Salvatore Tringali · Power semigroups

This page collects references related to power semigroups, from the earliest works to recent developments. I created it with the assistance of Zhengliang Yuan in the spring–summer of 2026, when Z.Y. was a first-year graduate student at the School of Mathematical Sciences, Hebei Normal University.

The bibliography is organized chronologically by decade. The list is intended as a working resource and may be incomplete. A scroll (📜) indicates that a copy of the paper is available to me, while a red cross (✘) means that I do not currently have access to it. On the other hand, a test tube (🧪) marks items that are available to me only as preprints (typically on arXiv), and a pile of books (📚) indicates that I have a copy of the volume (e.g., a journal issue or conference proceedings) containing the item.

Corrections, suggestions for additional items not currently included in the list, and help in obtaining copies of items marked with a red cross are very welcome (contact me).

Jump to decade: 1950s · 1960s · 1970s · 1980s · 1990s · 2000s · 2010s · 2020s

From 2020 to present

Shuolin LIU and Salvatore TRINGALI, Power Semigroups and Two Rigidity Theorems for Groups, 2026 preprint (arXiv)
Salvatore TRINGALI and Kerou WEN, On the automorphisms of the power semigroups of numerical semigroups, Trans. London Math. Soc. 13 (2026), paper No. e70030 (DOI · arXiv)
Sergey BERSHADSKY, Stanislav KUBLANOVSKY, and G. MASHEVITZKY, Lyapin's Notebook: A Collection of Unsolved Problems in Semigroup Theory, 2026 preprint (arXiv)
Salvatore TRINGALI, Power monoids and their arithmetic: a survey, Amer. Math. Monthly, to appear (arXiv)
Salvatore TRINGALI and Weihao YAN, Torsion groups and the Bienvenu–Geroldinger conjecture, Bull. London Math. Soc., to appear (arXiv)
Dein WONG, Songnian XU, Chi ZHANG, and Zhijun WANG, The automorphism group of the reduced finitary power monoid of the additive group of integers, 2026 preprint (arXiv)
Balint RAGO, The isomorphism problem for reduced finitary power monoids, 2026 preprint (arXiv)
Laura COSSU and Salvatore TRINGALI, On the arithmetic of power monoids, J. Algebra 686 (Jan 2026), 793–813 (DOI · zbMATH · arXiv)
Sergey G. BERSHADSKY and Stanislav I. KUBLANOVSKY, On power semigroups of groups, Semigroup Forum 111 (2025), No. 3, 543–560 (DOI · zbMATH)
Dein WONG, Songnian XU, Chi ZHANG, and Jinxing ZHAO, On automorphism groups of power semigroups over numerical semigroups, 2025 preprint (arXiv)
Balint RAGO, The automorphism group of reduced power monoids of finite abelian groups, 2025 preprint (arXiv)
Balint RAGO, A counterexample to an isomorphism problem for power monoids, Proc. Amer. Math. Soc. 154 (2026), 1855–1858 (DOI · arXiv)
Salvatore TRINGALI and Lingxi LI, On global isomorphisms and a closure property of semigroups, 2025 preprint (arXiv)
Andreas REINHART, On the system of length sets of power monoids, 2025 preprint (arXiv)
Salvatore TRINGALI and Kerou WEN, The Automorphism Group of the Finitary Power Monoid of the Integers under Addition, 2025 preprint (arXiv)
Baomin YU and Xianzhong ZHAO, Global determinism of completely regular semigroups, 2025 preprint (arXiv)
Pedro A. GARCÍA-SÁNCHEZ and Salvatore TRINGALI, Semigroups of ideals and isomorphism problems, Proc. Amer. Math. Soc. 153 (2025), No. 6, 2323–2339 (DOI · zbMATH · arXiv)
Salvatore TRINGALI and Weihao YAN, On power monoids and their automorphisms, J. Combin. Theory, Ser. A 209 (2025), #105961, 16 pp. (DOI · zbMATH · arXiv)
Salvatore TRINGALI and Weihao YAN, A conjecture of Bienvenu and Geroldinger on power monoids, Proc. Amer. Math. Soc. 153 (2025), No. 3, 913–919 (DOI · zbMATH · arXiv)
Jiya DANI, Felix GOTTI, Leo HONG, Bangzheng LI, and Shimon SCHLESSINGER, On finitary power monoids of linearly orderable monoids, 2025 preprint (arXiv)
Salvatore TRINGALI, “On the isomorphism problem for power semigroups”, pp. 429–437 in: Matej BREŠAR, Alfred GEROLDINGER, Bruce OLBERDING, and Daniel SMERTNIG (eds.), Recent Progress in Ring and Factorization Theory ^(*), Springer Proc. Math. Stat. 477, Springer, 2025 (DOI · zbMATH · arXiv)

^(*) Proc. Conf. “Rings and Factorizations” held at the Univ. of Graz (Graz, Austria) on July 10–14, 2023.
Victor GONZALEZ, Eddy LI, Henrick RABINOVITZ, Pedro RODRIGUEZ, and Marcos TIRADOR, On the atomicity of power monoids of Puiseux monoids, Internat. J. Algebra Comput. 35 (2025), No. 2, 167–181 (DOI · zbMATH · arXiv)
Pierre-Yves BIENVENU and Alfred GEROLDINGER, On algebraic properties of power monoids of numerical monoids, Israel J. Math. 265 (2025), 867–900 (DOI · zbMATH · arXiv)
Anay AGGARWAL, Felix GOTTI, and Susy LU, On primality and atomicity of numerical power monoids, 2024 preprint (arXiv)
Igor DOLINKA, Sergey V. GUSEV, and Mikhail V. VOLKOV, Semiring and involution identities of powers of inverse semigroups, Commun. Algebra 52 (2024), No. 5, 1922–1929 (DOI · zbMATH · arXiv)
Sergey V. GUSEV and Mikhail V. VOLKOV, Semiring and involution identities of power groups, J. Australian Math. Soc. 115 (2023), 354–374 (DOI · zbMATH · arXiv)
Indrani DUTTA and Sukhendu KAR, Ternary \(*\)-bands are globally determined, Ural Math. J. 9 (2023), No. 1, 64–77 (DOI · zbMATH)
Austin A. ANTONIOU and Salvatore TRINGALI, On the Arithmetic of Power Monoids and Sumsets in Cyclic Groups, Pacific J. Math. 312 (2021), No. 2, 279–308 (DOI · zbMATH · arXiv)
Junqing FENG, Guohong LIANG, and Hui XU, The power semiring of involution ternary semigroup (Chinese), J. Hebei Norm. Univ., Nat. Sci. Ed. 45 (2021), No. 2, 109–111 (DOI · zbMATH)

From 2010 to 2019

Baomin YU and Xianzhong ZHAO, The bands satisfying the strong isomorphism property, Semigroup Forum 98 (2019), No. 2, 327–337 (DOI · zbMATH)
Yushuang FAN and Salvatore TRINGALI, Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics, J. Algebra 512 (Oct. 2018), 252–294 (DOI · zbMATH · arXiv)
Baomin YU, Xianzhong ZHAO, and Aiping GAN, Global determinism of idempotent semigroups, Commun. Algebra 46 (2018), No. 1, 241–253 (DOI · zbMATH)
Aiping GAN, Xianzhong ZHAO, and Yong SHAO, Globals of Idempotent Semigroups, Commun. Algebra 44 (2016), No. 9, 3743–3766 (DOI · zbMATH)
Aiping GAN, MiaoMiao REN, and Xianzhong ZHAO, Global determinism of semigroups having regular globals, Period. Math. Hung. 72 (2016), No. 1, 12–22 (DOI · zbMATH)
Aiping GAN and Xianzhong ZHAO, Global determinism of Clifford semigroups, J. Australian Math. Soc. 97 (2014), 63–77 (DOI · zbMATH)
Igor DOLINKA, Power semigroups of finite groups and the INFB property, Algebra Universalis 63 (2010), 239–242 (DOI · zbMATH)

From 2000 to 2009

Benjamin STEINBERG, A note on the power semigroup of a completely simple semigroup, Semigroup Forum 76 (2008), No. 3, 584–586 (DOI · zbMATH)
Pingfeng LIN, Chuanjun ZHANG, and Minyao MA, Some properties of power semigroup of group (Chinese), J. Guizhou Norm. Univ., Nat. Sci. 24 (2006), No. 4, 86–90 (zbMATH)
György POLLÁK, “Infima in the power set of free semigroups”, pp. 281–286 in: Helmut JÜRGENSEN, Gerard J. LALLEMENT, and Hanns J. WEINERT (eds.), Semigroups Theory and Applications ^(*), Lecture Notes Math. 1320, Springer, 1988 (DOI · zbMATH)

^(*) Proc. 3rd Conf. “Semigroup Theory and Applications” held at the MFO (Oberwolfach, FRG) on Feb. 23–Mar. 1, 1986.
Peter GALLAGHER, On the finite and non-finite generation of finitary power semigroups, Semigroup Forum 71 (2005), No. 3, 481–494 (DOI · zbMATH)
Peter GALLAGHER and Nik RUŠKUC, Finitary power semigroups of infinite groups are not finitely generated, Bull. Lond. Math. Soc. 37 (2005), No. 3, 386–390 (DOI · zbMATH)
Jorge ALMEIDA, Some Key Problems on Finite Semigroups, Semigroup Forum 64 (2002), 159–179 (DOI · zbMATH)
Siniša CRVENKOVIĆ, Igor DOLINKA, and Milovan VINČIĆ, Involution semigroups are not globally determined, Semigroup Forum 62 (2001), No. 3, 477–481 (DOI)

From 1990 to 1999

David EASDOWN and Victoria GOULD, Orders in power semigroups, Glasg. Math. J. 38 (1996), No. 1, 39–47 (DOI · zbMATH)
Stojan BOGDANOVIĆ and Miroslav ĆIRIĆ, Power semigroups that are Archimedean. II, Filomat 10 (1996), 87–92 (zbMATH)
Stojan BOGDANOVIĆ and Miroslav ĆIRIĆ, Power semigroups that are Archimedean, Filomat 9 (1995), No. 1, 57–62 (zbMATH)
Jorge ALMEIDA, On the power semigroup of a finite semigroup, Port. Math. 49 (1992), No. 3, 295–331 (zbMATH)
Takayuki TAMURA, The regular \(\mathcal J\)-class of the power semigroup of a completely \(0\)-simple semigroup, Math. Nachr. 149 (1990), 7–27 (DOI · zbMATH)
Takayuki TAMURA and Kenya YAMAOKA, “Note on finite power semigroups”, pp. 11–15 in: EDS. MISSING (eds.), VOLUME TITLE MISSING ^(*), Kyoto Sangyo Univ., 1990

^(*) Proc. 13th Symposium on Semigroups held at Kyoto Sangyo Univ. (Kyoto, Japan) in [MORE PRECISE DATES MISSING] Nov 1989.

From 1980 to 1989

Yuji KOBAYASHI, Remarks on the Structure of Power Semigroups, J. Math. Tokushima Univ. 23 (1989), 1–6 (zbMATH)
Reuben SPAKE, Semilattice decomposition of the power semigroup of the group of integers, Math. Japon. 33 (1988), No. 4, 609–625 (zbMATH)
Morio SASAKI, Isomorphism problem of power semigroups of integers semigroups, Ann. Rep. Fac. Educ. Iwate Univ. 48 (Oct 1988), No. 1, 123–126
Takayuki TAMURA and Kenya YAMAOKA, The \(0\)-minimal ideal of the global of a combinatorial completely \(0\)-simple semigroup, Proc. Japan Acad., Ser. A 63 (1987), 364–366 (DOI · zbMATH)
Kenya YAMAOKA, “Note on a finite power semigroup of a regular semigroup satisfying \(xyzw = xzyw\)”, pp. 25–31 in: Reikichi YOSHIDA (ed.), Semigroup Theory and its Related Fields ^(*), Ritsumeikan Univ., 1987 (zbMATH)

^(*) Proc. 11th Symposium on Semigroups held at Ritsumeikan Univ. (Kyoto, Japan) in Nov 1987.
Jean-Éric PIN, “Power semigroups and related varieties of finite semigroups”, pp. 139–152 in: Simon M. GOBERSTEIN and Peter M. HIGGINS (eds.), Semigroups and Their Applications ^(*), Reidel Publishing Co., 1987 (zbMATH)

^(*) Proc. Intl. Conf. “Algebraic Theory of Semigroups and Its Applications” held at the California State Univ. (Chico, California) on April 10–12, 1986.
Takayuki TAMURA, “On the recent results in the study of power semigroups”, pp. 191–200 in: Simon M. GOBERSTEIN and Peter M. HIGGINS (eds.), Semigroups and Their Applications ^(*), Reidel Publishing Co., 1987 (zbMATH)

^(*) Proc. Intl. Conf. “Algebraic Theory of Semigroups and Its Applications” held at the California State Univ. (Chico, California) on April 10–12, 1986.
Jorge ALMEIDA, Power pseudovarieties of semigroups II, Semigroup Forum 33 (1986), 375–390 (DOI · zbMATH)
Jorge ALMEIDA, Power pseudovarieties of semigroups I, Semigroup Forum 33 (1986), 357–373 (DOI · zbMATH)
Takayuki TAMURA, Isomorphism problem of power semigroups of completely \(0\)-simple semigroups, J. Algebra 98 (1986), 319–361 (DOI · zbMATH)
Reuben SPAKE, “Idempotent-free Archimedean components of the power semigroup of the group of integers, an \(F\)-type semigroup”, pp. 55–62 in: Kiyoshi ISEKI and Shotaro TANAKA (eds.), Semigroups and Related Topics ^(*), Naruto University of Education, 1986 (zbMATH)

^(*) Proc. 9th Symposium on Semigroups and Related Topics held at the Naruto University of Education (Naruto City, Japan) on Nov 27–28, 1985.
Takayuki TAMURA, “On the power semigroup of a finite \(0\)-simple semigroup. Rectangle \(J\)-classes”, pp. 48–54 in: Kiyoshi ISEKI and Shotaro TANAKA (eds.), Semigroups and Related Topics ^(*), Naruto University of Education, 1986 (zbMATH)

^(*) Proc. 9th Symposium on Semigroups and Related Topics held at the Naruto University of Education (Naruto City, Japan) on Nov 27–28, 1985.
Reuben SPAKE, Idempotent-free Archimedean components of the power semigroup of the group of integers. I, Math. Japon. 31 (1986), 791–810 (zbMATH)
Reuben SPAKE, The semigroup of nonempty finite subsets of integers, Int. J. Math. Math. Sci. 9 (1986), 605–616 (zbMATH)
Takayuki TAMURA, “Archimedean components of the power semigroup of the infinite cyclic group”, pp. 62–70 in: EDS. UNKOWN (eds.), Semigroup Theory and its Related Fields ^(*), Shimane Univ., 1985 (zbMATH)

^(*) Proc. 8th Symposium on Semigroups held at Shimane Univ. (Matsue, Japan) in [EXACT DATES MISSING] 1984.
Matthew GOULD, Joseph A. ISKRA, and Constantine TSINAKIS, Globals of completely regular periodic semigroups, Semigroup Forum 29 (1984), 365–374 (DOI · zbMATH)
Stuart W. MARGOLIS and Jean-Éric PIN, Power monoids and finite \(J\)-trivial monoids, Semigroup Forum 29 (1984), 99–108 (DOI · zbMATH)
Matthew GOULD and Joseph A. ISKRA, Globally determined classes of semigroups, Semigroup Forum 28 (1984), 1–11 (DOI · zbMATH)
Stuart W. MARGOLIS and Jean-Éric PIN, Minimal noncommutative varieties and power varieties, Pacific J. Math. 111 (1984), 125–135 (DOI · zbMATH)
Jean-Éric PIN, Semigroupe des parties et relations de Green, Canadian J. Math. 36 (1984), 327–343 (DOI · zbMATH)
Takayuki TAMURA, On the power semigroup of the group of integers, Proc. Japan Acad., Ser. A 60 (1984), 388–390 (zbMATH)
Takayuki TAMURA, Errata to the paper “Power semigroups”, Math. Japon. 29 (1984), no. 4, 679
Takayuki TAMURA, Power semigroups of rectangular groups, Math. Japon. 29 (1984), no. 4, 671–678 (zbMATH)
Takayuki TAMURA, On chains whose power semigroups are lattices, Semigroup Forum 30 (1984), 35–40 (DOI · zbMATH)
Yuji KOBAYASHI, Semilattices are globally determined, Semigroup Forum 29 (1984), No. 1, 217–222 (zbMATH)
Richard D. BYRD, Justin T. LLOYD, Franklin D. PEDERSEN, and James W. STEPP, The automorphism group of some semigroups, Fundamenta Mathematicae 124 (1984), No. 2, 187–195 (DOI · zbMATH)
Stojan BOGDANOVIĆ, A note on power semigroups, Math. Japon. 28 (1983), 725–727 (zbMATH)
Takayuki TAMURA, “On semigroups of subsets of semigroups, II: Idempotents in power semigroups of finite \(0\)-simple semigroups”, pp. 12–19 in: EDS. UNKOWN (eds.), Semigroup Theory and its Related Fields ^(*), Tokyo Gakugei Univ., 1983 (zbMATH)

^(*) Proc. 7th symposium on Semigroups held at Tokyo Gakugei Univ. (Tokyo, Japan) in [EXACT DATES MISSING] 1983.
Richard D. BYRD, Justin T. LLOYD, and James W. STEPP, The automorphism group of the semigroup of finite complexes of a rank one torsion free Abelian group, Arch. Math. 39 (1982), 385–393 (DOI · zbMATH)
Stuart W. MARGOLIS, On \(M\)-varieties generated by power monoids, Semigroup Forum 22 (1981), 339–353 (DOI · zbMATH)
Franklin D. PEDERSEN and Walter S. SIZER, Subsemigroups of the finite complexes of a group, Semigroup Forum 20 (1980), 285–292 (DOI · zbMATH)
Jean-Éric PIN, Variétés de langages et monoïde des parties, Semigroup Forum 20 (1980), 11–47 (DOI · zbMATH)
Richard D. BYRD, Justin T. LLOYD, Roberto A. MENA, and J. Roger TELLER, Extensions of group retractions, Intl. J. Math. and Math. Sciences 3 (1980), No. 4, 719–730 (DOI · zbMATH)
Robert P. SULLIVAN, Regular power semigroups, Publ. Math. Debr. 27 (1980) 289–290 (DOI · zbMATH)

From 1970 to 1979

Howard STRAUBING, Recognizable sets and power sets of finite semigroups, Semigroup Forum 18 (1979), 331–340 (DOI · zbMATH)
August LAU, Finite Abelian semigroups represented into the power set of finite groups, Czechoslovak Math. J. 29 (1979), No. 1, 159–162 (DOI · zbMATH)
Mohan S. PUTCHA, Subgroups of the power semigroup of a finite semigroup, Canad. J. Math. 31 (1979), 1077–1083 (DOI · zbMATH)
Richard D. BYRD, Justin T. LLOYD, and James W. STEPP, The automorphism group of the semigroup of finite complexes of a periodic Abelian group, J. Australian Math. Soc., Ser. A 28 (1979), 15–22 (zbMATH)
Mohan S. PUTCHA, On the maximal semilattice decomposition of the power semigroup of a semigroup, Semigroup Forum 15 (1978), 263–267 (DOI · zbMATH)
Richard D. BYRD, Justin T. LLOYD, and James W. STEPP, Groups of complexes of a representable lattice-ordered group, Glasg. Math. J. 19 (1978), 135–139 (DOI · zbMATH)
Richard D. BYRD, Justin T. LLOYD, Roberto A. MENA, and J. Roger TELLER, Automorphisms of the semigroup of finite complexes of locally finite groups, J. Reine Angew. Math. 299/300 (1978), 151–160 (DOI · MR)
Richard D. BYRD, Justin T. LLOYD, Roberto A. MENA, and J. Roger TELLER, The lattice of solid \(\sigma\)-subgroups of a retractable group, Czechoslovak Math. J. 28 (1978), No. 103, 189–199 (DOI · zbMATH)
Richard D. BYRD, Justin T. LLOYD, Franklin D. PEDERSEN, and James W. STEPP, Automorphisms of the semigroup of finite complexes of a periodic locally cyclic group, Pacific J. Math. 72 (1977), No. 1, 27–39 (zbMATH)
Richard D. BYRD, Justin T. LLOYD, Roberto A. MENA, and J. Roger TELLER, “The lattice of \(\rho-\sigma\)-subgroups of a retractable group”, pp. 515–524 in: EDITORS MISSING (eds.), TITLE MISSING ^(*), Symposia Mathematica 21, Academic Press, 1977 (zbMATH)

^(*) Proc. Conf. “Gruppi e Anelli Ordinati” held at INDAM, Istituto Nazionale di Alta Matematica (Rome, Italy) on Dec 15–19, 1975.
Richard D. BYRD, Justin T. LLOYD, and Roberto A. MENA, On the retractability of some one-relator groups, Pacific J. Math. 72 (1977), 351–359 (DOI · zbMATH)
Richard D. BYRD, Justin T. LLOYD, Roberto A. MENA, and James W. STEPP, Retractable groups, Acta Math. Acad. Sci. Hung. 29 (1977), 219–233 (DOI · zbMATH)
Richard D. BYRD, Justin T. LLOYD, Franklin D. PEDERSEN, and James W. STEPP, Automorphisms of semigroups of complexes of Abelian groups, Bull. Amer. Math. Soc. 83 (1977), No. 2, 260–261 (DOI · zbMATH)
Russell G. BILYEU and August Y.W. LAU, Representations into the hyperspace of a compact group, Semigroup Forum 13 (1977), 267–270 (DOI · zbMATH)
Richard D. BYRD, Justin T. LLOYD, and James W. STEPP, Groups of complexes of a group, J. Nat. Sci. and Math. 15 (1975), 83–87 (MR)
Věra TRNKOVÁ, On a representation of commutative semigroups, Semigroup Forum 10 (1975), 203–214 (DOI · zbMATH)
Donald J. McCARTHY and David L. HAYES, Subgroups of the power semigroup of a group, J. Combin. Theory, Ser. A 14 (1973), 173–186 (DOI · zbMATH)
Eugenia M. MOGILJANSKAJA, Non-isomorphic semigroups with isomorphic semigroups of subsets, Semigroup Forum 6 (1973), 330–333 (zbMATH)
Eugenia M. MOGILJANSKAJA, The solution to a problem of Tamura (Russian), pp. 148–151 in: I. Ya. BAKELMAN et al. (eds.), Modern Analysis and Geometry, Sbornik Nauchnykh Trudov 541, Leningrad Gos. Ped. Inst., 1972.

From 1960 tyo 1969

Takayuki TAMURA and John SHAFER, Power semigroups II, Notices Amer. Math. Soc. 15 (1968), 395
John SHAFER, Note on power semigroups, Math. Japon. 12 (1967), 32
Takayuki TAMURA and John SHAFER, Power semigroups, Math. Japon. 12 (1967), 25–32 (zbMATH)
Takayuki TAMURA, Unsolved problems on semigroups, Seminar Reports of Math. Sci. Kyoto Surikaiseki Kenkyusho (1967), 1–35
Sándor LAJOS, O polugruppe podmnozhestv polugruppy ^(*) (Russian), Publ. Math. Debrecen 9 (1962), no. 3--4, 223--226 (DOI · zbMATH)

^(*) The official title is a transliteration of the original Russian title and can be translated into English as On the semigroup of subsets of a semigroup.

From 1950 to 1959

Paul DUBREIL, Contribution à la théorie des demi-groupes. III (French), Bull. Soc. Math. France 81 (1953), 289–306 (zbMATH)
Robert BALLIEU, Sur les groupes de parties d'un demi-groupe (French), Ann. Soc. Sci. Bruxelles, Sér. I 64 (1950), 139–147 (zbMATH)