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Journal articles on the topic 'Algorithmic and combinatorics of monoids'

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Published: 25 May 2024

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1

Cain, Alan J., António Malheiro, and Fábio M. Silva. "Combinatorics of patience sorting monoids." Discrete Mathematics 342, no. 9 (September 2019): 2590–611. http://dx.doi.org/10.1016/j.disc.2019.05.022.

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2

Abbes, S., S. Gouëzel, V. Jugé, and J. Mairesse. "Asymptotic combinatorics of Artin–Tits monoids and of some other monoids." Journal of Algebra 525 (May 2019): 497–561. http://dx.doi.org/10.1016/j.jalgebra.2019.01.019.

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3

DIEKERT, VOLKER, NICOLE ONDRUSCH, and MARKUS LOHREY. "ALGORITHMIC PROBLEMS ON INVERSE MONOIDS OVER VIRTUALLY FREE GROUPS." International Journal of Algebra and Computation 18, no. 01 (February 2008): 181–208. http://dx.doi.org/10.1142/s0218196708004366.

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Let G be a finitely generated virtually-free group. We consider the Birget–Rhodes expansion of G, which yields an inverse monoid and which is denoted by IM (G) in the following. We show that for a finite idempotent presentation P, the word problem of a quotient monoid IM (G)/P can be solved in linear time on a RAM. The uniform word problem, where G and the presentation P are also part of the input, is EXPTIME-complete. With IM (G)/P we associate a relational structure, which contains for every rational subset L of IM (G)/P a binary relation, consisting of all pairs (x,y) such that y can be obtained from x by right multiplication with an element from L. We prove that the first-order theory of this structure is decidable. This result implies that the emptiness problem for Boolean combinations of rational subsets of IM (G)/P is decidable, which, in turn implies the decidability of the submonoid membership problem of IM (G)/P. These results were known previously for free groups, only. Moreover, we provide a new algorithmic approach for these problems, which seems to be of independent interest even for free groups. We also show that one cannot expect decidability results in much larger frameworks than virtually-free groups because the subgroup membership problem of a subgroup H in an arbitrary group G can be reduced to a word problem of some IM (G)/P, where P depends only on H. A consequence is that there is a hyperbolic group G and a finite idempotent presentation P such that the word problem is undecidable for some finitely generated submonoid of IM (G)/P. In particular, the word problem of IM (G)/P is undecidable.

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4

Hurwitz, Carol M. "On the homotopy theory of monoids." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 47, no. 2 (October 1989): 171–85. http://dx.doi.org/10.1017/s1446788700031621.

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AbsractIn this paper, it is shown that any connected, small category can be embedded in a semi-groupoid (a category in which there is at least one isomorphism between any two elements) in such a way that the embedding includes a homotopy equivalence of classifying spaces. This immediately gives a monoid whose classifying space is of the same homotopy type as that of the small category. This construction is essentially algorithmic, and furthermore, yields a finitely presented monoid whenever the small category is finitely presented. Some of these results are generalizations of ideas of McDuff.

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5

BLANCHET-SADRI, F. "ALGORITHMIC COMBINATORICS ON PARTIAL WORDS." International Journal of Foundations of Computer Science 23, no. 06 (September 2012): 1189–206. http://dx.doi.org/10.1142/s0129054112400473.

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Algorithmic combinatorics on partial words, or sequences of symbols over a finite alphabet that may have some do-not-know symbols or holes, has been developing in the past few years. Applications can be found, for instance, in molecular biology for the sequencing and analysis of DNA, in bio-inspired computing where partial words have been considered for identifying good encodings for DNA computations, and in data compression. In this paper, we focus on two areas of algorithmic combinatorics on partial words, namely, pattern avoidance and subword complexity. We discuss recent contributions as well as a number of open problems. In relation to pattern avoidance, we classify all binary patterns with respect to partial word avoidability, we classify all unary patterns with respect to hole sparsity, and we discuss avoiding abelian powers in partial words. In relation to subword complexity, we generate and count minimal Sturmian partial words, we construct de Bruijn partial words, and we construct partial words with subword complexities not achievable by full words (those without holes).

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6

RENNER, LEX E. "DISTRIBUTION OF PRODUCTS IN FINITE MONOIDS I: COMBINATORICS." International Journal of Algebra and Computation 09, no. 06 (December 1999): 693–708. http://dx.doi.org/10.1142/s0218196799000394.

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7

Okniński, Jan, and Magdalena Wiertel. "Combinatorics and structure of Hecke–Kiselman algebras." Communications in Contemporary Mathematics 22, no. 07 (June 15, 2020): 2050022. http://dx.doi.org/10.1142/s0219199720500224.

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Hecke–Kiselman monoids [Formula: see text] and their algebras [Formula: see text], over a field [Formula: see text], associated to finite oriented graphs [Formula: see text] are studied. In the case [Formula: see text] is a cycle of length [Formula: see text], a hierarchy of certain unexpected structures of matrix type is discovered within the monoid [Formula: see text] and this hierarchy is used to describe the structure and the properties of the algebra [Formula: see text]. In particular, it is shown that [Formula: see text] is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand–Kirillov dimension one. This is used to characterize all Noetherian algebras [Formula: see text] in terms of the graphs [Formula: see text]. The strategy of our approach is based on the crucial role played by submonoids of the form [Formula: see text] in combinatorics and structure of arbitrary Hecke–Kiselman monoids [Formula: see text].

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8

ROSALES, J. C., P. A. GARCÍA-SÁNCHEZ, and J. I. GARCÍA-GARCÍA. "PRESENTATIONS OF FINITELY GENERATED SUBMONOIDS OF FINITELY GENERATED COMMUTATIVE MONOIDS." International Journal of Algebra and Computation 12, no. 05 (October 2002): 659–70. http://dx.doi.org/10.1142/s021819670200105x.

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We give an algorithmic method for computing a presentation of any finitely generated submonoid of a finitely generated commutative monoid. We use this method also for calculating the intersection of two congruences on ℕp and for deciding whether or not a given finitely generated commutative monoid is t-torsion free and/or separative. The last section is devoted to the resolution of some simple equations on a finitely generated commutative monoid.

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9

Garg, Vijay K. "Algorithmic combinatorics based on slicing posets." Theoretical Computer Science 359, no. 1-3 (August 2006): 200–213. http://dx.doi.org/10.1016/j.tcs.2006.03.005.

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10

Polo, Harold. "Approximating length-based invariants in atomic Puiseux monoids." Algebra and Discrete Mathematics 33, no. 1 (2022): 128–39. http://dx.doi.org/10.12958/adm1760.

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A numerical monoid is a cofinite additive submonoid of the nonnegative integers, while a Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. Using that a Puiseux monoid is an increasing union of copies of numerical monoids, we prove that some of the factorization invariants of these two classes of monoids are related through a limiting process. This allows us to extend results from numerical to Puiseux monoids. We illustrate the versatility of this technique by recovering various known results about Puiseux monoids.

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11

Li, Weimin. "Graphs with regular monoids." Discrete Mathematics 265, no. 1-3 (April 2003): 105–18. http://dx.doi.org/10.1016/s0012-365x(02)00625-8.

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12

Tsaranov, S. V. "Representation and Classification of Coxeter Monoids." European Journal of Combinatorics 11, no. 2 (March 1990): 189–204. http://dx.doi.org/10.1016/s0195-6698(13)80073-x.

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13

Šunić, Zoran. "Tamari lattices, forests and Thompson monoids." European Journal of Combinatorics 28, no. 4 (May 2007): 1216–38. http://dx.doi.org/10.1016/j.ejc.2006.02.001.

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14

Humphries, Stephen P., and Zane Kun Li. "Counting powers of words in monoids." European Journal of Combinatorics 30, no. 5 (July 2009): 1297–308. http://dx.doi.org/10.1016/j.ejc.2008.10.005.

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15

Kabil, Mustapha, Maurice Pouzet, and Ivo G. Rosenberg. "Free monoids and generalized metric spaces." European Journal of Combinatorics 80 (August 2019): 339–60. http://dx.doi.org/10.1016/j.ejc.2018.02.008.

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16

Ayyer, Arvind, Anne Schilling, Benjamin Steinberg, and Nicolas M. Thiéry. "Markov chains, ${\mathscr R}$-trivial monoids and representation theory." International Journal of Algebra and Computation 25, no. 01n02 (February 2015): 169–231. http://dx.doi.org/10.1142/s0218196715400081.

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We develop a general theory of Markov chains realizable as random walks on [Formula: see text]-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via Möbius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom–Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.

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17

Magid, Andy. "Differential Brauer monoids." Proceedings of the American Mathematical Society, Series B 10, no. 13 (April 12, 2023): 153–65. http://dx.doi.org/10.1090/bproc/162.

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The differential Brauer monoid of a differential commutative ring R R is defined. Its elements are the isomorphism classes of differential Azumaya R R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them, with entry-wise differentiation, are differentially isomorphic. The fine Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the fine Brauer monoids of R R and R D R^D and the submonoid of the Brauer monoid whose underlying Azumaya algebras are matrix rings.

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18

Fan, Yushuang, and Salvatore Tringali. "Power monoids: A bridge between factorization theory and arithmetic combinatorics." Journal of Algebra 512 (October 2018): 252–94. http://dx.doi.org/10.1016/j.jalgebra.2018.07.010.

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19

Zohre, Khaki, Saan Hossein Mohammadzadeh, and Nouri Leila. "Characterization of monoids by condition (P_E)." Quasigroups and Related Systems 30, no. 1(47) (May 2022): 91–100. http://dx.doi.org/10.56415/qrs.v30.07.

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In this paper first we recall condition (PE) and then will give general properties and a characterization of monoids for which all right acts satisfy this condition. Finally, we give a characterization of monoids, by comparing this property of their acts with some others.

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20

Kauers, Manuel, Peter Paule, and Greg Reid. "Workshop on Symbolic Combinatorics and Algorithmic Differential Algebra." ACM Communications in Computer Algebra 50, no. 1/2 (September 28, 2016): 27–34. http://dx.doi.org/10.1145/3003653.3003658.

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21

Kauers, Manuel, Peter Paule, and Greg Reid. "Workshop on symbolic combinatorics and algorithmic differential algebra." ACM Communications in Computer Algebra 50, no. 1 (April 27, 2016): 27–34. http://dx.doi.org/10.1145/2930964.2930968.

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22

Lee, Seung-Joo. "Combinatorics inN=1Heterotic Vacua." Advances in High Energy Physics 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/397895.

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We briefly review an algorithmic strategy to explore the landscape of heteroticE8×E8vacua, in the context of compactifying smooth Calabi-Yau threefolds with vector bundles. The Calabi-Yau threefolds are algebraically realised as hypersurfaces in toric varieties, and a large class of vector bundles are constructed thereon as monads. In the spirit of searching for standard-like heterotic vacua, emphasis is placed on the integer combinatorics of the model-building programme.

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23

Ateş, Fırat, Eylem G. Karpuz, Canan Kocapınar, and A. Sinan Çevik. "Gröbner–Shirshov bases of some monoids." Discrete Mathematics 311, no. 12 (June 2011): 1064–71. http://dx.doi.org/10.1016/j.disc.2011.03.008.

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24

Baginová Jajcayová, Tatiana. "Schützenberger automata for HNN-extensions of inverse monoids and their use in algorithmic questions." Information and Computation 269 (December 2019): 104448. http://dx.doi.org/10.1016/j.ic.2019.104448.

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25

Kambites, Mark. "Retracts of trees and free left adequate semigroups." Proceedings of the Edinburgh Mathematical Society 54, no. 3 (August 17, 2011): 731–47. http://dx.doi.org/10.1017/s0013091509001230.

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AbstractRecent research of the author has studied edge-labelled directed trees under a natural multiplication operation. The class of all such trees (with a fixed labelling alphabet) has an algebraic interpretation, as a free object in the class of adequate semigroups. We consider here a natural subclass of these trees, defined by placing a restriction on edge orientations, and show that the resulting algebraic structure is a free object in the class of left adequate semigroups. Through this correspondence we establish some structural and algorithmic properties of free left adequate semigroups and monoids, and consequently of the category of all left adequate semigroups.

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26

MARKUS-EPSTEIN, L. "STALLINGS FOLDINGS AND SUBGROUPS OF AMALGAMS OF FINITE GROUPS." International Journal of Algebra and Computation 17, no. 08 (December 2007): 1493–535. http://dx.doi.org/10.1142/s0218196707003846.

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Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper, we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for "sewing" on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus, all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems.

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27

AKER, KÜRŞAT, MAHIR BILEN CAN, and MÜGE TAŞKIN. "R-POLYNOMIALS OF FINITE MONOIDS OF LIE TYPE." International Journal of Algebra and Computation 20, no. 06 (September 2010): 793–805. http://dx.doi.org/10.1142/s0218196710005893.

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This paper studies the combinatorics of the orbit Hecke algebras associated with W × W orbits in the Renner monoid of a finite monoid of Lie type, M, where W is the Weyl group associated with M. It is shown by Putcha in [12] that the Kazhdan–Lusztig involution [6] can be extended to the orbit Hecke algebra which enables one to define the R-polynomials of the intervals contained in a given orbit. Using the R-polynomials, we calculate the Möbius function of the Bruhat–Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group.

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28

Chang, Ching-Lueh. "Hardness of learning loops, monoids, and semirings." Discrete Applied Mathematics 162 (January 2014): 149–58. http://dx.doi.org/10.1016/j.dam.2013.08.012.

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29

Nathanson, Melvyn B. "Free monoids and forests of rational numbers." Discrete Applied Mathematics 216 (January 2017): 662–69. http://dx.doi.org/10.1016/j.dam.2015.07.011.

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30

Putcha, Mohan S. "Bruhat-Chevalley Order in Reductive Monoids." Journal of Algebraic Combinatorics 20, no. 1 (July 2004): 33–53. http://dx.doi.org/10.1023/b:jaco.0000047291.42015.a6.

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31

Flores, Ramón, and Juan González-Meneses. "On lexicographic representatives in braid monoids." Journal of Algebraic Combinatorics 52, no. 4 (October 30, 2019): 561–97. http://dx.doi.org/10.1007/s10801-019-00913-7.

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32

Krob, Daniel, Jean Mairesse, and Ioannis Michos. "Computing the average parallelism in trace monoids." Discrete Mathematics 273, no. 1-3 (December 2003): 131–62. http://dx.doi.org/10.1016/s0012-365x(03)00233-4.

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33

Giscard, P. L., and P. Rochet. "Algebraic Combinatorics on Trace Monoids: Extending Number Theory to Walks on Graphs." SIAM Journal on Discrete Mathematics 31, no. 2 (January 2017): 1428–53. http://dx.doi.org/10.1137/15m1054535.

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34

Cain, Alan J., and António Malheiro. "Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids." Journal of Algebra 535 (October 2019): 159–224. http://dx.doi.org/10.1016/j.jalgebra.2019.06.025.

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35

Cieliebak, Mark, Thomas Erlebach, Zsuzsanna Lipták, Jens Stoye, and Emo Welzl. "Algorithmic complexity of protein identification: combinatorics of weighted strings." Discrete Applied Mathematics 137, no. 1 (February 2004): 27–46. http://dx.doi.org/10.1016/s0166-218x(03)00187-2.

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36

Cardinal, Jean, Erik D. Demaine, Martin L. Demaine, Shinji Imahori, Tsuyoshi Ito, Masashi Kiyomi, Stefan Langerman, Ryuhei Uehara, and Takeaki Uno. "Algorithmic Folding Complexity." Graphs and Combinatorics 27, no. 3 (March 17, 2011): 341–51. http://dx.doi.org/10.1007/s00373-011-1019-0.

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37

Goode, Elizabeth, and Dennis Pixton. "Recognizing splicing languages: Syntactic monoids and simultaneous pumping." Discrete Applied Mathematics 155, no. 8 (April 2007): 989–1006. http://dx.doi.org/10.1016/j.dam.2006.10.006.

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38

Coleman, Thomas D. H., David M. Evans, and Robert D. Gray. "Permutation monoids and MB-homogeneity for graphs and relational structures." European Journal of Combinatorics 78 (May 2019): 163–89. http://dx.doi.org/10.1016/j.ejc.2019.02.005.

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39

Ebanks, Bruce. "Some generalized quadratic functional equations on monoids." Aequationes mathematicae 94, no. 4 (November 5, 2019): 737–47. http://dx.doi.org/10.1007/s00010-019-00690-5.

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40

Rybalov, Alexander Nikolaevich. "On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two." Herald of Omsk University 25, no. 4 (December 28, 2020): 10–15. http://dx.doi.org/10.24147/1812-3996.2020.25(4).10-15.

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Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

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41

White, Jacob A. "Coloring complexes and combinatorial Hopf monoids." Journal of Combinatorial Theory, Series A 194 (February 2023): 105698. http://dx.doi.org/10.1016/j.jcta.2022.105698.

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42

Dvurečenskij, Anatolij, and Jiří Rachůnek. "Probabilistic averaging in bounded commutative residuated ℓ-monoids." Discrete Mathematics 306, no. 13 (July 2006): 1317–26. http://dx.doi.org/10.1016/j.disc.2005.12.024.

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43

Ahmed, Chwas, Paul Martin, and Volodymyr Mazorchuk. "On the number of principal ideals in d-tonal partition monoids." Annals of Combinatorics 25, no. 1 (January 8, 2021): 79–113. http://dx.doi.org/10.1007/s00026-020-00518-z.

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AbstractFor a positive integer d, a non-negative integer n and a non-negative integer $$h\le n$$ h ≤ n , we study the number $$C_{n}^{(d)}$$ C n ( d ) of principal ideals; and the number $$C_{n,h}^{(d)}$$ C n , h ( d ) of principal ideals generated by an element of rank h, in the d-tonal partition monoid on n elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions.

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44

Wiertel, Magdalena. "Irreducible representations of Hecke–Kiselman monoids." Linear Algebra and its Applications 640 (May 2022): 12–33. http://dx.doi.org/10.1016/j.laa.2022.01.007.

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45

Chapman, Scott T., Rolf Hoyer, and Nathan Kaplan. "Delta sets of numerical monoids are eventually periodic." Aequationes mathematicae 77, no. 3 (June 2009): 273–79. http://dx.doi.org/10.1007/s00010-008-2948-4.

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46

Sabour, Kh, B. Fadli, and S. Kabbaj. "Wilson’s functional equation on monoids with involutive automorphisms." Aequationes mathematicae 90, no. 5 (August 26, 2016): 1001–11. http://dx.doi.org/10.1007/s00010-016-0435-x.

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47

Bessy, S., and D. Rautenbach. "Algorithmic aspects of broadcast independence." Discrete Applied Mathematics 314 (June 2022): 142–49. http://dx.doi.org/10.1016/j.dam.2022.03.001.

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48

Bodlaender, Hans L. "The algorithmic theory of treewidth." Electronic Notes in Discrete Mathematics 5 (July 2000): 27–30. http://dx.doi.org/10.1016/s1571-0653(05)80116-7.

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49

Dourado, Mitre C., Fábio Protti, and Jayme L. Szwarcfiter. "Algorithmic Aspects of Monophonic Convexity." Electronic Notes in Discrete Mathematics 30 (February 2008): 177–82. http://dx.doi.org/10.1016/j.endm.2008.01.031.

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50

DIEKERT, VOLKER, and ALEXEI MYASNIKOV. "GROUP EXTENSIONS OVER INFINITE WORDS." International Journal of Foundations of Computer Science 23, no. 05 (August 2012): 1001–19. http://dx.doi.org/10.1142/s0129054112400424.

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Non-Archimedean words have been introduced as a new type of infinite words which can be investigated through classical methods in combinatorics on words due to a length function. The length function, however, takes values in the additive group of polynomials ℤ[t] (and not, as traditionally, in ℕ), which yields various new properties. Non-Archimedean words allow to solve a number of interesting algorithmic problems in geometric and algorithmic group theory. There is also a connection to logic and the first-order theory in free groups (Tarski Problems). In the present paper we provide a general method to use infinite words over a discretely ordered abelian group as a tool to investigate certain group extensions for an arbitrary group G. The central object is a group E (A, G) which is defined in terms of a non-terminating, but confluent rewriting system. The groupG as well as some natural HNN-extensions of G embed into E (A, G) (and still "behave like" G), which makes it interesting to study its algorithmic properties. In order to show that every group G embeds into E (A, G) we combine methods from combinatorics on words, string rewriting and group theory.

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