Algebra in the Stone–Čech Compactification: An Update
Keywords:
notions of size, Ramsey theory, semigroups, Stone–Čech compactificationAbstract
The first edition of Algebra in the Stone-Čech Compactification was published in 1998 and the second edition in 2012. Since that time there have been many new results published about the algebraic structure of the Stone-Čech compactification $\beta S$ of the discrete semigroup $S$ and the combinatorial applications of that structure, mostly in the area of Ramsey theory. We present here, with proofs so far as possible, what we believe to be some of the most significant of these new results.
References
Ben Barber, Neil Hindman, and Imre Leader, Partition regularity in the rationals, J. Comb. Theory (Series A) 120 (2013), no. 7, 1590-1599.
Ben Barber, Neil Hindman, Imre Leader, and Dona Strauss, Distinguishing subgroups of the rationals by their Ramsey properties, J. Comb. Theory (Series A) 129 (2015), 93-104. (MR3275116)
Ben Barber, Neil Hindman, Imre Leader, and Dona Strauss, Partition regularity without the columns property, Proc. Amer. Math. Soc. 143 (2015), no. 8, 33873399. (MR3348781)
Ben Barber and Imre Leader, Partition regularity with congruence conditions, J. Comb. 4 (2013), no. 3, 293-297.
E. Bayatmanesh and M. Akbari Tootkaboni, Central sets theorem near zero, Topology Appl. 210 (2016), 70-80.
Vitaly Bergelson, Ergodic Ramsey theory - an update in Ergodic Theory of $\mathbb{Z}^d$ Actions (Warwick, 1993-1994). Ed. Mark Pollicott and Klaus Schnidt. London Math. Soc. Lecture Note Ser., 228. Cambridge: Cambridge University Press, 1996. 1-61.
Vitaly Bergelson and Daniel Glasscock, On the interplay between additive and multiplicative largeness and its combinatorial applications J. Combin. Theory Ser. A 172(2020), 105203,60 pp.
Vitaly Bergelson and Neil Hindman, Some new multi-cell Ramsey theoretic results, Proc. Amer. Math. Soc. Ser. B 8 (2021), 358-370.
Vitaly Bergelson, Neil Hindman, and Kendall Williams, Polynomial extensions of the Milliken-Taylor theorem, Trans. Amer. Math. Soc. 366 (2014), no. 11, 5727-5748.
Vitaly Bergelson, John H. Johnson, Jr., and Joel Moreira, New polynomial and multidimensional extensions of classical partition results, J. Combin. Theory Ser. A 147 (2017), 119-154.
Vitaly Bergelson and Alexander Leibman, Polynomial extensions of van der Waerden's and Szemeréds's theorems, J. Amer. Math. Soc. 9 (1996), no. 3, 725-753.
V. Bergelson and A. Leibman, Set-polynomials and polynomial extension of the Hales-Jewett theorem, Ann. of Math. (2) 150 (1999), no.1, 33-75.
V. Bergelson and A. Leibman, $I P_r^*$-recurrence and nilsystems, Adv. Math. 339 (2018), 642-656.
Vitaly Bergelson and Randall Mc Cutcheon, An ergodic IP polynomial Szemerédì theorem, Mem. Amer. Math. Soc. 146 (2000), no. 695, viii+106 pp.
Vitaly Bergelson and Joel Moreira, Ergodic theorem involving additive and multiplicative groups of a field and $\{x+y, x y\}$ patterns, Ergodic Theory Dynam. Systems 37 (2017), no. 3, 673-692.
Vitaly Bergelson and Joel Moreira, Measure preserving actions of affine semigroups and $\{x+y, x y\}$ patterns, Ergodic Theory Dynam. Systems 38 (2018), no. 2,473-498.
Vitaly Bergelson and Donald Robertson, Polynomial multiple recurrence over rings of integers, Ergodic Theory Dynam. Systems 36 (2016), no. 5, 1354-1378.
Vitaly Bergelson and Donald Robertson, Polynomial recurrence with large intersection over countable fields, Israel J. Math. 214 (2016), no. 1, 109-120.
Vitaly Bergelson and Rigoberto Zelada, Iterated differences sets, Diophantine approximations and applications, J. Combin. Theory Ser. A 184 (2021), Paper No. 105520,51 pp.
Tanushree Biswas, Dibyendu De, and Ram Krishna Paul, Matrices centrally image partition regular near 0, New York J. Math. 21 (2015), 601-613.
Garith Botha and Yevhen Zelenyuk, On closed left ideal decompositions of $G^*$, Topology Appl. 160 (2013), no. 1, 133-136.
Garith Botha, Yevhen Zelenyuk, and Yuliya Zelenyuk, Finer closed left ideal decompositions of $G^*$, Topology Appl. 165 (2014), 58-63.
A. Bouziad and M. Filali, The Stone-Crech compactification of a topological group as a semigroup and the SIN property, Houston J. Math. 38 (2012), no. 4, 1329-1341.
Matt Bowen and Marcin Sabok, Monochromatic products and sums in the rationals. Available at https://arXiv.org/pdf/2210.12290[math.CO].
W. R. Brian, $P$-sets and minimal right ideals in $\mathbb{N}^*$ Fund. Math. 229 (2015), no. 3, 277-293.
William R. Brian, Ideals and idempotents in the uniform ultrafilters, Topology Appl. 237 (2018), 53-66.
Will Brian and Neil Hindman, Factoring a minimal ultrafilter into a thick part and a syndetic part, Fund. Math. 252 (2021), 121-145.
Tom Brown, Monochromatic solutions of exponential equations, Integers 15A (2015), Paper No. A 2, 9 pp.
Michelangelo Bucci, Svetlana Puzynina, and Luca Q. Zamboni, Central sets generated by uniformly recurrent words, Ergodic Theory Dynam. Systems 35 (2015), no. $3,714-736$.
Lorenzo Carlucci and David Fernández-Bretón, The adjacent Hindman's theorem for uncountable groups, Colloq. Math. 173 (2023), no. 2, 273-284.
Aninda Chakraborty and Sayan Goswami, Richness of arithmetic progressions in commutative semigroups, Integers 20 (2020), Paper No. A24, 9 pp.
Aninda Chakraborty and Sayan Goswami, Polynomial central set theorem near zero, Semigroup Forum 102 (2021), no. 2, 568-574.
Sukrit Chakraborty and Sourav Kanti Patra, In finite image partition regular matrices - solution in $C$-sets, Bull. Braz. Math. Soc. (N.S.) 52 (2021), no. 2, 253-265.
Jonathan Chapman, Partition regularity and multiplicatively syndetic sets, Acta Arith. 196 (2020), no. 2, 109-138.
Cory Christopherson and John H. Johnson, Jr., Algebraic characterizations of some relative notions of size, Semigroup Forum 104 (2022), no. 1, 28-44.
H. Garth Dales, Dona E. Strauss, Yevhen Zelenyuk, and Yuliya Zelenyuk, Radicals of some semigroup algebras, Semigroup Forum 87 (2013), no. 1, 80-96.
Dennis Davenport, Neil Hindman, Imre Leader, and Dona Strauss, Multiply partition regular matrices, Discrete Math. 322 (2014), 61-68.
Dibyendu De and Subhajit Jana, Image partition regularity over the Gaussian integers, Integers 17 (2017), Paper No. A59, 20 pp.
Dibyendu De and Ram Krishna Paul, Combined additive and multiplicative properties near zero, New York J. Math. 18 (2012), 353-360.
Pintu Debnath and Sayan Goswami, Abundance of arithmetic progressions in some combinatorially rich sets by elementary means, Integers 21 (2021), Paper No. A 105, 7 pp.
Pintu Debnath and Sayan Goswami, Dynamical IP*-sets in weak rings, Topology Appl. 303 (2021), Paper No. 107854, 7 pp.
Mauro Di Nasso, Infinite monochromatic patterns in the integers, J. Combin. Theory Ser. A 189 (2022), Paper No. 105610, 28 pp.
Mauro Di Nasso and Mariaclara Ragosta, Monochromatic exponential triples: An ultrafilter proof. To appear in Proc. Amer. Math. Soc.
DOI: https://doi.org/10.1090/proc/16547.
David J. Fernández-Bretón, Every strongly summable ultrafilter on $\bigoplus \mathbb{Z}_2$ is sparse, New York J. Math. 19 (2013), 117-129.
David J. Fernández-Bretón, Strongly summable ultrafilters, union ultrafilters, and the trivial sums property, Canad. J. Math. 68 (2016), no. 1, 44-66.
David J. Fernández-Bretón, Hindman's theorem is only a countable phenomenon, Order 35 (2018), no. 1, 83-91.
David José Fernández-Bretón, Stable ordered union ultrafilters and $\operatorname{cov}(\mathcal{M})\lt\mathfrak{c}$, J. Symb. Log. 84 (2019), no. 3, 1176-1193.
David J. Fernández-Bretón, Using ultrafilters to prove Ramsey-type theorems, Amer. Math. Monthly 129 (2022), no. 2, 116-131.
David J. Fernández-Bretón and Martino Lupini, Strongly productive ultrafilters on semigroups, Semigroup Forum 92 (2016), no. 1, 242-257.
Harry Furstenberg, Recurrence in Ergodic Theory and Combinatorical Number Theory. Princeton, NJ: Princeton University Press, 1981.
H. Furstenberg and Y. Katznelson, A density version of the Hales-Jewett theorem, J. Anal. Math. 57 (1991), 64-119.
Arthur D. Grainger, The cardinality of $\beta_A\left(S_J\right)$, Semigroup Forum 85 (2012), no. 2, 213-226.
David S. Gunderson, Neil Hindman, and Hanno Lefmann, Some partition theorems for infinite and finite matrices, Integers 14 (2014), Paper No. A12, 20 pp .
Arpita Ghosh, A generalized central sets theorem in partial semigroups, Semigroup Forum 100 (2020), no. 1, 169-179.
Neil Hindman, Partitions and sums and products of integers, Trans. Amer. Math. Soc. 247 (1979), 227-245.
Neil Hindman, Problems and new results in the algebra of $\beta \mathbb{N}$ and its application to Ramsey theory in Unsolved Problems on Mathematics for the 21st Century. Ed. Jair Minoro Abe and Shotaro Tanaka. Amsterdam: IOS Press, 2001. 295-305. Neil Hindman, Partition regularity of matrices, Integers 7 (2007), no. 2, A-18, 34 pp .
Neil Hindman, Maria-Romina Ivan, and Imre Leader, Some new results on monochromatic sums and products in the rationals, New York J. Math. 29 (2023), 301-322.
Neil Hindman and John H. Johnson, Jr., Images of $C$ sets and related large sets under nonhomogeneous spectra, Integers 12B (2012), Paper No. 2, 25 pp.
Neil Hindman and Lakeshia Legette Jones, Idempotents in $\beta S$ that are only products trivially, New York J. Math. 20 (2014), 57-80.
Neil Hindman, Lakeshia Legette Jones, and Monique Agnes Peters, Left large subsets of free semigroups and groups that are not right large, Semigroup Forum 90 (2015), no. 2, 374-385.
Neil Hindman, Lakeshia Legette Jones, and Dona Strauss, The relationships among many notions of largeness for subsets of a semigroup, Semigroup Forum 99 (2019), no. 1, 9-31.
Neil Hindman, Imre Leader, and Dona Strauss, Extensions of infinite partition regular systems, Electron. J. Combin. 22 (2015), no. 2, Paper 2.29, 28 pp.
Neil Hindman, Imre Leader, and Dona Strauss, Duality for image and kernel partition regularity of infinite matrices, J. Comb. 8 (2017), no. 4, 653-672.
Neil Hindman, Imre Leader, and Dona Strauss, Pairwise sums in colourings of the reals, Abh. Math. Semin. Univ. Hambg. 87 (2017), no. 2, 275-287.
Neil Hindman, Amir Maleki, and Dona Strauss, Central sets and their combinatorial characterization, J. Combin. Theory Ser. A 74 (1996), no. 2, 188-208.
Neil Hindman and Kendra Pleasant, Central Sets Theorem for arbitrary adequate partial semigroups, Topology Proc. 58 (2021), 183-206.
Neil Hindman, Juris Steprāns, and Dona Strauss, Semigroups in which all strongly summable ultrafilters are sparse, New York J. Math. 18 (2012), 835-848. (MR2991425)
Neil Hindman and Dona Strauss, Density and invariant means in left amenable semigroups, Topology Appl. 156 (2009), no. 16, 2614-2628.
Neil Hindman and Dona Strauss, Algebra in the Stone-Cech Compactification:
Theory and Applications. 2nd ed., Berlin/Boston: Walter de Gruyter, 2012.
Neil Hindman and Dona Strauss, The center and extended center of the maximal groups in the smallest ideal of $\beta \mathbb{N}$, Topology Proc. 42 (2013), 107-119. (MR2979964)
Neil Hindman and Dona Strauss, Separating Milliken-Taylor systems in $\mathbb{Q}$, J. Comb. 5 ( 2014), 305-333. (MR3274959)
Neil Hindman and Dona Strauss, Separating linear expressions in the Stone-Cech compactification of direct sums, Topology Appl. 213 (2016), 199-211. (MR3563080)
Neil Hindman and Dona Strauss, Topological properties of some algebraically defined subsets of $\beta \mathbb{N}$, Topology Appl. 220 (2017), 43-49. (MR3619279)
Neil Hindman and Dona Strauss, The scarcity of products in $\beta S \backslash S$, Topology Appl. 220 (2017), 50-64. (MR3619280)
Neil Hindman and Dona Strauss, Infinite compact sets of idempotents in $\beta S$, Semigroup Forum 95 (2017), no. 2, 415-417. (MR3715850)
Neil Hindman and Dona Strauss, Long increasing chains of idempotents in $beta G$, Fund. Math. 240 (2018), no. 1, 1-13. (MR3720916)
Neil Hindman and Dona Strauss, Sets and mappings in $beta S$ which are not Borel, New York J. Math. 24 (2018), 689-701.
Neil Hindman and Dona Strauss, Some new examples of infinite image partition regular matrices, Integers 19 (2019), Paper No. A5, 17 pp. (MR3901617)
Neil Hindman and Dona Strauss, Image partition regularity of matrices over commutative semigroups, Topology Appl. 259 (2019), 179-202. (MR3958267)
Neil Hindman and Dona Strauss, Some new results about the smallest ideal of $\beta S$, New York J. Math. 25 (2019), 897-913. (MR4012572)
Neil Hindman and Dona Strauss, Image partition regular matrices and concepts of largeness, New York J. Math. 26 (2020), 230-260.
Neil Hindman and Don a Strauss Some new results about the ubiquitous semigroup $\mathbb{H}$, Fund. Math. 251 (2020), no. 1, 87-108. (MR4128473)
Neil Hindman and Dona Strauss, Some properties of Cartesian products and Stone-Ćech compactifications, Topology Proc. 57 (2021), 279-304.
Neil Hindman and Dona Strauss, Minimal left ideals of $\beta S$ with isolated points, New York J. Math. 27 (2021), 417-436. (MR4226153)
Neil Hindman and Dona Strauss, Algebraic products of tensor products, Semigroup Forum 103 (2021), 888-898.
Neil Hindman and Dona Strauss, Strongly image partition regular matrices, Integers 21A (2021), Paper No. A15, 18 pp.
Neil Hindman and Dona Strauss, Some new results about left ideals of $\beta S$, New York J. Math. 28 (2022), 970-992.
Neil Hindman and Dona Strauss, Image partition regular matrices and concepts of largeness, II, Topology Proc. 61(2023), 49-76.
Neil Hindman, Dona Strauss, and Luca Q. Zamboni, Recurrence in the dynamical system $\left(X,\left\langle T_s\right\rangle_{s \in S}\right)$ and ideals of $\beta S$, Indag. Math. (N.S.) 29 (2018), no. 1, 293-312.
Neil Hindman, Dona Strauss, and Luca Q. Zamboni, Combining extensions of the Hales-Jewett theorem with Ramsey theory in other structures, Electron. J. Combin. 26 (2019), no. 4, Paper No. 4.23, 33 pp.
Neil Hindman, Dona Strauss, Yevhen Zelenyuk, Longer chains of idempotents in $beta G$, Fund. Math. 220 (2013), no. 3, 243-261.
Bernard Host, A short proof of a conjecture of Erdôs proved by Moreira, Richter and Robertson, Discrete Anal. 2019, Paper No. 19, 10 pp.
John H. Johnson, Jr., A new and simpler noncommutative central sets theorem, Topology Appl. 189 (2015), 10-24.
John H. Johnson, Jr., and Florian Karl Richter, Revisiting the nilpotent polynomial Hales-Jewett theorem, Adv. Math. 321 (2017), 269-286.
Valent in Keyantuo and Yevhen Zelenyuk, Right ideal decompositions of $G^*$, Topology Appl. 210 (2016), 90-96.
Péter Komjáth, Imre Leader, Paul A. Russell, Saharon Shelah, Dániel T. Soukup, and Zoltán Vidnyánszky, Infinite monochromatic sumsets for colourings of the reals, Proc. Amer. Math. Soc. 147 (2019), no. 6, 2673-2684.
Peter Krautzberger, On rapid idempotent ultrafilters, Semigroup Forum 89 (2014), no. 3, 692-696.
Lorenzo Luperi Baglini, Partition regularity of polynomial systems near zero, Semigroup Forum 103 (2021), no. 1, 191-208.
Lorenzo Luperi Baglini and Paulo Henrique Arruda, Rado equations solved by linear combinations of idempotent ultrafilters, Topology Appl. 305 (2022), Paper No. 107897, 15 pp .
Randall Mc Cutcheon, A combinatorial proof of a stronger dense Hindman theorem, Colloq. Math. 162 (2020), no. 2, 303-310.
Randall McCutcheon and Alist air Windsor, $D$ sets and a Sárközy theorem for countable fields, Israel J. Math. 201 (2014), no. 1, 123-146.
Randall McCutcheon and Jee Zhou, $D$ sets and IP rich sets in $\mathbb{Z}$, Fund. Math. 233 (2016), no. 1, 71-82.
Heike Mildenberger, On Milliken- Taylor ultrafilters, Notre Dame J. Form. Log. 52 (2011), no. 4, 381-394.
Joel Moreira, Monochromatic sums and products in N, Ann. of Math. (2) 185 (2017), no. 3, 1069-1090.
Joel Moreira, Florian K. Richter, and Donald Robertson, A proof of a sumset conjecture of Erdôs, Ann. of Math. (2) 189 (2019), no. 2, 605-652.
Sourav Kanti Patra and Swapan Kumar Ghosh, Concerning partition regular matrices, Integers 17 (2017), Paper No. A63, 16 pp.
Sourav Kanti Patra and Md. Moid Shaikh, Monochromatic sums equal to products near zero, Integers 20 (2020), Paper No. A66, 11 pp.
Dev Phulara, A generalized central sets theorem and applications, Topology Appl. 196 (2015), 92-105.
D. H. J. Polymath, A new proof of the density Hales-Jewett theorem, Ann. of Math. (2) 175 (2012), no. 3, 1283-1327.
Julian Sahasrabudhe, Exponential patterns in arithmetic Ramsey theory, Acta Arith. 182 (2018), no. 1, 13-42.
Julian Sahasrabudhe, Monochromatic solutions to systems of exponential equations, J. Combin. Theory Ser. A 158 (2018), 548-559.
A. Sárközy, On difference sets of sequences of integers. III, Acta Math. Acad. Sci. Hungar. 31 (1978), no. 3-4, 355-386.
Alessandro Sisto, Exponential triples, Electron. J. Combin. 18 (2011), no. 1, Paper 147, 17 pp .
Boris Sobot, Divisibility in the Stone-Cech compactification, Rep. Math. Logic No. 50 (2015), 53-66.
Boris Šobot, T-divisibility of ultrafilters, Ann. Pure Appl. Logic 172 (2021), no. 1, Paper No. 102857, 13 pp.
Boris Šobot, More about divisibility in $\beta \mathbb{N}$, MLQ Math. Log. Q. 67 (2021), no. 1, 77-87.
Sławomir Solecki, Monoid actions and ultrafilter methods in Ramsey theory, Forum Math. Sigma 7 (2019), Paper No. e2, 40 pp.
Dona Strauss, $\mathbb{N}^*$ does not contain an algebraic and topological copy of $\beta \mathbb{N}, \mathrm{J}$. London Math. Soc. (2) 46 (1992), 463-470.
Henry Towsner, A simple proof and some difficult examples for Hindman's theorem, Notre Dame J. Form. Log. 53 (2012), no. 1, 53-65.
Yevhen Zelenyuk, $K(\beta S)$ is not closed, Topology Appl. 158 (2011), no. 13, 1721-1723.
Yevhen Zelenyuk, Principal left ideals of $\beta G$ may be both minimal and maximal, Bull. Lond. Math. Soc. 45 (2013), no. 3, 613-617.
Yevhen Zelenyuk, Left maximal idempotents in $G^*$, Adv. Math. 262 (2014), 593-603.
Yevhen Zelenyuk, Discontinuity of multiplication and left translations in $\beta G$, Proc. Amer. Math. Soc. 143 (2015), no. 2, 877-884.
Yevhen Zelenyuk, Left maximal and strongly right maximal idempotents in $G^*$, J. Symb. Log. 82 (2017), no. 1, 26-34.
Yevhen Zelenyuk, Idempotents in $\beta G \backslash G$ with only trivial divisors, Fund. Math. 248 (2020), no. 2, 205-218.
Yevhen Zelenyuk, Elements of order 2 in $\beta \mathbb{N}$, Fund. Math. 252 (2021), no. 3, 355-359.
Yevhen Zelenyuk, Finite semigroups in $\beta \mathbb{N}$ and Ramsey theory, Bull. Lond. Math. Soc. 53 (2021), no. 3, 710-722.
Yevhen Zelenyuk, Elements of finite order in $\beta \mathbb{N}$, Adv. Math. 408 (2022), Paper No. 108608, 12 pp.
Yevhen Zelenyuk, Increasing sequences of principal left ideals of $\beta \mathbb{Z}$ are finite, Fund. Math. 258 (2022), no. 3, 225-235.
Yevhen Zelenyuk and Yuliya Zelenyuk, Three-element bands in $\beta \mathbb{N}$, New York J. Math. 21 (2015), 1263-1267.
Yevhen Zelenyuk and Yuliya Zelenyuk, Dynamical decompositions of $\beta X \backslash X$, Ergodic Theory Dynam. Systems 37 (2017), no. 1, 324-336.
Yevhen Zelenyuk and Yuliya Zelenyuk, Direct products of null semigroups and rectangular bands in $\beta \mathbb{N}$, New York J. Math. 28 (2022), 610-616.
