-inverse is idempotent in a monoid. Let S be a monoid and a, b, c ? S. Firstly, we give several characterizations for the idempotency of a||(b,c) as follows: a||(b,c) exists and is idempotent if and only if cab = cb, cS = cbS, Sb = Scb if and only if both a||(b,c) and 1||(b,c) exist and a||(b,c) = 1||(b,c), which establish the relationship between a||(b,c) and 1||(b,c). They imply that a||(b,c) merely depends on b, c but is independent of a when a||(b,c) exists and is idempotent. Particularly, when b = c, more characterizations which ensure the idempotency of a||b by inner and outer inverses are given. Finally, the relationship between a||b and a||bn for any n ? N+ is revealed.

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Imam, A. T., S. Ibrahim, G. U. Garba, L. Usman, and A. Idris. "Quasi-idempotents in finite semigroup of full order-preserving transformations." Algebra and Discrete Mathematics 35, no. 1 (2023): 62–72. http://dx.doi.org/10.12958/adm1846.

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Let Xn be the finite set {1,2,3· · ·, n} and On defined by On={α∈Tn:(∀x, y∈Xn), x⩽y→xα⩽yα}be the semigroup of full order-preserving mapping on Xn. A transformation α in On is called quasi-idempotent if α=α2=α4. We characterise quasi-idempotent in On and show that the semigroup On is quasi-idempotent generated. Moreover, we obtained an upper bound forquasi-idempotents rank of On, that is, we showed that the cardinality of a minimum quasi-idempotents generating set for On is less than or equal to ⌈3(n−2)2⌉ where ⌈x⌉ denotes the least positive integerm such that x⩽m<x+ 1.

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50

Zhao, Ping. "On the semigroups of order-preserving transformations generated by idempotents of rank n −1." Journal of Algebra and Its Applications 16, no. 02 (February 2017): 1750023. http://dx.doi.org/10.1142/s0219498817500232.

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Let [Formula: see text] be the semigroup of all singular order-preserving mappings on the finite set [Formula: see text]. It is known that [Formula: see text] is generated by its set of idempotents of rank [Formula: see text], and its rank and idempotent rank are [Formula: see text] and [Formula: see text], respectively. In this paper, we study the structure of the semigroup generated by any nonempty subset of idempotents of rank [Formula: see text] in [Formula: see text]. We also calculate its rank and idempotent rank.

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